Grassmannian Geometry of Framed Curve Spaces Thomas

Grassmannian Geometry

of Framed Curve Spaces

by

Thomas Richard Needham

(Under the direction of Jason Cantarella)

Abstract

We develop a general framework for solving a variety of variational and computer vision prob-

lems involving framed space curves. Our approach is to study the global Riemannian and

symplectic geometry of the moduli space of similarity classes of framed loops in R3. We show

that this space is an infinite-dimensional Frechet manifold with a natural Kahler structure.

The proof uses novel coordinates on the space of framed paths, which are used to locally

identify the moduli space with the Grassmannian of 2-planes in an infinite-dimensional com-

plex vector space. Results on the geometry of framed loop space are obtained, including

a characterization of its sectional curvatures and an in-depth description of some natural

Hamiltonian group actions on the space. We give a variety of applications of this structure,

such as a classification of critical points of a generalization of the Kirchhoff elastic energy

functional and a shape recognition algorithm for, e.g., protein backbones. We show connec-

tions between previous results by various authors on infinite-dimensional Kahler geometry,

fluid dynamics and moduli spaces of linkages.

Index words: Infinite-dimensional geometry, symplectic geometry, Riemanniangeometry, elastic shape matching



by


B.S., University of Wisconsin-Milwaukee, 2007

M.S., University of Wisconsin-Milwaukee, 2009

A Dissertation Submitted to the Graduate Faculty

of The University of Georgia in Partial Fulfillment

of the

Requirements for the Degree

Doctor of Philosophy

Athens, Georgia

2016

c©2016


All Rights Reserved



by


Approved:

Major Professor: Jason Cantarella

Committee: Joseph H.G. FuDavid T. GayWilliam H. KazezMichael Usher

Electronic Version Approved:

Suzanne BarbourDean of the Graduate SchoolThe University of GeorgiaMay 2016

Acknowledgments

This thesis is dedicated to my wonderful wife Chelsie. She provided unwavering support

throughout my work in graduate school; in particular, she was completely indispensable in

keeping me sane throughout the stressful process of writing a dissertation and venturing into

the academic job market. I would never have achieved these goals without her, and I can’t

thank her enough for her patience and encouragement.

I am also, of course, deeply indebted to my advisor Jason Cantarella. He spent countless

hours discussing the material in this work with me and was consistently supportive, while

never failing to push me to work just beyond my abilities. I hope that I have, at least in

some small part, internalized some of his amazing mathematical intuition. Overall, I can’t

imagine a better advisor.

There are many, many more people who deserve thanks, and I will mention a few of them

here. I would like to thank my committee members Joseph Fu, David Gay, William Kazez

and Michael Usher, each of whom has taught me interesting mathematics at some point in

my time at UGA. I would also like to thank Simon Foucart, who has graciously mentored me

in the field of compressive sensing. Countless other mathematicians at UGA and elsewhere

have helped me throughout my time in graduate school, and special thanks goes to Sybilla

Beckmann, Sa’ar Hersonsky, Ken Millett, Clayton Shonkwiler and Dennis Sullivan. I would

also like to thank the office staff at the UGA math department, especially Laura Ackerley.

iv

Finally, I would like to thank the many close friends that I have made in Athens, both inside

and outside of the math department.

v

Contents

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Main Results and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Preliminaries and Notation 15

2.1 Classical Geometry of Framed Curves . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Frechet Spaces and Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Preliminaries from Differential Geometry . . . . . . . . . . . . . . . . . . . . 35

2.4 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Complex Coordinates for Framed Paths and Loops 44

3.1 Framed Path Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 Framed Loop Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 The Geometry of Spaces of Framed Paths and Loops 82

4.1 The Symplectic Structure of Framed Loop Space . . . . . . . . . . . . . . . . 82

4.2 Principal Bundle Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.3 Symplectic Reduction by the action of LS1{S1 . . . . . . . . . . . . . . . . . 111

4.4 The actions of Diff`pr0, 2sq and Diff`pS1q . . . . . . . . . . . . . . . . . . . . 130

4.5 Riemannian Geometry of Framed Path and Loop Spaces . . . . . . . . . . . 143

vi

5 Application: Critical Points of Energy Functionals 159

5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5.2 Weighted Total Twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

5.3 Total Elastic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6 Application: Shape Recognition 196

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

6.2 Shape Matching for Framed Paths . . . . . . . . . . . . . . . . . . . . . . . . 199

Bibliography 217

vii

List of Figures

3.1 An element of AC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.2 A knot diagram with Wr “ 3´ 5 “ ´2 . . . . . . . . . . . . . . . . . . . . . . . 69

3.3 Links determined by (a) pγ0, V0q, (b) A perturbation of pγ0, V0q along the homotopy,

and (c) pγ1, V1q. In each image, the blue loop is the image of γ and the red loop is

the image of γ ` εV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1 The complex structure ofM. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.2 Regions described in the proof of Lemma 4.3.3. . . . . . . . . . . . . . . . . 116

4.3 Geodesic in M joining a p2, 3q-torus knot to a p2, 5q-torus knot. . . . . . . . 146

5.1 A torus knot realized as a pushoff of a critical point of ĂTw and the corre-

sponding Clifford torus knot as a stereographic projection of Φ2,3. . . . . . . 169

5.2 The 1-parameter family of critical points with h “ 2, k “ 1. . . . . . . . . . 194

5.3 The 1-parameter family of critical points with h “ 3, k “ 2. . . . . . . . . . 195

6.1 Geodesic before reparameterization. . . . . . . . . . . . . . . . . . . . . . . . 215

6.2 Geodesic after reparameterization. . . . . . . . . . . . . . . . . . . . . . . . . 215

6.3 Geodesic distances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

viii

Chapter 1

Introduction

This chapter introduces the results of the thesis. We begin with a general discussion of the

motivations for this work. We then give a more specific outline of the structure of the paper

and highlight the main results.

1.1 Overview

1.1.1 Motivating Questions

The overarching goal of this thesis is to develop a convenient framework to answer a diverse

collection of questions such as:

1. What is the natural notion of shape similarity for protein backbones?

2. What are the equilibrium shapes of an extensible elastic rod?

3. How should one efficiently compute the gradient flow of a knot energy functional such

as Mobius Energy (see [61])? What does “gradient flow” mean in this setting?

The common thread between these questions is that they can be rephrased as questions

about the space of shapes of framed curves in R3. By a framed curve in R3, we mean a

1

pair pγ, V q consisting of a parameterized open path or closed loop γ in R3 together with a

unit normal vector field V along γ. The shape of a framed curve refers to an equivalence

class with respect to some “shape-preserving” group action—e.g., the Euclidean group of

rotations and translations, or the diffeomorphism group of reparameterizations.

Thus it remains to explain what is meant by the space of shapes of framed curves. We will

show in Chapter 3 that many natural collections of shapes of framed paths and loops admit

manifold structures. In this setting, manifold means a topological space which is locally

modelled on a function space such that the transition functions are smooth in an appropriate

sense. The function space formalism that we adopt here is that of (tame) Frechet spaces.

These are infinite-dimensional vector spaces which are flexible enough to describe spaces of

smooth maps (as opposed to Hilbert or Banach spaces), but which have enough structure to

admit an inverse function theorem (as opposed to more general topological vector spaces).

We work with spaces of smooth framed curves with the viewpoint that spaces of framed

curves of lower regularity are realized as completions of the smooth spaces. Thus any results

obtained in the smooth category should generalize without much work to, say, the L2 category

or a Sobolev regularity category. For example, the results in this thesis can be generalized

to spaces of absolutely continuous framed curves almost immediately. Another reason for

working in the smooth category is that we wish to study actions of diffeomorphism groups,

and topological groups of Ck-diffeomorphisms do not admit Lie group structures [62].

From this perspective we are able to outline solution strategies for the above questions

as follows:

1. The space of protein backbone shapes can be viewed as the infinite-dimensional mani-

fold of equivalence classes of framed paths, where equivalence is with respect to transla-

tion, rotation, scaling and reparameterization. This manifold has a natural Riemannian

metric, so shape similarity of protein backbones can be measured by geodesic distance.

2

2. An extensible elastic rod in equilibrium can be viewed as a critical point of a certain

functional on the manifold of Euclidean similarity classes of framed paths or loops.

Thus the goal is to classify the critical point set of this functional.

3. A knot energy functional can be viewed as a function on the manifold of Euclidean

similarity classes of framed loops. This manifold also admits a natural metric, so that

gradient is well-defined. Then the task is to determine the existence of the gradient

flow and to develop effective methods of computing it.

Now that we have strategies for attacking the questions, our concern is whether these

strategies are actually tractable. For example, the usefulness of the solution strategy for

Question 1 depends on our ability to compute geodesic distances efficiently. While the

(short time) existence of geodesics in a finite-dimensional Riemannian manifold is trivial,

the geodesic equation on an infinite-dimensional manifold becomes a PDE—thus even short

time solutions are not guaranteed to exist. Moreover, even if solutions do exist, actually

computing geodesic distance could conceivably be quite difficult. We contend that the main

results of this thesis also make these solution strategies tractable. The reason is that we do

not only show that many shape spaces of framed paths and loops admit manifold structures

and natural Riemannian metrics; we also show that the shape spaces are isometric to infinite-

dimensional versions of classical manifolds such as spheres and Grassmannians. Despite

their infinite-dimensionality, these classical manifolds are remarkably easy to work with.

For example, the infinite-dimensional Grassmannians that appear admit completely explicit

geodesics.

1.1.2 Complex Coordinates for Framed Curves

As is frequently the case in differential geometry, a key step in unlocking the structure of the

manifolds of interest is to find an appropriate coordinate system. Our approach to studying

3

manifolds of shapes of framed curves is to develop a convenient coordinate system where

a pair of complex-valued functions represents a framed path or loop. The idea of using

functional coordinates for shape spaces of curves is actually quite classical, and we now draw

parallels with the classical approach.

It is a basic fact from elementary differential geometry that the curvature κ and torsion τ

of a space curve completely determine the curve up to Euclidean isometry, provided κ ą 0. A

natural question, attributed independently to Fenchel [18] and to Chern [34] is the following:

What are necessary and sufficient conditions on κ and τ that ensure that the corresponding

space curve is periodic? Certainly a necessary condition is that κ and τ are themselves

periodic, but periodicity of κ and τ is far from sufficient (e.g. constant κ and τ produce a

non-periodic helix). Hwang Cheng Chung gave a complete answer to this question in [34],

but the sufficient conditions include that an infinite series of integrals of functions of κ and

τ vanish. Various other answers exist (e.g., [22, 60]), but the sufficient conditions given are

similarly impractical. The general consensus is that the question has no practical answer.

This situation can be interpreted as follows. One can view pκ, τq as a global system of

coordinates on the infinite-dimensional manifold of isometry classes of arclength parameter-

ized paths in R3 with nonvanishing curvature. This has the benefit of endowing an seemingly

complicated infinite-dimensional manifold with the differential structure of an open subset

of a vector space; that is, we identify the space

tγ P C8pr0, 1s,R3q | }γ1} ” 1, κ ą 0u{ttranslation, rotationu

with

tpκ, τq P C8pr0, 1s,Rą0q ˆ C8pr0, 1s,Rq | κ ą 0u Ă C8pr0, 1s,R2

q.

4

However, the lack of reasonable conditions on κ and τ which guarantee periodicity of a curve

means that there are no constraint equations in κ and τ that can be used to identify the

submanifold consisting of closed loops.

In this thesis we introduce a global coordinate system of pairs of complex-valued func-

tions, denoted pφ, ψq, on the space of framed paths. We show that pφ, ψq determine a

smoothly closed framed loop if and only if they satisfy two simple constraint equations, in

stark contrast with the classical pκ, τq-coordinates. The simple constraint equations turn

the moduli space of smoothly closed framed loops into a finite-codimension submanifold of

a vector space, and this allows us to work with it in an extraordinarily hands-on way.

1.1.3 Why Framed Curves?

Framed paths and loops are ubiquitous in biology and physics. They are frequently used to

model protein backbones [28, 33, 72], circular DNA [13, 19, 31], magnetic field lines [58] and

oriented trajectories [25], and they are the central objects of study in the theory of Kirchhoff

elastic rods [66]. The main purpose of this paper is to develop a natural framework for

studying the dynamics of these physical objects in terms of the geometry and topology of

moduli spaces of framed paths and loops.

Framed Curves are also interesting from a pure mathematics perspective; e.g. it is well-

known that every orientable 3-manifold can be obtained by surgery on a framed link in the

3-sphere [46]. The geometric structures of a variety of interesting spaces of moduli spaces of

loops in 3-manifolds have been studied extensively in, e.g., [8, 49, 55]. We show in this thesis

that, in the case of R3, geometric structures on loop spaces can be seen as arising from more

complicated structures on the moduli space of framed loops. We conjecture that this is also

true in the general 3-manifold setting, and this will be the subject of future work.

5

1.2 Main Results and Outline

1.2.1 Some Definitions

Before giving a detailed outline of the thesis, some more precise definitions are in order. We

define a framed path to be a pair pγ, V q, where γ is a parameterized immersed path in R3

and V is a unit normal vector field along γ. For concreteness, we take the domain of any

parameterized path to be r0, 2s—this convention is a normalization whose convenience will

be made apparent later on. A relatively framed path is an equivalence class of a framed path

with respect to the circle-action of rotating all frame vectors simultaneously around γ by

the same angle. A framed loop (respectively, relatively framed loop) is an ordered pair of

maps pγ, V q : S1 Ñ R3 ˆ R3, such that γ is an immersion and V is a normal vector field

along γ (respectively, an equivalence class of normal vector fields up to the frame rotation

circle-action). We identify the circle S1 with the smooth quotient r0, 2s{p0 „ 2q so that the

collection of framed loops can be considered as a subset of the collection of framed paths;

i.e., a framed loop is a framed path which happens to smoothly close. To distinguish this

realization of S1 from the more standard case, will use the notation Sn to denote the standard

radius-1 n sphere embedded in Rn`1.

Let M denote a finite-dimensional manifold. Throughout the rest of the thesis, we will

use the notation

PM :“ C8pr0, 2s,Mq and LM :“ C8pS1,Mq

for the path space and loop space of M , respectively.

1.2.2 Chapter 2

Chapter 2 collects some background material. We summarize some results from the classical

theory of the geometry of framed curves in R3. We then give some background on the theory

6

of Frechet spaces, paying special attention to the most relevant cases of path and loop spaces

of a finite-dimensional manifold. Next we collect some results on Riemannian and symplectic

geometry, both in the classical setting and in the infinite-dimensional setting. Finally, we

give a brief description of the geometry of the quaternions. This chapter also serves to set

much of the basic notation that will be used throughout the thesis.

1.2.3 Chapter 3

The first original results of the thesis appear in this chapter. We define the moduli space of

framed paths

S :“ tframed pathsu{ttranslation, scalingu,

where scaling refers to scaling the length of the base curve (for concreteness, we take all

framed loops to have fixed length 2). The space S is shown to have the structure of a

manifold modelled on Frechet spaces. It is also shown to admit a natural Riemannian metric.

This is constructed by first defining a 4-parameter family of metrics ga,b,c,d on S called the

framed curve elastic metrics. This definition is a direct generalization of the elastic metrics

ga,b defined on spaces of plane curves by Mio, Srivastava and Joshi [57]. It is then shown that

g1,1,1,1 arises as a pullback metric from a natural embedding of S into an infinite-dimensional

Lie group with a left-invariant metric (Proposition 3.1.5).

Next we utilize the well-known Hopf map, which gives an isometric double cover SUp2q Ñ

SOp3q, to prove that, with respect to g1,1,1,1, S has the geometry of an infinite-dimensional

sphere.

Theorem (Theorem 3.1.12). The moduli space of framed paths S is isometrically double

covered by an open subset of the L2 sphere in PC2.

7

Next we move on to the moduli space of framed loops

M :“ trelatively framed loopsu{ttranslation, scaling, rotationu

consisting of Euclidean similarity classes of relatively framed loops. This space has two

connected components, which are indexed by mod-2 self-linking number (Corollary 3.2.5).

The moduli space admits a natural metric induced by g1,1,1,1, and our next main theorem is

roughly stated as:

Theorem (Theorem 3.2.8). Each component of the moduli space of relatively framed loops

M is isometric to an open subset of the Grassmann manifold of complex 2-planes in LC.

Theorems 3.1.12 and 3.2.8 serve as the basis for the rest of our results. We recall that the

basic premise of the thesis is that many diverse questions can be phrased as questions about

the geometry of moduli spaces such as S and M. These are, on first glance, quite abstract

manifolds, but the results of this chapter show that (with respect to natural Riemannian

metrics), these spaces actually have surprisingly well-behaved geometry.

We also mention here that these theorems generalize some remarkable work of Younes,

Michor, Shah and Mumford [76]. It was shown in [76] that various moduli spaces of immersed

plane curves admit a natural metric (an example of one of the elastic metrics of Mio et. al.)

with respect to which they are isometric to classical manifolds (e.g. real infinite-dimensional

Grassmannians). The authors were primarily interested in the topic for its application to

shape recognition. We are also interested in this application and it is treated in Chapter

6. Theorems 3.1.12 and 3.2.8 generalize the work of Younes et. al. in two ways. First,

the shape spaces they consider embed as totally geodesic submanifolds of the shape spaces

considered here. Second, the proofs give some extra insight into why the results of [76] are

true, and demonstrate the dimensions where such strategies could potentially work—these

are the dimensions where Hopf fibrations exist.

8

1.2.4 Chapter 4

In this chapter we more thoroughly investigate the geometry of S and M. We begin by

noting that

Gr2pLCq :“ tcomplex 2-planes in LCu

is a complex Kahler manifold. We show that the Kahler structure is obtained by viewing

Gr2pLCq as a Kahler reduction of LC2 with its natural L2 Kahler structure. From this we

are able to describe an interesting complex structure on the moduli space of framed loops

M (Corollary 4.1.8).

Next we study the action of the Lie group LS1{S1 onM. Recall that we use the notation

S1 to specifically denote the standard unit circle embedded in C « R2. The action of this

Lie group is by adjusting the framing of (an equivalence class of) a framed loop. Let

B :“ t(unframed) immersed loops in R3u{ttranslation, rotation, scalingu.

We show that B is a Frechet manifold (Lemma 4.2.1) and then prove the following theorem.

Theorem (Theorem 4.2.5). Framed loop space M has the structure of a principal bundle

over B with structure group LS1{S1.

This theorem has a variety of applications. An immediate corollary (Corollary 4.2.6)

roughly states that any curve framing algorithm (see, e.g., [26]) satisfying some reasonable

conditions is bound to fail on some subset of the space of immersed curves.

Since M has a Kahler structure, a natural question to ask is whether the action of the

Lie group LS1{S1 is Hamiltonian. We show that it is by explicitly defining its momentum

map. Moreover, we show that symplectic reduction by this group produces a space which

9

has been previously studied. This is the moduli space of Millson and Zombro [55] defined by

MMZ :“ tarclength-parameterized loops in R3u{ttranslation, rotationu.

Millson and Zombro showed that MMZ admits a symplectic structure by showing that it is

realized as a symplectic reduction of the loop space of the 2-sphere by the action of SOp3q.

We give an alternate description of the symplectic structure.

Theorem (Theorem 4.3.7). The space of Millson and ZombroMMZ is realized as symplectic

reduction of M by the action of LS1{S1.

Next we examine the action of the group Diff`pS1q of orientation-preserving diffeomor-

phisms onM by reparameterization. The action is not free, and because of this we focus on

subgroup Diff`0 pS1q of basepoint-preserving diffeomorphisms, which does have a free action

onM. We show that the spaceM{Diff`0 pS1q is a manifold (Proposition 4.4.3) and that the

tangent spaces toM split orthogonally with respect to the action of Diff`0 pS1q (Proposition

4.4.4).

We conclude this chapter by studying the Riemannian geometry ofM and S. In particu-

lar, we are able to explicitly describe the geodesics of these moduli spaces using the fact that

they are isometric to classical manifolds. Such a description is extremely useful for shape

recognition applications, as we shall see in Chapter 6. This description is also used to prove

a result on the sectional curvatures of M and its quotient by Diff`pS1q.

Theorem (Theorem 4.5.8). The spaces M and M{Diff`0 pS1q both have non-negative sec-

tional curvatures with respect to their natural metrics.

In shape recognition applications, one typically wants to optimize geodesic distance over

reparameterizations, and for this reason it is useful to know that the quotient is nonnegatively

curved. Furthermore, nonnegative sectional curvature is important for numerical stability

10

if one were to use a Newton’s method-type approximation for the gradient flow of some

functional onM. The proof of this theorem uses a variety of results of independent interest.

These include that the exponential map of M is well-defined (Proposition 4.5.6) and that

for any collection of 2-planes in Gr2pLCq, there exists a totally geodesic copy of a finite-

dimensional complex Grassmannian containing the collection (Proposition 4.5.2). The latter

result somewhat formalizes the notion that the geometry of Gr2pLCq can be understood from

a finite-dimensional perspective.

1.2.5 Chapter 5

Chapter 5 turns towards applications to energy functionals on the moduli space M. First

we study the weighted total twist functional, ĂTw : M Ñ R, which measures the total

accumulated twisting of the normal vector field along its base curve, weighted by the pa-

rameterization speed of the curve. We show that ĂTw is a moment map for a natural circle

action onM (Proposition 5.2.2). The circle which acts is a subgroup of Diff`pS1q, acting by

rotating the base point of a based framed loop. We use this fact to characterize the critical

points of ĂTw:

Theorem (Theorem 5.2.8). The critical points of ĂTw : M Ñ R are equivalence classes

of framed loops pγ, V q such that γ is an arclength parameterized, length-2, multiply covered

round circle and V has constant twist rate. Thus a small pushoff γ ` εV forms a multiply-

covered torus knot and every torus knot type is realized as such a pushoff for some critical

point. Each critical framed loop has a complex-coordinate representative as a torus knot on

the standard Clifford torus in S3.

11

Next we introduce the total elastic energy functional, E :MÑ R, which is a straightfor-

ward generalization of classical Kirchhoff elastic energy

EKir “

ż

S1

κ2` tw2 ds

of a framed loop pγ, V q. Here κ is the curvature of γ, tw is the twist rate of V around γ

and ds is measure with respect to arclength of γ (see Section 2.1.2 for the definition of twist

rate and Section 5.1.1 for a short history of the study of Kirchhoff elastic energy and related

functionals). Kirchhoff elastic energy should be viewed as the potential energy of a uniform

inextensible elastic rod, whereas our generalization can be viewed as the potential energy of

an extensible rod whose bending and twisting tension depends on how it is stretched.

Total elastic energy has an incredibly natural representation in our coordinate system.

We are able to completely classify the critical points of E, and in fact we can give explicit

parameterizations of all critical points in our coordinate system. This is due to the following

theorem, which is stated here in a nontechnical form.

Theorem (Theorem 5.3.10). Each critical point of E :MÑ R has a unique representation

in complex coordinates as a pair of eigenfunctions of the operator φ ÞÑ φ2.

To get a more geometric picture of the types of critical points of E, we show:

Theorem (Theorem 5.3.16). Let h, k be integers such that gcdph, h`kq “ gcdpk, h`kq “ 1.

The critical point set of E :MÑ R contains a 1-parameter family of equivalence classes of

framed loops pγu, Vuq such that

(i) pγ0, V0q is an arclength parameterized h-times-covered round circle linked k-times,

(ii) γε is an ph, h` kq-torus knot for sufficiently small ε ą 0,

(iii) γ1´ε is a p´k, h` kq torus knot for sufficiently small ε ą 0,

12

(iv) pγ1, V1q is an arclength parameterized k-times-covered round circle linked ´h-times.

Moreover, there exists a unique u P p0, 1q such that γu is not embedded.

This result can be seen as an analog of Ivey and Singer’s classification of critical points

of the classical Kirchhoff elastic energy functional [35]. The Ivey-Singer result shows that

critical points of the classical elastic energy come in 1-parameter families. Their result was

concerned with arclength parameterized framed loops, and we see here that allowing for

arbitrary parameterizations has produced a richer critical point set, as one might expect.

1.2.6 Chapter 6

In this chapter we address the application of this framework to shape recognition problems.

We first address the geodesic distance problem in the quotient space S{SOp3q, which can be

interpreted as optimizing geodesic distance in the L2-sphere of PC2. Optimizing registration

over rotations is an important part of any shape recognition algorithm, and is typically

treated computationally via matrix decompositions (e.g., [68]). In our setup, the problem

has a pleasing closed form solution, which is most naturally stated by identifying PC2 with

the path space of the quaternions H.

Theorem (Theorem 6.2.2). Geodesic distance in S{SOp3q has an explicit closed form. In

quaternionic coordinates q0 and q1, the distance is given by

min

distpq0, q1 ¨ pq | p P S3Ă H

(

“?

2 arccos1

2

›

›

›

›

ż 2

0

q1 ¨ q0 dt

›

›

›

›

H,

where dist denotes geodesic distance in the radius-?

2 L2-sphere of PH.

We then turn to computing distance in S{Diff`pr0, 2sq; i.e., we wish to optimize geodesic

distance over reparameterizations. Here a technical issue arises in that the full L2-sphere

of PC2 modulo the induced action of Diff`pr0, 2sq is a non-Hausdorff space. This is treated

13

by replacing this quotient space by the space of L2-closures of Diff`pr0, 2sq-orbits. We show

that optimizing over reparameterizations in this new space produces a well-defined distance

metric (Proposition 6.2.5).

The chapter concludes with a desription of a computational algorithm for optimizing

over reparameterizations. This optimization is approximate (by necessity), as we replace the

group of diffeomorphisms with the set of increasing piecewise-linear homeomorphisms with

vertices on a fixed grid.

14

Chapter 2

Preliminaries and Notation

2.1 Classical Geometry of Framed Curves

In this section we recall basic facts about the differential geometry of framed space curves,

following [15] and [27]. We also set some of the notation to be used throughout the rest of

the thesis.

2.1.1 Definitions

A smooth space curve is a C8 map γ : Σ Ñ R3, where Σ is a one-dimensional manifold,

perhaps with boundary. We will almost exclusively consider the case where either Σ “ r0, 2s

(this particular closed interval is convenient for normalization purposes), or Σ “ S1, where

S1 denotes the round circle of length 2. We will identify S1 with the smooth quotient

S1« r0, 2s{p0 „ 2q « R{2Z.

Those space curves with domain r0, 2s will be called paths and those with domain S1 will be

called loops.

15

Our primary interest will be immersed space curves, where γ1ptq ‰ 0 for all t P Σ. An

immersed space curve has a well-defined rank-2 normal bundle, which is necessarily trivial.

A trivialization of the unit normal bundle of a particular curve γ is called a framing. For

concreteness, we realize a framing as a smooth map V : Σ Ñ R3 satisfying

〈γ1ptq, V ptq〉 “ 0 and }V ptq} “ 1 @ t P Σ.

In the above and throughout the thesis, 〈¨, ¨〉 refers to the standard Euclidean inner product

on R3—the various other inner products which will arise will be given specialized notation.

Likewise, } ¨ } will be reserved for Euclidean norm in R3. A pair pγ, V q will be called a framed

curve (respectively, framed path if the domain is r0, 2s or framed loop if the domain is S1).

Although it is implicit in the definition, we emphasize that in the case that γ is a loop, we

require that V is also a closed loop.

We note that by our conventions, a pair of framed curves pγ1, V1q and pγ2, V2q which

differ only by a reparameterization—that is, pγ2ptq, V2ptqq “ pγ1 ˝ ρptq, V1 ˝ ρptqq for some

orientation-preserving diffeomorphism ρ of Σ—are considered to be distinct framed curves.

It should also be noted that other authors use other definitions of a curve framing. For

example, [27] takes a framed curve to be a triple pγ, V,W q, where γ and V fill the same

role as our definition and W “ γ1{}γ1} ˆ V , while [67] defines a framing of γ to be a one-

parameter family of planes Π such that γ1ptq P Πptq for all t. Each of these definitions is

clearly equivalent to the one that we will use, perhaps up to choice of orientation.

2.1.2 Geometric Invariants and Examples

Before giving some fundamental examples of framed space curves, it will be useful to define

the various geometric invariants of a framed space curve—these are geometric invariants in

the sense that they do not depend on a particular parameterization. First, we introduce the

16

invariants of a smooth immersed space curve γ. The curvature of γ, denoted κ, is defined

by the formula

κptq :“

›

›

›

›

d

dsT ptq

›

›

›

›

,

where

d

ds:“

1

}γ1ptq}

d

dt

denotes derivative with respect to arclength and

T ptq :“γ1ptq

}γ1ptq}

denotes the unit tangent vector to γptq. Working this out, κ can be written more explicitly

as

κ2 :“

›

›

›

›

γ2

}γ1}2

›

›

›

›

2

´

⟨γ2

}γ1}2, T

⟩2

.

In the above, we supress explicit dependence on t P Σ. This convention will be adopted

frequently. Note that if γ is arclength-parameterized—i.e., }γ1} ” 1—then this formula

reduces to κ “ }γ2}.

Example 2.1.1. For a curve γ with nonvanishing curvature, we have our first example of a

framing. Define the principal normal vector to γ to be

N :“1

κ

d

dsT “

1

κ

ˆ

γ2

}γ1}2´

⟨γ2

}γ1}2, T

⟩T

˙

.

The principal normal N is unit length, by design. Then pγ,Nq defines a framed curve, called

the Frenet frame of γ. Since N is defined pointwise by a geometric invariant of γ, it is clear

that if γ is a closed loop, then N is as well, so that pγ,Nq is well-defined according to our

conventions.

17

One should observe that if we treat N as the fundamental object, then κ is recovered by

the formula

κ “

⟨d

dsT,N

⟩.

Similarly, another geometric invariant of a space curve γ with nonvanishing curvature is its

torsion τ , defined by

τ :“

⟨d

dsN,B

⟩,

where the vector B :“ T ˆ N is called the binormal vector of γ. We have now defined the

ingredients necessary to state the fundamental theorem of space curves.

Theorem 2.1.1 (Fundamental Theorem of Space Curves, [15], Section 1-5). Let κ, τ :

r0, 2s Ñ R be arbitrary smooth functions with κ ą 0. There is an arclength-parameterized

space curve γ : r0, 2s Ñ R3 with curvature κ and torsion τ which is unique up to translations

and rigid rotations.

Proof. By the above definitions, the problem of recovering γ can be rephrased as solving the

vector ODE

d

ds

¨

˚

˚

˚

˚

˝

T

N

B

˛

‹

‹

‹

‹

‚

“

¨

˚

˚

˚

˚

˝

0 κ 0

´κ 0 τ

0 ´τ 0

˛

‹

‹

‹

‹

‚

¨

˚

˚

˚

˚

˝

T

N

B

˛

‹

‹

‹

‹

‚

,

and then defining γ by the formula

γptq “

ż t

0

T`

t˘

dt.

By existence and uniqueness of solutions of ODEs, the first step is possible up to choice of

initial position pT p0q, Np0q, Bp0qq, hence we have uniqueness only up to rotations. Moreover,

implicit in our definition of γ was the choice of starting point at the origin, hence uniqueness

up to translations.

18

Thus pκ, τq can be seen as global coordinates on the space

tarclength parameterized paths with nonzero curvatureu{ttranslation, rotationu.

However, these coordinates are not suitable to describe the subspace of closed loops and this

was a major motivation for the new coordinate system developed in this thesis (see Section

3.1 for more details).

Now we consider a general framed curve pγ, V q. The definitions of κ and τ given above

immediately generalize to this setting. It will be convenient to use the notation W :“ T ˆV .

Then we define the Darboux curvatures κ1 and κ2 of pγ, V q by the formulas

κ1 “ κ1pγ, V q :“

⟨d

dsT, V

⟩and κ2 “ κ2pγ, V q :“

⟨d

dsT,W

⟩

and the twist rate tw of pγ, V q by

tw “ twpγ, V q :“

⟨d

dsV,W

⟩.

Note that in the case that V “ N , κ1 “ κ, κ2 “ 0 and tw “ τ .

Example 2.1.2. Using the definition of tw, we obtain a second example of a framing for an

open curve γ, called the Bishop framing [4]. This is the framing V obtained by demanding

that the twist rate of pγ, V q is identically zero. This amounts to solving a simple ODE, and

“the” Bishop framing is only well-defined up to an initial choice of V p0q. In general, we will

use the term relative framing to refer to an equivalence class of framings up to choice of initial

frame vector. Note that for a closed loop the Bishop framing will not necessarily be closed,

so the Bishop framing is not always well-defined for loops. Indeed, it follows easily from the

Calugareanu-White-Fuller Theorem [20] that a closed loop admitting a Bishop framing is

highly non-generic.

19

Relatively framed curves can be defined more geometrically as follows. For any given

framed curve pγ, V q, we have a circle action defined by

θ ¨ pγ, V q :“ pγ, cos θV ` sin θW q, θ P r0, 2πs{p0 „ 2πq.

A relatively framed curve is an equivalence class of a framed curve under this action. For a

framed curve pγ, V q, we denote the corresponding relatively framed curve by pγ, rV sq. The

geometric invariants of θ ¨ pγ, V q are easily computed:

κ1pθ ¨ pγ, V qq “

⟨d

dsT, cos θV ` sin θW

⟩“ cos θκ1pγ, V q ` sin θκ2pγ, V q.

Similarly,

κ2pθ ¨ pγ, V qq “ ´ sin θκ1pγ, V q ` cos θκ2pγ, V q

and

twpθ ¨ pγ, V qq “ twpγ, V q.

Thus twist rate tw is well-defined even for relatively framed curves, whereas the Darboux cur-

vatures κ1, κ2 are well-defined only up to phase. We define the projective Darboux curvatures

to be the equivalence class of function pairs

rκ1, κ2s :“ tpcos θκ1 ` sin θκ2,´ sin θκ1 ` cos θκ2q | θ P r0, 2πqu. (2.1)

Theorem 2.1.2 (Fundamental Theorem for Relatively Framed Curves). Let r, κ1, κ2, tw :

r0, 2s Ñ R be arbitrary smooth functions with r ą 0. There is a relatively framed space curve

pγ, rV sq : r0, 2s Ñ R3ˆR3 satisfying }γ1} “ r with projective Darboux curvatures rκ1, κ2s and

twist rate tw which is unique up to translations and rigid rotations.

20

Proof. Choosing a representative of rκ1, κ2s, say pκ1, κ2q, we solve the system of ODEs

d

dt

¨

˚

˚

˚

˚

˝

T

V

W

˛

‹

‹

‹

‹

‚

“ r

¨

˚

˚

˚

˚

˝

0 κ1 κ2

´κ1 0 tw

´κ2 ´tw 0

˛

‹

‹

‹

‹

‚

¨

˚

˚

˚

˚

˝

T

V

W

˛

‹

‹

‹

‹

‚

,

then define γ by

γptq “

ż t

0

r`

t˘

T`

t˘

dt.

The ODE solving step is possible by existence of solutions of ODE’s and is unique up to

choice of initial conditions pT p0q, V p0q,W p0qq, hence we have uniqueness up to rotations.

We have also implicitly chosen γp0q “ 0, hence uniqueness up to translations. Finally, a

different choice of representative of rκ1, κ2s would have produced a different framing in the

class rV s, thus the solution is only well-defined as a relatively framed curve.

Moreover, if we considered the r in the theorem up to constant multiplication, i.e.,

r1 „ r2 ô Dx ą 0 s.t. r1ptq “ x ¨ r2ptq @ t,

then the resulting framed curve would be well-defined up to scaling. This conclusion moti-

vates the definition of the moduli space that we will study for the majority of this thesis,

the moduli space of framed loops :

M :“ trelatively framed loopsu{ttranslation, scaling, and rotationu.

See Section 3.2.3 for a description of the geometry of M.

21

2.2 Frechet Spaces and Manifolds

As opposed to the finite-dimensional setting, there are a variety of flavors of model spaces for

infinite-dimensional manifolds with distinct topological properties. The infinite-dimensional

manifolds that we use in this work will be modeled on Frechet spaces. The following sections

outline some of the basic background material on Frechet spaces and manifolds, following

the standard references [24] and [39].

The theory of Frechet spaces is vast and sometimes forbiddingly technical. To keep the

conversation grounded, we will frequently refer back to a pair of relevant (and more down-

to-Earth) classes of examples: path and loop spaces of Rn or a finite-dimensional manifold

M .

2.2.1 Tame Frechet Spaces

Let V denote a topological vector space over R or C. By convention, we will also assume

that the topology on V is Hausdorff.

A seminorm on V is a map V Ñ Rě0 which has all of the properties of a norm except

nonzero vectors may take the value zero (in particular, any norm is also a seminorm). We say

that V is locally convex if there exists a family of seminorms t}¨}kukPK (with K not necessarily

countable) on V such that the topology on V is generated by the family of seminorms. This

means that the topology of V is the coarsest one in which, for any fixed v0 P V , every map

of the form

V Ñ Rě0 : v ÞÑ }v ´ v0}k

is continuous. A base for this topology is given by sets of the form

UH,εpv0q “ tv P V | }v ´ v0}h ă ε @h P Hu,

22

where H runs over all finite subsets of K and ε ą 0. An equivalent characterization is that

a sequence vn P V converges to v P V if and only if

}vn ´ v}k Ñ 0 as nÑ 8 @ k.

A simple exercise shows that our assumption that V is Hausdorff implies that any family

of seminorms generating the topology of V must satisfy the property

}v}k “ 0 @ k P K ñ v “ 0.

We can characterize the metrizability of a locally convex space as follows.

Lemma 2.2.1. The topology of V can be generated by a countable family of seminorms

t} ¨ }kukPN if and only if V is metrizable.

Proof. If the topology of V can be generated by a countable family of seminorms, we define

a metric by the formula

dpv, wq :“ÿ

kPN

2´k}v ´ w}k

1` }v ´ w}k. (2.2)

Conversely, suppose that V is metrizable with metric d. Then we define a family of seminorms

} ¨ }k on V by

}v}k “ inftλ P Rą0 | v P Bp0, λ{kqu,

where Bp0, λ{kq is the d-ball around the origin of radius λ{k.

If V is a Hausdorff, locally convex, metrizable vector space which is complete as a metric

space, then it is called a Frechet space. The space is complete as a metric space if and only

if every Cauchy sequence of points vn P V satisfying

}vi ´ vj}k Ñ 0 as i, j Ñ 8 @ k

23

converges in V .

Example 2.2.1 (Trivial Example: Banach spaces.). Any Banach space is trivially a Frechet

space with countable family of seminorms consisting of only the Banach norm.

Example 2.2.2 (Main Example: The loop and path spaces of Rn.). The space C8pS1,Rnq is

a vector space with pointwise addition and scalar multiplication. The Whitney C8 topology

is generated by the family of seminorms

}γ}k :“kÿ

j“0

maxt}γpjqptq}Rn | t P S1u, k “ 0, 1, 2, . . . ,

where } ¨ }Rn is the Euclidean norm, γpjq denotes the j-th derivative of γ, and γp0q “ γ. Note

that each } ¨ }k is actually a norm, and is in particular the standard norm on the Banach

space CkpS1,Rnq.

We claim that this space is Frechet. To see this, let γn be a Cauchy sequence in

C8pS1,Rnq. Using the filtration

C8pS1,Rnq Ă Ck

pS1, Rnq Ă Ck´1

pS1,Rnq Ă ¨ ¨ ¨ Ă C1

pS1,Rnq Ă C0

pS1,Rnq

and the fact that } ¨ }k is the standard Banach norm, we conclude that γn is Cauchy in

CkpS1,Rnq for all k. Thus γn converges in each CkpS1,Rnq and these limits must agree, so

that γn converges to a smooth function.

We can similarly define a Frechet structure on the path space of Rn, C8pr0, 2s,Rnq. We

use the special notations LRn for C8pS1,Rnq and PRn for C8pr0, 2s,Rnq with this topology.

Example 2.2.3 (Important Example: Fast-falling sequences.). Let pV , } ¨ }Vq be a Banach

space. The collection of all sequences tvnu in V such that the sum

}tvnu}k :“ÿ

n

ekn}vn}V

24

converges for all k forms a Frechet space with seminorms } ¨ }k. This is known as the space

of fast-falling sequences in V—see [24], Section II, Example 1.1.2. We will denote the space

of fast falling sequences ΣpVq. Note that the seminorms of ΣpVq have the property that

}tvnu}1 ď }tvnu}2 ď }tvnu}3 ď ¨ ¨ ¨

for all tvnu P ΣpVq.

Remark 2.2.2. We will later define a tame Frechet space to be a Frechet space which

is “close enough” (in a precise sense) to a space of fast-falling sequences for some Banach

space. The category of tame Frechet spaces is convenient in that it admits an inverse function

theorem. The heuristic reason is that spaces of fast-falling sequences are close enough to

Banach spaces that they admit an inverse function theorem, and tame Frechet spaces come

with built-in estimates which allow the inversion result to translate.

A graded Frechet space is a Frechet space V together with a choice of countable family

of seminorms t} ¨ }ku satisfying

}v}1 ď }v}2 ď }v}3 ď ¨ ¨ ¨

for all v P V .

Example 2.2.4. The loop space LRn with the family of seminorms of 2.2.2 is graded.

We note that any Frechet space admits a grading. Indeed, for any Frechet space, choose

an arbitrary countable family of seminorms t} ¨ }ku generating its topology. We then define

a new family of seminorms t} ¨ }1ku by the formula

} ¨ }1k “

kÿ

j“1

} ¨ }j.

25

This new family of seminorms is graded and generates the same topology as the original

family.

A linear map L between graded Frechet spaces pV , t} ¨ }kuq and pV 1, t} ¨ }1kuq is called tame

if there exists r P N and a sequence of real numbers ck P R such that

}Lv}1k ď ck}v}k`r

for all v P V and all sufficiently large k. We say that a graded Frechet space V is a tame direct

summand of another graded Frechet space V 1 if there exist tame linear maps L : V Ñ V 1

and L1 : V 1 Ñ V satisfying L1 ˝ L “ IdV .

Example 2.2.5. The linear map L : LRn Ñ LRn defined by Lpγq “ γphq is tame for any h.

It satisfies

}Lpγq}k ď }γ}k`h

for all k.

A graded Frechet space V is said to be tame if there exists a Banach space W such

that V is a tame direct summand of ΣpWq. Tame Frechet spaces form a category of infinite-

dimensional topological vector spaces which are simultaneously flexible enough to treat many

interesting spaces of smooth maps and well-behaved enough to satisfy an inverse function

theorem, which we state in Section 2.2.4.

Example 2.2.6. Let M be a compact (hence finite-dimensional) manifold and E a vector

bundle over M . Consider the space of smooth sections ΓpM,Eq. Choosing a Riemannian

metric gM on M and a metric gE and connection ∇ on E, this space admits a family of

seminorms t} ¨ }ku defined by

}σ}k :“kÿ

j“1

suppPM

›

›

›

`

∇jσ˘

p

›

›

›

gEbjgM,

26

where ∇j denotes the j-th covariant derivative,

gE bj gM “ gE b gM b ¨ ¨ ¨ b gM

is the induced metric on the vector bundle E bj TM defined on pure tensors by

gE bj gMpσ1 bX11 b ¨ ¨ ¨ bX

j1 , σ2 bX

12 b ¨ ¨ ¨X

j2q

“ gEpσ1, σ2qgMpX1

1 , X12 q ¨ ¨ ¨ g

MpXj

1 , Xj2q

and extended linearly, and } ¨ }gEbjgM is the induced norm. With this family of seminorms,

ΓpM,Eq is a tame Frechet space—see [24], Section II, Corollary 1.3.9 for details.

2.2.2 Calculus in Frechet Spaces and Frechet Manifolds

Unlike the Banach space setting, the total derivative of a map between Frechet spaces is not

well-defined. Going back to a more basic concept, we can define the derivative of a path into

a Frechet space γ : RÑ V in the usual way:

γ1ptq :“ limεÑ0

γpt` εq ´ γptq

ε,

provided the limit exists. We can similarly define higher-order derivatives, and we define a

path to be smooth if all of its derivatives exist. We now define a map P : O Ă V Ñ V 1 from

an open subset of a Frechet space into another Frechet space to be smooth if for every path

γ : RÑ O, the composition P ˝ γ : RÑ V 1 is smooth.

Remark 2.2.3. In the case that V and V 1 are finite-dimensional, this definition of smooth-

ness agrees with the usual one. Surprisingly, this fact is nontrivial and was not proved until

1967 [5].

27

Although total derivatives are not defined for maps between Frechet spaces, it still makes

sense to define directional derivatives. For a smooth map P : O Ă V Ñ V 1 and a pair of

vectors v1 P O, v2 P V we define the derivative of P at v1 in the v2-direction in the expected

way:

DP pv1qpv2q :“d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

P ˝ γpεq P V 1,

where γpεq is the path in V defined by v1` εv2. This is sometimes referred to as the Gateaux

derivative.

A (not necessarily linear) map P from an open subset U of a graded Frechet space V into

another graded Frechet space V 1 satisfies a tame estimate on U if

}P pvq}1k ď ckp1` }v}k`rq,

for some r P N and some sequence of real numbers ck for all sufficiently large k and for all

v P U . A map is said to be tame if it satisfies a tame estimate near each point.

Example 2.2.7. Any continuous map P from Rn with its usual norm into a graded Frechet

space V is tame. Indeed, let x0 P Rn and pick a compact set X containing a neighborhood

of x0. Since P is continuous, }P p¨q}k is a continuous map X Ñ R for all k. Thus there exist

ck such that

}P pxq}k ď ck ď ckp1` }x}q

for all x in X.

We define a Frechet manifold to be a Hausdorff topological space equipped with an open

cover by charts such that each chart is homeomorphic to an open subset of a Frechet space

and such that all transition maps (defined in the usual way) are smooth. The Frechet spaces

in the definition are called model spaces for the manifold. A tame Frechet manifold is a

28

Frechet manifold whose model spaces are tame and whose transition maps are both smooth

and tame.

Example 2.2.8 ([24], Chapter II, Corollary 2.3.2). Every space of smooth maps from a

compact manifold into another finite-dimensional manifold is a tame Frechet manifold.

Let F be a Frechet manifold with charts Pα : Oα Ă F Ñ Vα. A submanifold of F is a

subset G such that for each Oα with Oα X G ‰ H, the model space Vα splits as

Vα “ V1 ‘ V2

and Pα|OαXG has image in V1 ‘ t0u.

Let F be a Frechet manifold and let f P F . The tangent space to F at f is defined in

the same way as the finite-dimensional case. That is, the tangent space is the collection of

derivatives

d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

γpεq,

where γpεq is a path in F satisfying γp0q “ f . The derivative is calculated in local coordi-

nates, and is well-defined by smooth compatibility of the charts of F . Each tangent space

to F of course forms a vector space, denoted TfF , and the collection of all tangent spaces

forms a vector bundle, denoted TF .

Finally, a map P : F Ñ G between Frechet manifolds is called smooth if it is smooth in

local coordinates. Smooth maps induce linear maps on tangent spaces called derivatives as

follows. For f P F , let δf P TfF (we will frequently adopt variational notation for tangent

vectors). Then

δf “d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

γpεq

29

for some path γ. Then we define the directional derivative DP pfqpδfq by

DP pfqpδfq :“d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

P ˝ γpεq P TP pfqG.

2.2.3 Extended Examples: Path and Loop Spaces

The main Frechet spaces and manifolds which appear in this thesis are path and loop spaces

or submanifolds thereof. Thus we take extra care to explore these examples and to set

notation.

Recall that for a finite-dimensional vector space V, we define

PV :“ C8pr0, 2s,Vq, and

LV :“ C8pS1,Vq

and that these are tame Frechet spaces.

Similarly, for a finite-dimensional manifold M , we define the path space of M and loop

space of M to be the spaces

PM :“ C8pr0, 2s,Mq, and

LM :“ C8pS1,Mq,

respectively. These spaces are tame Frechet manifolds. Indeed, this follows from the much

more general theorem of Example 2.2.8:

Theorem 2.2.4 ([24], Section II, Corollary 2.3.2). Let M and M 1 be smooth finite-dimensional

manifolds with M compact. Then the space of smooth maps C8pM,M 1q is a tame Frechet

manifold.

30

We will give explicit manifold charts for the specific case of path spaces, as the ideas in

the proof will be useful later on. The proof sketched here follows [70], and we skip technical

details.

Proposition 2.2.5. The path space PM is a Frechet manifold for any finite-dimensional

manifold M . Moreover, if M is n-dimensional then PM is locally modelled on its tangent

spaces, which take the form

TγPM “ Γpr0, 2s, γ˚TMq,

where γ P PM and γ˚TM Ñ r0, 2s is the pullback bundle. Each tangent space is isomorphic

to PRdimpMq.

Similarly, the loop space LM is a Frechet manifold, locally modelled on its tangent spaces

TγLM “ ΓpS1, γ˚TMq « LRdimpMq.

Proof sketch. We prove the statement regarding path spaces; the loop space proof is essen-

tially the same. We define a local addition on M to be a smooth map ν : TM Ñ M such

that

(i) The composition of ν with the zero section is the identity on M , and

(ii) there exists an open neighborhood Udiag of the diagonal in M ˆM such that the map

π ˆ ν : TM Ñ M ˆM is a diffeomorphism onto Udiag, where π : TM Ñ M is the

natural projection.

We will denote the restriction of ν to the fiber over p PM by νp : TpM ÑM . A local addition

is essentially a generalization of the exponential mapping on a Riemannian manifold. It is a

fact that any M admits a local addition (see [39], Section 42.4).

We will use a fixed local addition ν : TM Ñ M to define charts pPγ,Oγq on PM . Let

Udiag Ă M ˆM be the image of the diffeomorphism from the definition of local addition.

31

For fixed γ P PM , define Oγ Ă PM by

Oγ :“ tβ P PM | pγ, βq P PUdiagu.

Let γ˚TM Ñ r0, 2s denote the pullback bundle of TM by γ : r0, 2s Ñ M . We define

Pγ : Γpr0, 2s, γ˚TMq Ñ Oγ as follows: For α P Γpr0, 2s, γ˚TMq, define

Pγpαqptq :“ νγptqpαptqq.

Choosing a global trivialization of the vector bundle α˚TM Ñ r0, 2s, we have Γpr0, 2s, α˚TMq «

PRdimpMq, and we have an atlas of charts on PM .

Next we need to show that the transition charts for this atlas are smooth. Let β, γ P PM ,

and consider the map

τβγ :“ P´1β ˝ Pγ : P´1

γ pOβ XOγq Ă Γpr0, 2s, γ˚TMq Ñ P´1β pOβ XOγq Ă Γpr0, 2s, β˚TMq.

Let α P P´1γ pOβ XOγq. Then

τβγpαqptq “ ν´1βptq ˝ νγptqpαptqq.

Let αpuq denote a smooth path in Γpr0, 2s, γ˚TMq with parameter u. Then

τβγpαpuqqptq “ ν´1βptq ˝ νγptqpαpuqptqq,

which can be considered as a smooth map from R ˆ r0, 2s into M . Smoothness in the

u-parameter means that our transition functions are smooth, by definition.

32

Our next task is to explore the functorial properties of P and L. For example, given a

smooth map of finite-dimensional manifolds f : M Ñ M 1, we obtain new smooth maps of

Frechet manifolds denoted

Pf : PM Ñ PM 1 and

Lf : LM Ñ LM 1.

The path space version is defined by the formula

Pfpαqptq :“ fpαptqq

and the loop space version is defined similarly. These new maps behave well with respect to

differentiation in the case that M is an open subset of a vector space.

Lemma 2.2.6. Let f : M Ñ M 1 be a smooth map between (finite-dimensional) manifolds,

where M is an open submanifold of a vector space. Let α P PM and δα P TαPM . Then the

derivative of Pf at α in the direction δα satisfies

DPfpαqpδαqptq “ Dfpαptqqpδαptqq.

A similar formula holds for Lf : LM Ñ LM 1.

Proof. This is a simple calculation using the vector space structure of M :

DPfpαqpδαqptq “ d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

fpαptq ` εδαptqq

“ Dfpαptqqpδαptqq.

33

These functors also respect principal bundle structures. We first note that for a smooth

finite-dimensional Lie group G, we obtain new infinite-dimensional Lie groups PG and LG,

where group composition is applied pointwise. Then we have the following proposition.

Proposition 2.2.7 (See [70], Section 4.3). Let G ãÑ F Ñ M be a principal fiber bundle.

Then applying the loop space functor to these maps produces a new principal fiber bundle

LG ãÑ LF Ñ LM . A similar statement holds for the path space functor.

We close this subsection by noting that our identification of S1 with r0, 2s{p0 „ 2q has a

special relevance. This is predicated upon our choice of defining paths to have domain r0, 2s,

which is merely a convenient normalization for the results of this thesis. The relevance is

that it will be convenient to think of the loop space LM as the embedded submanifold of the

path space PM consisting of paths which happen to smoothly close. We record this obvious

but useful fact below.

Proposition 2.2.8. The inclusion map LM ãÑ PM is a smooth embedding of Frechet

manifolds.

2.2.4 The Nash-Moser Inverse Function Theorem

We are now prepared to state the Nash-Moser Inverse Function Theorem. This is a hard

theorem whose proof is well beyond the scope of this thesis. Hamilton’s treatise [24] is

devoted to this theorem and its many applications, while a condensed version of the proof

can be found [39], Chapter 10, Section 51.

Theorem 2.2.9 (Nash-Moser Inverse Function Theorem, [24], Section III, Theorem 1.1.1).

Let V and V 1 be tame Frechet spaces and P : V Ñ V 1 a smooth tame map. Let O Ă V be an

open set such that for each v P O, the derivative

DP pvq : V Ñ V 1

34

has a unique inverse, denoted

pDP q´1pvq : V 1 Ñ V .

If

pDP q´1 : O ˆ V 1 Ñ V

is a smooth tame map, then P is locally invertible and its inverse is smooth and tame.

We will frequently make use of a simpler version of the theorem, which assumes that

the range of the map is finite-dimensional. This simpler version more closely resembles the

classical inverse function theorem which holds in the Banach space category.

Theorem 2.2.10 ([24], Section III, Theorem 2.3.1). Let V be a tame Frechet space and

P : O Ă V Ñ V be a smooth map of an open subset of V into a finite-dimensional vector

space. If the derivative DP pv0q is surjective for some v0 P O, then the level set

F “ tv P O | P pvq “ P pv0qu

is a smooth tame submanifold in a neighborhood of f0 with tangent space

TvF “ kernelpDP pvqq.

2.3 Preliminaries from Differential Geometry

In this section we define symplectic and Riemannian structures for Frechet manifolds. Note

that these definitions are more-or-less formal, and we do not claim that any general the-

orems from finite-dimensional differential geometry automatically apply (in fact, we will

point out some finite-dimensional theorems which are known to fail in infinite-dimensions).

35

Nevertheless, these geometric structures frequently provide intuition for specific cases of

infinite-dimensional manifolds and we will find them quite useful throughout the thesis.

2.3.1 Symplectic and Riemannian Structures

We define a weak symplectic structure on a Frechet space V to be a 2-form ω on V satisfying:

1. ω is closed. This means dω “ 0, where d is the exterior derivative defined by the

formula

dωvpδv1, δv2, δv3q “

Dω‚pδv2, δv3qpfqpδv1q ´Dω‚pδv1, δv3qpfqpδv2q `Dω‚pδv1, δv2qpfqpδv3q

for any triple of vectors δvj P TvV « V . In the formula we are treating the δvj as

constant vector fields near v. The expression ω‚pδv2, δv3q (for example) is considered

as a map from V to R and

Dω‚pδv2, δv3qpvqpδv1q

as the derivative of this map at v in the δv1 direction.

2. ω is nondegenerate. That is, ω induces an injection from TvV into T ˚v V for each v P V

via the formula

δv ÞÑ ωvpδv, ¨q. (2.3)

This definition also works for a Frechet submanifold F of a Frechet space V , where

tangent spaces TfF are naturally identified with linear subspaces of TfV « V . The definition

similarly extends to the base space of certain submersions with total space a submanifold

of a Frechet space and with finite-dimensional fiber. These are the only cases of symplectic

Frechet manifolds which appear in this thesis, so we will not worry about extending the

36

definition to more general situations where technicalities involving the existence of a Lie

bracket arise. For a more general treatment, see [39], Section 48.

In the case that V is finite-dimensional, this agrees with the standard definition of a

symplectic structure. Moreover, the injection defined by (2.3) will be an isomorphism in

this case. A symplectic structure on a general Frechet manifold is called strong if (2.3) is an

isomorphism—this situation will not come up again in this thesis.

An almost complex structure on a Frechet manifold F is a smooth choice of involution

J : TfF Ñ TfF , J2“ Íd.

Such an involution is called a complex structure if it arises via holomorphic charts on F

taking values in a complex Frechet space. In this case the almost complex structure is also

frequently referred to as integrable. We note that in finite-dimensions the integrability of an

almost complex structure J is equivalent to the vanishing of a certain tensor defined in terms

of J , called the Nijenhuis tensor. The definition of the Nijenhuis tensor extends to infinite-

dimensional manifolds, but a vanishing Nijenhuis tensor no longer implies the integrability

of the almost complex structure (see [43]).

A Riemannian metric on F is a smoothly-varying choice g of inner product on each

tangent space

gf p¨, ¨q : TfF ˆ TfF Ñ R.

The Riemannian metrics used in this thesis will always be weak in the sense that they induce

injections

TfF Ñ T ˚f F : δf ÞÑ gf pδf, ¨q,

but we do not requre them to induce bijections (in the case that the induced map is a

bijection, the metric is called strong). The definition of a Levi-Civita connection for a

37

metric g extends without alteration to the infinite-dimensional setting, although we make

no claim of existence or uniqueness for such a connection. We define a path γ in F to be a

geodesic with respect to g if

∇γ1ptqγ1ptq “ 0

for all t, where ∇ denotes a Levi-Civita connection for g. We once again do not claim

the existence or uniquenss of geodesics (such a claim can frequently be translated into a

statement about the existence and uniqueness of a solution to a PDE).

Finally, we define a Kahler structure on F to be a triple pg, ω, Jq, where g is a weak

Riemannian metric, ω is a weak sympectic structure, and J is a complex structure such that

the three structures are related by the formula

gp¨, ¨q “ ωp¨, J ¨q.

2.3.2 Marsden-Weinstein Reduction

In this section we describe the process of reducing a finite-dimensional symplectic manifold

by the Hamitonian action of a Lie group, introduced by Marsden and Weinstein [48]. The

definition given here is somewhat nonstandard, but it is easy to see that it is equivalent to

the usual definition in the finite-dimensional case. The reason for the departure is that it

will allow us to avoid working with the dual to a Frechet vector space later on in the thesis.

LetG be a finite-dimensional Lie group that freely and properly acts on a finite-dimensional

symplectic manifold pM,ωq. Let g denote the Lie algebra of G. For each ξ P g, there is an

associated vector field Xξ on M describing the infinitesimal action of ξ. Let 〈¨, ¨〉g be any

choice of inner product on g. A moment map for the action of G is a smooth map

µ : M Ñ g,

38

which satisfies

Dp 〈µppq, ξ〉g pY |pq “ ωppY |p, Xξ|pq (2.4)

for all p P Σ, ξ P g and vector fields Y on Σ. The left side of the equation should be

understood as the derivative of the function

p ÞÑ 〈µppq, ξ〉g (2.5)

at the point p in the direction Y |p.

Remark 2.3.1. In the standard definition of a moment map, µ is g˚-valued, and the unnat-

ural introduction of an auxilliary inner product on g is avoided. As previously mentioned,

our goal is to avoid working wth dual spaces. The Lie groups that we will work with come

endowed with preferred metrics so that this definition will be natural enough in practice.

In the case that a moment map exists, it is a fact that for any regular value ξ P g, µ´1pξq

is a smooth submanifold of Σ such that the restriction of ω degenerates along exactly the

G-orbits. Therefore one obtains a new manifold, called the symplectic reduction of Σ by G

and denoted

Σ �G :“ µ´1pξq{G,

which has a canonical symplectic form. If pM,ωq is a Kahler manifold and G acts by

isometries with respect to the Riemannian structure and preserves the complex structure,

then the symplectic reduction is also a Kahler manifold and we refer to it as the Kahler

reduction of M by G ([54], Chapter 8).

The Kahler reduction construction extends without issue to the case that the original

manifold and the Lie group are infinite dimensional. However, the general finite-dimensional

theorems (or general theorems in the Banach space category as in, e.g., [73]) no longer apply,

and it must be checked by hand that the resulting space is indeed a Kahler manifold.

39

2.4 Quaternions

Throughout the thesis, it will be convenient for calculations to identify C2 with the algebra

of quaternions. The quaternions are elements of the real vector space

H :“ spanRt1, i, j,ku,

which is endowed with an algebra structure by extending the relations

i2 “ j2“ k2

“ ijk “ ´1

linearly. Elements of the quaternions will typically be denoted by q “ Q0`Q1i`Q2j`Q3k.

In this notation, the real part of q is

Repqq “ Q0 P R

and the imaginary part of q is

Impqq “ Q1i`Q2j`Q3k.

We will quite frequently identify the purely imaginary quaternions with R3 so that Impqq P

R3. The quaternionic conjugate of q P H is the quaternion

q “ Q0 ´Q1i´Q2j´Q3k.

40

We identify C2 with H via the map

C2Ø H

pz, wq Ø z ` wj (2.6)

where we also identify the complex i with the quaternionic i.

Let 〈¨, ¨〉C2 denote the standard Hermitian inner product on C2 and let } ¨ }C2 denote

the corresponding norm. From the identification (2.6), this endows H with a Hermitian

inner product and norm, denoted 〈¨, ¨〉H and } ¨ }H, respectively. The real part of 〈¨, ¨〉C2 is a

Euclidean inner product on C2, and by extension Re 〈¨, ¨〉H is a Euclidean inner product on

H. A simple calculation shows that the Euclidean product on H is given by

Re 〈p, q〉H “ Reppqq.

However, we should be careful to note that in general 〈p, q〉H ‰ pq (the latter is not even

complex-valued).

The identification (2.6) restricts to an identification of the unit 3-sphere S3 Ă C2 with

the unit quaternions

tq P H | }q}H “ 1u.

The product of a pair of unit quanterions is still a unit quaternion, thus this endows S3 with

a Lie group structure. We endow S3 with its standard, constant curvature-1 Riemannian

metric by restricting the real part of the Hermitian inner product on C2. We denote this

metric gS3

and note that it is left-invariant with respect to the Lie group structure of S3.

We also endow the unit quaternions with the Riemannian metric obtained by restricting the

real part of 〈¨, ¨〉H. By the above discussion, it is clear that, with respect to these metrics,

S3 and the unit quaternions are isometric.

41

We conclude this section by identifying S3 and the unit quaternions with a third well-

known Lie group, the special unitary group SUp2q. Elements of SUp2q are of the form

¨

˚

˝

z w

´w z

˛

‹

‚

,

where }pz, wq}C2 “ 1. The mapping

S3Ñ SUp2q

pz, wq ÞÑ

¨

˚

˝

z w

´w z

˛

‹

‚

(2.7)

is a diffeomorphism. Moreover, we claim that it is an isometry. That is, the standard

left-invariant metric on the Lie group SUp2q is

gSUp2qpξ1, ξ2q :“

1

2tracepξ1ξ

˚2 q, for ξ1, ξ2 P TASUp2q,

where TASUp2q is isomorphic to the Lie algebra of skew-Hermitian 2 ˆ 2 complex matrices

sup2q via the identification

ξ P TASUp2q Ø A´1ξ P sup2q.

Then we have the following.

Lemma 2.4.1. The map (2.7) is an isometry with respect to gS3

and gSUp2q.

Proof. For the remainder of this proof, we denote the map (2.7) by f . Since each metric is

left-invariant and f maps p1, 0q P S3 to the idenity matrix in SUp2q, it suffices to prove the

claim at the identity element. The first step is to compute the derivative of f at p1, 0q P S3

42

in the direction of pa, bq P T1S3:

Dfpp1, 0qqpa, bq “d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

fp1` εa, εbq

“d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

¨

˚

˝

1` εa εb

´εb 1` εa

˛

‹

‚

“

¨

˚

˝

a b

´b a

˛

‹

‚

.

Next we compute the pullback of gSUp2q by f :

f˚gSUp2q1 ppa1, b1q, pa2, b2qq “ g

SUp2qId pDfpp1, 0qqpa1, b1q, Dfpp1, 0qqpa2, b2qq

“ gSUp2q

¨

˚

˝

¨

˚

˝

a1 b1

´b1 a1

˛

‹

‚

,

¨

˚

˝

a2 b2

´b2 a2

˛

‹

‚

˛

‹

‚

“1

2trace

¨

˚

˝

¨

˚

˝

a1 b1

´b1 a1

˛

‹

‚

¨

˚

˝

a2 b2

´b2 a2

˛

‹

‚

˚˛

‹

‚

“1

2

`

a1a2 ` b1b2 ` b1b2 ` a1a2

˘

“ Re`

a1a2 ` b1b2

˘

“ gS3

ppa1, b1q, pa2, b2qq.

43

Chapter 3

Complex Coordinates for Framed

Paths and Loops

This chapter describes the complex coordinate system for framed paths and loops which

will be used throughout the rest of the thesis. We will first describe the moduli spaces of

framed paths and loops of interest. These moduli spaces are endowed with Riemannian

metrics, which are natural from a variety of perspectives. Finally, we will establish our first

main results, which isometrically parameterize moduli spaces of framed paths and loops by

infinite-dimensional versions of classical Riemannian manifolds.

3.1 Framed Path Space

As described in Section 1.1.2, our goal is to develop a coordinate system for spaces of sim-

ilarity classes of framed curves in analogy with the classical approach of using curvature

κ and torsion τ as a global coordinate system for the space of arclength parameterized,

nonvanishing curvature curves in R3 up to ambient Euclidean isometry. In Section 3.1.3 we

describe how to associate a framed path to a pair of smooth complex-valued functions, which

44

we denote

pφ, ψq : r0, 2s Ñ C2zt0u.

Such a pair will also frequently be denoted Φ “ pφ, ψq. In contrast with pκ, τq-coordinates,

the conditions on pφ, ψq that ensure periodicity of the resulting framed path are easy to

check, and this makes the submanifold of framed loops quite convenient to work with. We

note that the construction of a framed curve from pφ, ψq given here is not new—see, e.g.,

[22, 27]. The novel contributions of this thesis are the interpretation of pφ, ψq as global

coordinates on the moduli space of framed paths and the resulting description of the space

of framed loops as a submanifold.

In this section, we introduce the space of framed paths. We represent a framed path

pγ, V q as an element of PpSOp3q ˆ R`q via the map

pγ, V q ÞÑ ppT, V, T ˆ V q, }γ1}q , where T :“γ1

}γ1}. (3.1)

Up to translation, this map is a bijection with inverse

ppU, V,W q, rq ÞÑ

ˆ

t ÞÑ

ż t

0

rU dt, V

˙

, (3.2)

and we identify

rS :“ tpγ, V q | γp0q “ 0u

« tframed pathsu{ttranslationu

« PpSOp3q ˆ R`q. (3.3)

From the discussion in Section 2.2.3, we conclude that rS naturally has the structure of a

tame Frechet manifold. We claim that it also admits a natural Riemannian metric, which

45

we describe in the following section. Since we wish to define a Riemannian metric, it will be

useful to have an explicit description of the tangent spaces of rS. Throughout the thesis we

denote tangent vectors to an infinite-dimensional manifold using variational notation; e.g.,

a tangent vector to pγ, V q will be denoted pδγ, δV q.

Lemma 3.1.1. The tangent space at pγ, V q P rS is the space of vectors

pδγ, δV q P C8pr0, 2s,R3ˆ R3

q

satisfying the constraints

δγp0q “ 0,⟨d

dtδγ, V

⟩`

⟨d

dtγ, δV

⟩“ 0, and

〈δV, V 〉 “ 0.

Proof. We first note that we can consider elements of rS as paths

pγ, V q : r0, 2s Ñ R3ˆ R3

satisfying the constraint equations

γp0q “ 0, (3.4)

〈γ1, V 〉 “ 0, and (3.5)

}V }2 “ 1. (3.6)

46

A tangent vector at pγ, V q is thus an equivalence class of derivatives of the form

pδγ, δV q “d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

pγε, Vεq,

where pγε, Vεq is a one-parameter family of paths r0, 2s Ñ R3 ˆ R3 satisfying constraint

equations (3.4)-(3.6), as well as pγ0, V0q “ pγ, V q. From (3.4), we conclude

δγp0q “d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

γεp0q “ 0.

From (3.5), we obtain

d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

〈γ1ε, Vε〉 “ 0,

which implies

〈δγ1, V 〉` 〈γ1, δV 〉 “⟨d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

γ1ε, V0

⟩`

⟨γ10,

d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

Vε

⟩“ 0.

Similarly, the third constraint defining rS implies the third constraint defining the tangent

space.

The framed path space rS admits an R`-action by scaling of the base curve: for x P R`

and pγ, V q P rS, the action is defined by

x ¨ pγ, V q :“ pxγ, V q (3.7)

We are particularly interested in the quotient space

S :“ tframed pathsu{ttranslation, scalingu,

47

which we refer to as the moduli space of framed paths. This space is of interest because

it is more relevant for shape recognition applications and has especially nice geometry. In

particular, we will see in Section 3.1.4 that it is isometric to an L2 sphere. The R`-action

admits a global slice chart, and we will identify S with the concrete model space of framed

paths of length 2,

S «!

pγ, V q P rS | lengthpγq “ 2)

.

Using this identification, we show that S has the structure of a manifold.

Proposition 3.1.2. The moduli space of framed paths S is a tame Frechet submanifold of

rS with tangent spaces of the form

Tpγ,V qS “"

pδγ, δV q P Tpγ,V q rS |ż 2

0

〈δγ1, T 〉 dt “ 0

*

, (3.8)

where T “ γ1{}γ1}.

Proof. By transfer of structure via the identification (3.3), it suffices to show that the space

"

pA, rq P PpSOp3q ˆ R`q |ż 2

0

r dt “ 2

*

(3.9)

is a tame Frechet manifold. As previously noted, PpSOp3qˆR`q is a tame Frechet manifold.

The set (3.9) is the inverse image of 2 of the map

` : PpSOp3q ˆ R`q Ñ R

pA, rq ÞÑ

ż 2

0

rptq dt.

48

This map is smooth and has 2 as a regular value. Indeed, taking a variation pδA, δrq of pA, rq

where δA is arbitrary and δr is constantly 1, we have

D`pA, rqpδA, δrq “d

dε

ˇ

ˇ

ˇ

ˇ

0

`pA` εδA, r ` εδrq “d

dε

ˇ

ˇ

ˇ

ˇ

0

ż 2

0

prptq ` εδrq dt “

ż 2

0

δr dt “ 2 ‰ 0.

Thus S is a submanifold by Corollary 2.2.10. Moreover, in pγ, V q-coordinates δr “ 〈δγ1, T 〉,

whence

D`pγ, V qpδγ, δV q “

ż 2

0

〈δγ1, T 〉 dt

and we conclude that the space (3.8) is the kernel of D`pγ, V q.

3.1.1 The Natural Metric on Framed Path Space

In this section we define a Riemannian metric on

rS “ tframed pathsu{ttranslationu

which is invariant under scaling, rotations and reparameterizations. Invariance under these

transformations is essential for the metric to be useful for shape recognition algorithms.

Using the identification (3.3), we first define the metric on PpSOp3q ˆ R`q.

Recall that the standard left-invariant metric on the Lie group SOp3q is

pξ1, ξ2q ÞÑ1

2tracepξ1ξ

T2 q, for ξ1, ξ2 P TASOp3q,

where TASOp3q is isomorphic to the Lie algebra of skew-symmetric 3ˆ 3 matrices sop3q via

the identification

ξ P TASOp3q Ø A´1ξ P sop3q.

We denote this metric by gSOp3q.

49

Similarly, the standard left-invariant metric on R` (a Lie group under multiplication) is

given by

px1, x2q ÞÑx1 ¨ x2

r2, for x1, x2 P TrR`,

where TrR` is isomorphic to the abelian Lie algebra R via the identification

x P TrR` Øx

rP R.

We endow the product Lie group SOp3q ˆ R` with the product metric

gSOp3qˆR`pA,rq ppξ1, x1q, pξ2, x2qq :“

1

2tracepξ1ξ

T2 q `

x1 ¨ x2

r2.

Applying the isomorphisms

TASOp3q « sop3q and TrR` « R

pointwise produces an isomorphism

TpA,rqPpSOp3q ˆ R`q « Ppsop3q ˆ Rq

for each pA, rq P PpSOp3q ˆ R`q. The natural reparametrization-invariant L2-type metric

on PpSOp3q ˆ R`q is given by

gPpSOp3qˆR`qpA,rq ppδA1, δr1q, pδA2, δr2qq :“

ż 2

0

gSOp3qˆR`pAptq,rptqq ppδA1ptq, δr1ptqq, pδA2ptq, δr2ptqqq ds,

where

ds “ dsprq :“ rptqdt

50

corresponds to measure with respect to arclength in pγ, V q-coordinates. Integrating against

this measure is necessary to ensure that the metric is invariant under reparametrizations.

By construction, we also have that gPpSOp3qˆR`q is invariant with respect to the action of R`

defined by

x ¨ pAptq, rptqq :“ pAptq, x ¨ rptqq, x P R`.

In pγ, V q-coordinates, this action corresponds to scaling the base curve γ, as described in

(3.7). Therefore the metric restricts to a well-defined metric on the moduli space of framed

paths

S “ tframed pathsu{ttranslation, scalingu.

To simplify notation, we use the notation

gS :“1

4gPpSOp3qˆR`q

ˇ

ˇ

ˇ

rSor

1

4gPpSOp3qˆR`q

ˇ

ˇ

ˇ

S.

There are two important remarks to be made here:

1. The slight abuse of notation employed here by using the same notation for either

restricted metric has the benefit of avoiding an unwieldy list of specialized notations.

Such abuses will occur frequently throughout the remainder of the thesis.

2. The extra factor of 1{4 in this definition is a normalization in anticipation of cleaning

up notation later on, and has no deep conceptual signifcance.

The metric gS has an interesting form when expressed in pγ, V q-coordinates. It is related

to a well-studied family of metrics called the elastic metrics [57] on the Frechet space

Nplanar :“ tImmersions r0, 2s Ñ R2u{ttranslation, scalingu.

51

This family of metrics ga,b, a, b ě 0 are defined as follows. Let γ be an element of the space

and let δγ1 and δγ2 be tangent vectors. Then ga,b is defined by the formula

ga,bγ pδγ1, δγ2q “1

2`pγq

ż 2

0

a

⟨d

dsδγ1, N

⟩⟨d

dsδγ2, N

⟩` b

⟨d

dsδγ1, T

⟩⟨d

dsδγ2, T

⟩ds,

where d{ds is derivative with respect to arclength, N is the unit normal to γ, T is the unit

tangent to γ, and `pγq is the length of γ. The first term in the integrand compares bending

of the variations and the second term compares stretching. We note that this metric is in

fact defined on the space of immersions r0, 2s Ñ R2, but that it is invariant to translation

and scaling (invariance to scaling is facilitated by the `pγq´1 factor before the integral)

so that it induces a well-defined metric on Nplanar. Moreover, the metrics are invariant to

rotations of R2 and reparameterization—such symmetries are necessary for shape recognition

applications.

This definition cannot be extended to the full space of immersed curves in R3, since

the normal vector to a parameterized space curve γ is not well-defined when γ2 vanishes.

However, the analogues of elastic metrics can be defined for framed space curves. Let pγ, V q

be a framed curve, let T denote the unit tangent to γ, and let W “ T ˆ V . Let pδγj, δVjq,

j “ 1, 2, be tangent vectors to pγ, V q. We define the family of framed curve elastic metrics

ga,b,c,d, a, b, c, d ě 0, on S by the formula

ga,b,c,dpγ,V q ppδγ1, δV1q, pδγ2, δV2qq

“1

2`pγq

ż 2

0

a

⟨d

dsδγ1, V

⟩⟨d

dsδγ2, V

⟩` b

⟨d

dsδγ1,W

⟩⟨d

dsδγ2,W

⟩` c

⟨d

dsδγ1, T

⟩⟨d

dsδγ2, T

⟩` d 〈δV1,W 〉〈δV2,W 〉 ds.

The first and second terms in ga,b,c,d compare the bending deformation of γ under the vari-

ations δγj, the third term compares the stretching deformation of γ under the δγj and the

52

last term compares the rotational deformation of the framing V under the variations δVj.

As in the planar case, we note that these metrics are actually defined on the space of framed

paths, but that they are invariant to translation and scaling so that they each induce a

well-defined metric on rS and S. Each metric is also invariant under rotations of R3 and

reparameterizations.

Remark 3.1.3. The somewhat unnatural-looking factor of 1{2 in front of ga,b and ga,b,c,d is

a convenient normalization whose purpose will become apparent in Section 3.1.4.

The next proposition shows that the natural metric metric gS from the perspective of the

Lie group embedding is also a particularly nice example of this generalized family of elastic

metrics; namely it is equal to g1,1,1,1. First, we prove a lemma.

Lemma 3.1.4. The formula for the metric g1,1,1,1 restricted to S reduces to

g1,1,1,1pγ,V q ppδγ1, δV1q, pδγ2, δV2qq “

1

4

ż 2

0

⟨d

dsδγ1,

d

dsδγ2

⟩` 〈δV1,W 〉〈δV2,W 〉 ds.

Proof. First note that by restricting to S, we have `pγq “ 2, so the 1{2`pγq factor in front of

the integral sign reduces to a constant factor of 1{4. Then note that, since pT ptq, V ptq,W ptqq

forms an orthonormal frame for all t, we can express the arclength derivative of any curve

variation δγ as

d

dsδγ “

⟨d

dsδγ, T

⟩T `

⟨d

dsδγ, V

⟩V `

⟨d

dsδγ,W

⟩W.

The result then follows by an elementary computation using the orthonormality of pT, V,W q

and the normalization a “ b “ c “ d “ 1.

Proposition 3.1.5. In pγ, V q-coordinates, the natural metric gS restricted to S is equal to

the framed curve elastic metric g1,1,1,1.

53

Proof. For the remainder of the proof, let f denote the identification (3.2); i.e.,

f : PpSOp3q ˆ R`q Ñ rS

ppU, V,W q, rq ÞÑ

ˆ

t ÞÑ

ż t

0

rU dt, V

˙

.

Our goal is to compute the pullback of g1,1,1,1 by f .

Let A “ pU, V,W q P PSOp3q and r P PR`. By the discussion above, tangent vectors to

PpSOp3q ˆ R`q at pA, rq take the form pAξ, xq, where ξ P Psop3q and x P PR. It will be

useful to explicitly write ξ as the skew-symmetric matrix

ξ “

¨

˚

˚

˚

˚

˝

0 ´ξ3 ξ2

ξ3 0 ´ξ1

´ξ2 ξ1 0

˛

‹

‹

‹

‹

‚

,

where ξj P PR. Then

Aξ “ pξ3V ´ ξ2W,´ξ3U ` ξ1W, ξ2U ´ ξ1V q.

Treating PpSOp3q ˆ R`q as an embedded subset of PppR3q3 ˆ R`q in the obvious way,

we are abe to compute the derivative of f as

DfpA, rqpAξ, xq “d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

fpA` εAξ, r ` εxq

“d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

ˆ

t ÞÑ

ż t

0

pr ` εxqpU ` εpξ3V ´ ξ2W qq dt, V ` εp´ξ3U ` ξ1W q

˙

“

ˆ

t ÞÑ

ż t

0

xU ` rpξ3V ´ ξ2W q dt,´ξ3U ` ξ1W

˙

.

54

If we denote the last line by pδγ, δV q and the image fpA, rq by pγ, V q, then we have

d

dsδγ “

1

}γ1}

d

dt

ˆ

t ÞÑ

ż t

0

xU ` rpξ3V ´ ξ2W q dt

˙

“1

rpxU ` rpξ3V ´ ξ2W qq

“x

rU ` ξ3V ´ ξ2W.

We can now compute f˚g1,1,1,1. Let pA, rq P PpSOp3q ˆ R`q and denote the image of

pA, rq under f by pγ, V q and let pAξj, xjq, j “ 1, 2, denote elements of TpA,rqPpSOp3q ˆ R`q

with images pδγj, δVjq under DfpA, rq. The entries of ξj will be denoted ξkj , k “ 1, 2, 3.

Applying Lemma 3.1.4 and the previous calculation, we obtain

pf˚g1,1,1,1qpA,rqppAξ1, x1q, pAξ2, x2qq “ g1,1,1,1

pγ,V q ppδγ1, δV1q, pδγ2, δV2qq

“1

4

ż 2

0

⟨d

dsδγ1,

d

dsδγ2

⟩` 〈δV1,W 〉〈δV2,W 〉 ds

“1

4

ż 2

0

⟨x1

rU ` ξ3

1V ´ ξ11W,

x2

rU ` ξ3

2V ´ ξ22W⟩

`⟨´ξ3

1U ` ξ11W,W

⟩ ⟨´ξ3

2U ` ξ12W,W

⟩ds

“1

4

ż 2

0

x1x2

r2` ξ3

1ξ32 ` ξ

21ξ

22 ` ξ

11ξ

12 ds

“1

4

ż 2

0

1

2tracepξ1ξ

T2 q `

x1x2

r2ds

“ gSpA,rqppAξ1, x1q, pAξ2, x2qq.

3.1.2 The Hopf Map

To obtain complex coordinates on framed path space which were promised in Section 3.1,

we recall the definition of the frame-Hopf map FrameHopf : C2 Ñ R3ˆ3 (e.g., [27]), defined

55

by the formula

FrameHopfpz, wq :“

¨

˚

˚

˚

˚

˝

|z|2 ´ |w|2 2Impzwq ´2Repzwq

2Impzwq Repz2 ` w2q Impz2 ` w2q

2Repzwq Imp´z2 ` w2q Repz2 ´ w2q

˛

‹

‹

‹

‹

‚

.

The first part of this section outlines some of the important properties of the frame-Hopf

map.

The frame-Hopf map has a natural representation when written in quaternionic coordi-

nates. Under the identification C2 « H described in Section 2.4, the frame-Hopf map takes

the form

FrameHopfpqq “ pqiq, qjq, qkqq P timaginary quaternionsu3 « pR3q3.

Another important fact about the frame-Hopf map is that the first column of the frame-

Hopf map gives an explicit form of the Hopf fibration S1 ãÑ S3 Ñ S2 when restricted to

S3 Ă C2 (actually, each column gives a form of the Hopf fibration, but we only treat the first

column for the sake of simplicity). This fact and can be deduced from the following pair of

lemmas. Each lemma can be verified by an elementary computation, which we omit.

Lemma 3.1.6. The first column of the frame-Hopf map, FrameHopf1, is invariant under

the diagonal Up1q-action on C2 by multiplication; that is, for all eiθ P Up1q,

FrameHopf1peiθz, eiθwq “ FrameHopf1pz, wq.

56

Lemma 3.1.7. The columns of the frame-Hopf map square norms. That is, for any pz, wq P

C2,›

›FrameHopfjpz, wq›

› “ }pz, wq}2C2 ,

where FrameHopfj, j “ 1, 2, 3,, denotes the j-th column of the frame Hopf map.

Thus the image of the restriction of FrameHopf1 to S3 is S2 and the map has circle fibers,

thereby establishing our earlier claim.

Recall from Section 2.4 that S3 inherits a Lie group structure via identification with the

Lie group of unit quaternions. Also recall that S3 is naturally identified with the Lie group

SUp2q. The following lemma is well-known (e.g., [21] Section I.1.4) and very useful. It may

be verified by an elementary computation.

Lemma 3.1.8. The restriction of the frame-Hopf to S3 Ă C2 gives a Lie group homomor-

phism onto SOp3q Ă R3ˆ3 which satisfies

FrameHopfpz, wq “ FrameHopfpz1, w1q ô pz1, w1q “ ˘pz, wq.

Thus the frame-Hopf map induces a homomorphic double covering SUp2q Ñ SOp3q.

Moreover, the frame-Hopf map has the following local (almost) isometry property.

Lemma 3.1.9. The restriction of the frame Hopf map

FrameHopf : S3Ñ SOp3q

satisfies

FrameHopf˚gSOp3q“ 4gS

3

.

57

Thus the induced frame-Hopf homomorphism SUp2q Ñ SOp3q is also a local isometry up to

a factor of 4.

Proof. We work in quaternionic coordinates. For the remainder of the proof we use the

notation

Impqq :“ pq1, q2, q3q P R3

for q “ Q0 `Q1i`Q2j`Q3k P H.

Since FrameHopf is known to be a homomorphism and the metrics involved are left-

invariant, it suffices to prove the claim at 1 P S3 Ă H. The derivative of FrameHopf at 1 in

the direction q is given by

DFrameHopfp1qpqq “ 2pImpiqq, Impjqq, Impkqqq.

Note that in quaternionic coordinates, the tangent space of S3 at 1 is the space

T1S3“ tq P H | Re 〈1, q〉H “ 0u “ tq “ Q0 `Q1i`Q2j`Q3k P H | Repqq “ Q0 “ 0u.

58

For p, q P T1S3 the pullback metric is given by

FrameHopf˚gSOp3q1 pq, pq “ g

SOp3qId pDFrameHopfp1qpqq,DFrameHopfp1qppqq

“ gSOp3qId p2pImpiqq, Impjqq, Impkqqq, 2pImpipq, Impjpq, Impkpqqq

“ 4 ¨1

2trace

¨

˚

˚

˚

˚

˝

¨

˚

˚

˚

˚

˝

0 Q3 ´Q2

´Q3 0 Q1

Q2 ´Q1 0

˛

‹

‹

‹

‹

‚

¨

˚

˚

˚

˚

˝

0 P3 ´P2

´P3 0 P1

P2 ´P1 0

˛

‹

‹

‹

‹

‚

T˛

‹

‹

‹

‹

‚

“ 4 ¨1

2¨ 2pQ1P1 `Q2P2 `Q3P3q

“ 4Repqpq

“ 4gS3

1 pp, qq.

Thus the first claim holds. The second follows immediately from Lemma 2.4.1, which says

that SUp2q and S3 are isometric with respect to their standard metrics.

3.1.3 Complex Coordinates for Framed Paths

In this section we finally describe our complex coordinate system for the moduli space of

framed paths. It follows from Lemma 3.1.8 that the frame-Hopf map induces a smooth

double-cover

H : C2zt0u Ñ SOp3q ˆ R`

defined by

Hpz, wq :“

ˆ

1

|z|2 ` |w|2FrameHopfpz, wq, |z|2 ` |w|2

˙

,

where Hpz, wq “ Hpz1, w1q if and only if pz, wq “ ˘pz1, w1q. Indeed,

H “ H2 ˝ H1,

59

where H1 is the diffeomorphism

H1 : C2zt0u Ñ S3

ˆ R`

pz, wq ÞÑ

˜

1a

|z|2 ` |w|2pz, wq,

a

|z|2 ` |w|2

¸

,

and H2 is the smooth double cover

H2 : S3ˆ R` Ñ SOp3q ˆ R`

ppz, wq, rq ÞÑ pFrameHopfpz, wq, r2q.

It follows from Proposition 2.2.7 that applying the path functor to H produces a smooth

double-cover

H : PpC2zt0uq Ñ PpSOp3q ˆ R`q,

which we still denote H by an abuse of notation. From the above, we see that Φ1 “

pφ1, ψ1q, Φ2 “ pφ2, ψ2q P PpC2zt0uq correspond to the same framed path if and only if

Φ1 ” ˘Φ2. By composing H with the identification (3.3) of PpSOp3q ˆ R`q with rS, we

obtain a smooth double cover of framed path space, denoted

pH : PpC2zt0uq Ñ rS “ tframed pathsu{ttranslationu.

Thus a pair pφ, ψq of complex valued functions determines a unique framed path, up to trans-

lation. In the following section, we show that we restrict pH to the L2-sphere in PpC2zt0uq,

then the image is the moduli space of framed paths S. Moreover, we show that this double

covering is an isometry.

60

3.1.4 The L2-Sphere

We endow PC2 with the L2 Hermitian inner product

〈Φ1,Φ2〉L2 :“

ż 2

0

〈Φ1ptq,Φ2ptq〉C2 dt,

where 〈¨, ¨〉C2 denotes the standard Hermitian inner product on C2. The real part of the L2

Hermitian inner product determines a real-valued inner product on PC2. Since this real part

will frequently be of central interest, we introduce the notation

gL2

:“ Re 〈¨, ¨〉L2 ,

which we call the L2 Riemannian metric. We refer to it as a metric since it will soon be

restricted to various submanifolds of PC2, where it will induce a weak Riemannian metric.

We will also use gL2

to denote the L2 Riemannian metric on PH

gL2

‚ pq, pq :“ Re

ż 2

0

qp dt.

These metrics agree under the identification C2 « H described in Section 2.4, so no confusion

should arise from the conflation of notations.

Our first goal is to show that the map pH defined in the previous section is an isometry.

Proposition 3.1.10. The map

pH : PpC2zt0uq Ñ rS

is a local isometry with respect to gL2

and gS “ g1,1,1,1.

61

Proof. It suffices to show that the map

H : PpC2zt0uq Ñ PpSOp3q ˆ R`q

is a local isometry, as the result then follows immediately from Proposition 3.1.5.

Throughout the proof, it will be convenient for calculations to use quaternionic notation.

We write H “ PH2 ˝ PH1, where

PH1 : PpHzt0uq Ñ PpS3ˆ R`q

q ÞÑ

ˆ

1

}q}Hq, }q}H

˙

,

and

PH2 : PpS3ˆ R`q Ñ PpSOp3q ˆ R`q

pq, rq ÞÑ pPFrameHopfpqq, r2q.

are path space versions of the maps defined in the previous section, written in quaternionic

coordinates.

Our first goal is to compute the pullback PH˚2gS . Lemma 2.2.6 implies

DPFrameHopfpqqpδqqptq “ DFrameHopfpqptqqpδqptqq

for any q P PS3 Ă PH and any δq P TqPS3. Similarly, if s : R` Ñ R` denotes the squaring

map r ÞÑ r2, then

DPsprqpδrqptq “ 2rptqδrptq.

62

Thus the variation of PH2 at pq, rq in the direction pδq, δrq is given by the formula

DPH2pq, rqpδq, δrqptq “ pDFrameHopfpqptqqpδqptqq, 2rptqδrptqq .

We will for the moment shorten our notation FrameHopf to F in order to avoid an unwieldy

calculation of the pullback metric:

`

PH˚2gS˘

pq,rqppδq1, δr1q, pδq2, δr2qq

“1

4

ż 2

0

gSOp3qFpqptqq pDFpqptqqpδq1ptqq, DFpqptqqpδq2ptqqq `

4rptq2δr1ptqδr2ptq

prptq2q2ds

“1

4

ż 2

0

ˆ

4gS3

qptq pδq1ptq, δq2ptqq `4δr1ptqδr2ptq

rptq2

˙

rptq2dt (3.10)

“

ż 2

0

Repδq1δq2q ¨ r2` δr1δr2 dt,

where (3.10) follows by Lemma 3.1.9 and the definition of ds.

Next we use Lemma 2.2.6 once again to compute the variation of PH1 at q in the direction

δq to be

DH1pqqpδqq “

ˆ

}q}δq ´ qRepqδqq{}q}

}q}2,Repqδqq

}q}

˙

.

63

In the above and for the remainder of this proof, we will denote the quaternionic norm by

} ¨ } in order to simplify notation. We conclude the proof by calculating the pullback metric:

`

H˚gS˘

qpδq1, δq2q

“`

H˚1`

H˚2gS˘˘

qpδq1, δq2q

“`

H˚2gS˘

pq{}q},}q}qpDH1pqqpδq1q, DH1pqqpδq2qq

“

ż 2

0

Re

ˆ

}q}δq1 ´ qRepqδq1q{}q}

}q}2¨}q}δq2 ´ qRepqδq2q{}q}

}q}2

˙

}q}2

`Repqδq1q

}q}¨

Repqδq2q

}q}dt

“

ż 2

0

Repδq1δq2q `Repqδq1qRepqδq2q

}q}2

´ Re

ˆ

δq1qRepqδq2q ` δq2qRepqδq2q

}q}2

˙

`Repqδq1qRepqδq2q

}q}2dt

“

ż 2

0

Re`

δq1δq2

˘

dt

“ gL2

q pδq1, δq2q.

Going through the calculations in the proof of Proposition 3.1.10, we obtain the following

useful corollary. This will later be used in Section 4.2 to describe the framed curve geometric

invariants of Section 2.1.2 in our complex coordinate system.

64

Corollary 3.1.11. Let Φ P PpC2zt0uq correspond to q P PpHzt0uq under the identification

C2zH and to the framed path pγ, V q under pH. Let the variations δΦj, j “ 1, 2, of Φ correspond

to the variations δqj of q and pδγj, δVjq of pγ, V q. Then we have the following equality as

functions:

1

4

ˆ⟨d

dsδγ1,

d

dsδγ2

⟩` 〈δV1,W 〉〈δV2,W 〉

˙

}γ1} “ Re 〈δΦ1, δΦ2〉C2 “ Re 〈δq1, δq2〉C2 .

Now we arrive at our first main result, which says that the moduli space of framed paths

S is isometrically double covered by an open subset of an L2 sphere. We use the notation

SpPC2q :“

!

Φ P PpC2zt0uq | }Φ}L2 “

?2)

,

for the radius-?

2 L2 sphere of PC2. We are particularly interested in a dense open subset,

denoted

S˝pPC2q :“ tΦ P SpPC2

q | Φptq ‰ 0 @ tu.

Theorem 3.1.12. The map pH restricts to an isometric double cover

pH|S˝pPC2q : S˝pPC2q Ñ S

with respect to the metrics gL2

and gS .

Proof. Recall that the moduli space S is concretely realized as

S :“ tpγ, V q P rS | lengthpγq “ 2u.

65

By Proposition 3.1.10, we only need to show that

pH´1pSq “ S˝pPC2

q Ă PC2.

Indeed, let pγ, V q P S and Φ P PpC2zt0uq such that pHpΦq “ pγ, V q. Then

2 “

ż 2

0

}γ1} dt “

ż 2

0

}FrameHopf1pΦq} dt “

ż 2

0

}Φ}2C2 dt (3.11)

“ }Φ}2L2 ,

where the last equality in (3.11) follows from Lemma 3.1.7.

This theorem can be phrased differently as follows.

Corollary 3.1.13. The moduli space of framed paths S is isometric to

S˝pPC2q{pΦ „ ´Φq,

an open subset of the projective space of real lines in PC2 « PR4 with its induced Riemannian

metric.

3.2 Framed Loop Space

As stated in Section 3.1, the benefit of the complex coordinates pφ, ψq for framed paths is

that the necessary and sufficient conditions for the periodicity of the resulting framed path

are surprisingly nice. Before describing the conditions, we require a new definition: a path

66

-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

Figure 3.1: An element of AC.

Φ P PC2 is called smoothly antiperiodic if

dk

dtk

ˇ

ˇ

ˇ

ˇ

t“2

Φ “ ´dk

dtk

ˇ

ˇ

ˇ

ˇ

t“0

Φ for all k “ 0, 1, 2, . . .

(see Figure 3.1). The space of antiperiodic paths in C2, or the antiloop space of C2, is denoted

AC2.The antiloop space is a complex tame Frechet vector space. We define ApC2zt0uq and

AC similarly.

Lemma 3.2.1. A path Φ “ pφ, ψq P PpC2zt0uq corresponds to a framed loop under pH if and

only if

(1) The path Φ is either smoothly closed or smoothly antiperiodic, and

(2) the maps φ and ψ have the same L2 norm and are L2-orthogonal:

ż 2

0

|φ|2 dt “

ż 2

0

|ψ|2 dt and

ż 2

0

φψ dt “ 0.

Proof. Let Φ “ pφ, ψq be either a loop or an antiloop. It follows easily from the definition

of H that this is a necessary and sufficient condition for HpΦq to be smoothly periodic; i.e.,

67

that

ppT, V,W q, rq :“ HpΦq P LpSOp3q ˆ R`q.

Then pHpΦq is a smoothly periodic framed loop if and only if

ż 2

0

rT dt “ 0. (3.12)

The key point here is that T and r are smooth loops, so the only obstruction to the curve as-

sociated to pT, rq being smoothly closed is that its endpoints match up—hence the vanishing

integral condition. Equation (3.12) is written in pφ, ψq-coordinates as

ż 2

0

|φ|2 ´ |ψ|2 dt “

ż 2

0

2Impφψq dt “

ż 2

0

2Repφψq dt “ 0,

which are exacly the conditions given in (2).

Since LpC2zt0uq and ApC2zt0uq are disjoint, the lemma implies that the space

tframed loopsu{ttranslationu

is disconnected. Indeed, the space of framed loops has two components, which are indexed

by mod-2 self-linking number. This fact is well-known, but a proof is seemingly absent from

the literature. The details of the proof are given in the next section.

3.2.1 Connected Components of Framed Loop Space

We use the classical concepts of writhe, twist, and linking number. Basic definitions are

given here for the convenience of the reader and to set notation (see, e.g., [14] for more

details).

68

+

-‐

-‐

+

-‐

-‐+-‐

+ -‐

Figure 3.2: A knot diagram with Wr “ 3´ 5 “ ´2

Recall that the writhe of an oriented knot diagram is a signed count of its crossings—see

Figure 3.2. We define the writhe of a (parameterized) knot γ, denoted Wrpγq, to be the

average writhe of all knot diagrams for γ which are obtained by projecting to a plane.

The total twist Twpγ, V q of pγ, V q is defined by the formula

Twpγ, V q :“1

2π

ż 2

0

twpγ, V q ds

“1

2π

ż 2

0

⟨d

dsV, T ˆ V

⟩ds,

where

d

ds“

1

}γ1}, T “

γ1

}γ1}, ds :“ }γ1}dt

and twpγ, V q is twist rate, as defined in Section 2.1.2.

Finally, we define the self-linking number of a framed knot pγ, V q, denoted Lkpγ, V q, to

be the topological linking number of the link formed by the disjoint knots γ and γ ` εV ,

where ε ą 0 is sufficiently small.

69

Note that Wr and Lk are only defined in the case that γ is embedded, while Tw makes

sense for general pγ, V q. In fact, one can check using the explicit formula for Tw that it is a

continuous functional on framed loop space—this observation will be useful in a moment.

We will employ the Calugareanu-White-Fuller Theorem [20], which says that for a framed

loop pγ, V q with embedded base curve,

Lkpγ, V q “ Twpγ, V q `Wrpγq.

We use the notation

Lk2pγ, V q :“ Lkpγ, V q mod 2

for the mod-2 linking number of pγ, V q. We will show that this quantity is well-defined even

for framed loops with nonembedded base curves.

The following lemma is well well known, but we include the proof for completeness.

Lemma 3.2.2. Let pγ0, V0q and pγ1, V1q be a pair of homotopic framed loops such that γ0

and γ1 are embedded loops. Then Lk2pγ0, V0q “ Lk2pγ1, V1q.

Proof. By standard genericity arguments and induction, it suffices to prove the lemma in

the case that there is a homotopy pγu, Vuq of framed loops such that γu is embedded for all

u ‰ 1{2, and that γ1{2 has a single transverse self-intersection.

By the isotopy invariance of Lk, the continuity of Tw, and the Calugareanu-White-Fuller

Theorem, a discontinuity in Lk can only occur due to a jump in Wr at u “ 1{2. By taking

ε sufficently small, we ensure that for almost every projection direction, the projection of

γ1{2´ε differs from the projection of γ1{2`ε by switching exactly one crossing. This has the

effect of changing Wr by ˘2, and we conclude that Lk changes by ˘2 at exactly u “ 1{2,

i.e.

Lk2pγ0, V0q “ Lk2pγ1, V1q.

70

Remark 3.2.3. It follows immediately from the lemma that Lk2pγ, V q determines the ho-

motopy class of pT, V,W q in SOp3q, where T “ γ1{}γ1} and W “ T ˆ V . Computing any

example shows that

Lk2pγ, V q “ 0 ô pT, V,W q generates π1pSOp3qq.

Thus the space of framed loops has at least two connected components. We now prove the

claim stated above: the space of framed loops has exactly two components. It is convenient

to identify

TS2zS2

0 « SOp3q ˆ R`

pW, ξq Ø

ˆˆ

ξ

}ξ},W,

ξ

}ξ}ˆW

˙

, }ξ}

˙

,

where S20 is the image of the zero section in TS2. Then the map (3.1), restricted to loops,

becomes

tframed loopsu Ñ LpTS2zS2

0q

pγ, V q ÞÑ pV, γ1q.

Note that the map is not onto—a general loop pW, ξq P LpTS2zS20q does not satisfy

ş2

0ξ dt “ 0,

hence does not correspond to a framed loop. The fact that a general homotopy in TS2zS20

can always be adjusted to lie in the image of this map is the content of the proposition:

Proposition 3.2.4. Let pγj, Vjq for j “ 0, 1 be framed loops such that there exists a ho-

motopy pWu, ξuq between their images pVj, γ1jq in LpTS2zS2

0q. Then pγ0, V0q and pγ1, V1q are

homotopic.

71

This proposition has the flavor of an h-principle [17, 23] (in particular, it is reminiscent of

the Smale-Hirsch theorem), but we are unaware of any general result that implies it. In any

case, we give a simple proof based on the classical proof of the Whitney-Graustein theorem

[75].

Proof. Without loss of generality, we assume that γjp0q “ 0 and }γ1j} ” 1 for j “ 0, 1.

Let pWu, ξuq be a homotopy of loops in TS2zS20 such that pWj, ξjq “ pVj, γ

1jq for j “ 0, 1.

Since TS2zS20 is homotopy equivalent to the unit tangent bundle of S2, we may assume that

}ξu} ” 1 for all u.

We construct a homotopy of framed loops pγu, Vuq by first defining

γuptq :“

ż t

0

ξuptq dt´t

2

ż 2

0

ξuptq dt.

Then γup0q “ γup2q “ 0 for all u and

γ1uptq “ ξuptq ´1

2

ż 2

0

ξuptq dt.

Assuming for now that ξu is nonconstant for all u, we conclude that γ1uptq ‰ 0 for all u, t.

This follows since the average value of a nonconstant spherical loop lies in the interior of the

sphere.

Now we note that Wuptq and γ1uptq are linearly indpendent for all u, t. Indeed,

Wu ˆ γ1u “ Wu ˆ ξu ´W ˆ

1

2

ż 2

0

ξuptq dt,

which is nonzero since the first term is a unit vector and the second has strictly less than

unit length (once again assuming that ξu is nonconstant for all u). Thus we can define Vu

72

by

Vu :“Wu ´ 〈Wu, γ

1u〉

γ1u}γ1u}

›

›

›Wu ´ 〈Wu, γ1u〉

γ1u}γ1u}

›

›

›

.

Then Vu is a normal vector field along the loop γu and we have produced a homotopy pγu, Vuq

between our original framed loops.

Finally, we note that ξu may be assumed to be nonconstant for all u without loss of

generality, perhaps after a small perturbation of our initial homotopy relative to its endpoints.

Corollary 3.2.5. Framed loop space has two connected components. The component in

which a framed curve pγ, V q with embedded base curve γ lies is determined by its mod-2

linking number. It follows that the definition of Lk2 extends to nonembedded framed loops.

Proof. The first statement follows from the proposition, as

LpTS2zS2

0q « LpSOp3q ˆ R`q

has two components—one for each element of

π1pSOp3q ˆ R`q « π1pSOp3qq « Z2.

The second statement follows immediately from Lemma 3.2.2. The third statement follows

by defining the mod-2 linking number of a nonembedded framed loop pγ, V q to be the mod-2

linking number of any embedded framed loop which is homotopic to pγ, V q.

Example 3.2.1. This definition of Lk2 is sometimes counterintuitive. For example, let

γ0ptq “1

2πp0,´ sinp2πtq, cosp2πtq ´ 1q, V0ptq “ p1, 0, 0q.

73

The image of γ0 is a doubly-covered circle, and any pushoff in the V0-direction is disjoint

from γ0 — see Fig. 3.3(a). Thus, for this particular nonembedded framed curve, the usual

self-linking number is well-defined; it is Lkpγ0, V0q “ 0.

On the other hand, pγ0, V0q is homotopic to pγ1, V1q, where

γ1ptq “1

πp0, cospπtq ´ 1, sinpπtqq, V1ptq “ p´ sinpπtq, cos2

pπtq, cospπtq sinpπtqq,

which has as its image an embedded framed curve with Lkpγ1, V1q “ 1 — see Fig. 3.3(c).

An explicit homotopy is given by

¨

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˝

¨

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˝

pu´ 1qu cospπtq

πp1´ u` u2q

cospπtqpu` pu´ 1q sinpπtqq

π?

1´ u` u2

sinpπtqpu` pu´ 1q sinpπtqq

π?

1´ u` u2

˛

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‚

,

¨

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˝

´´1` u` u sinpπtq?

1´ 2u` 2u2

upu` u cosp2πtq ` 2pu´ 1q sinpπtqq

2´ 4u` 4u2

u cospπtqp1´ u` u sinpπtqq

1´ 2u` 2u2

˛

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‚

˛

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‚

.

We conclude that Lk2pγ0, V0q “ 1 according to our definition, even though Lkpγ0, V0q “ 0.

3.2.2 The Stiefel Manifolds

For V “ LC or AC, we denote the Stiefel manifold of L2-orthonormal 2-frames in V by

St2pVq :“

"

pφ, ψq P V2|

ż 2

0

|φ|2dt “

ż 2

0

|ψ|2dt “ 1 and

ż 2

0

φψ dt “ 0

*

.

74

(a) (b) (c)

Figure 3.3: Links determined by (a) pγ0, V0q, (b) A perturbation of pγ0, V0q along the homotopy,and (c) pγ1, V1q. In each image, the blue loop is the image of γ and the red loop is the image ofγ ` εV .

This space is indeed a manifold—it is the level set of a regular value of the smooth map

LC2Ñ R4

pφ, ψq ÞÑ

ˆż 2

0

|φ|2 dt,

ż 2

0

|ψ|2 dt,Re

ż 2

0

φψ dt, Im

ż 2

0

φψ dt

˙

, (3.13)

hence it is a tame Frechet manifold by Corollary 2.2.10.

Recall that we denote the L2 Hermitian inner product on PC2 by 〈¨, ¨〉L2 . Likewise, we

denote the L2 Hermitian inner product on PC by

〈φ, ψ〉L2 :“

ż 2

0

φψ dt,

and the induced norm denoted } ¨ }L2 . Moreover, the restrictions of these inner products and

norms to LC, AC, LC2 and AC2 will still be denoted 〈¨, ¨〉L2 and } ¨ }L2 . The meaning of the

notation should always be clear from context.

75

We are interested in a particular open subset of the Stiefel manifold, called the stable

Stiefel manifold of V and denoted

St˝2pVq :“ tpφ, ψq P St2pVq | pφptq, ψptqq ‰ p0, 0q @ tu.

The use of the word stable refers to the fact that this is the stable submanifold (in the sense

of GIT quotients [73]) with respect to the action of the Lie group LS1 given by poinwise

multiplication. This fact is further explored in Section 4.3 below.

A simple but important fact is that the Stiefel manifolds are diffeomorphic via the map

St2pLCq Ñ St2pACq

pφptq, ψptqq ÞÑ exppíπt{2qpφptq, ψptqq. (3.14)

This diffeomorphism restricts to a diffeomorphism of the stable Stiefel manifolds.

It is clear that the Stiefel manifold St2pVq is connected. Indeed, let Φ0 “ pφ0, ψ0q and

Φ1 “ pφ1, ψ1q be elements of St2pVq and assume for now that

spantφ0, ψ0, φ1, ψ1u “:W

is 4-dimensional. Any isometry of C4 withW (with respect to the standard Hermitian inner

product on C4 and the restriction of the L2 Hermitian inner product to W) induces an

embedding of the standard complex Stiefel manifold

St2pC4q “ tpz, wq P pC4

q2| z and w are Hermitian orthonormalu

into St2pVq. Moreover, we can consider Φ0 and Φ1 as elements of the embedded St2pC4q.

Thus we can find a homotopy Φu between Φ0 and Φ1 which lies in the connected submanifold

76

St2pC4q, as this classical manifold is well-known to be connected. The same argument works

if dimpWq “ 2 or 3. A less obvious fact is that the stable Stiefel manifolds are connected.

Lemma 3.2.6. The stable Stiefel manifold St˝2pVq is connected.

Proof. In light of the diffeomorphism (3.14), it suffices to prove the claim for V “ LC .Let

Φ0,Φ1 P St˝2pLCq. In particular, Φ0,Φ1 P St2pLCq Ă LC2. By the above discussion, there

exists a homotopy rΦu in St2pLCq which joins Φ0 and Φ1, but such a homotopy may not stay

in StS2 pLCq. To remedy this, we consider rΦu as a homotopy of loops in C2. We perturb rΦu,

relative to Φ0 and Φ1, to obtain a new homotopy pΦu “ ppφu, pψuq which satisfies

(i) pΦuptq ‰ 0 for all t, u and

(ii) pφu and pψu are linearly independent as elements of LC for all u,

where we appeal to the fact that a small perturbation will preserve the stable condition that

the coordinates of pΦu are linearly independent.

Now we can orthonormalize the coordinate functions of pΦu with respect to 〈¨, ¨〉L2 :

φu :“pφu

}pφu}L2

, ψu :“

pψu ´⟨pψu, φu

⟩L2φu

›

›

›

pψu ´⟨pψu, φu

⟩L2φu

›

›

›

L2

.

The orthonormalization process does not introduce new simultaneous zeroes for φu and ψu,

thus the new homotopy Φu :“ pφu, ψuq lies in StS2 pLCq for all u and it still joins Φ0 and

Φ1.

Next we note that St˝2pVq is a subset of

S˝pPC2q “ tΦ P PC2

| }Φ}L2 “?

2 and Φptq ‰ 0 @ tu.

By Theorem 3.1.12, S˝pPC2q covers the space S of framed paths of length 2 via the restriction

of pH. From Lemma 3.2.1 we conclude that the orthonormal pairs pφ, ψq P LpC2zt0uq Y

77

ApC2zt0uq (i.e., pairs which lie in a Stiefel manifold) correspond exactly (under a further

restriction of pH to framed loops of length 2. By considering only framed loops of length 2,

we have chosen a horizontal slice with respect to the R` action on framed loop space by

scaling of the base curve.

Finally, we note that St˝2pVq Ă V2 Ă PC2, so that the Stiefel manifold has a natural

metric obtained by restricting the real part of the L2 Hermitian inner product—we will

continue to denote this restricted Riemannian metric by gL2. The restricted map

pH : S˝pPC2q Ñ S

was shown to be a local isometry with respect to gL2

and gS , thus its restriction to the Stiefel

manifolds is still a local isometry. Putting all of this together, we have:

Proposition 3.2.7. The restricted map

pH : St˝2pLCq \ St˝2pACq Ñ tframed loopsu{ttranslation, scalingu

is a smooth double covering and a local isometry with respect to the restricted metrics gL2

and gS .

The Lk2 “ 1 component of framed loop space is covered by St˝2pLCq and the Lk2 “ 0

component is covered by St˝2pACq.

Proof. Combining the preceding discussion with Corollary 3.2.5 and Lemma 3.2.6, the only

thing we need to check is that an element of St˝2pLCq maps under pH to a framed loop with

Lk2 “ 1. Indeed, from

Φptq “1?

2pcospπtq ´ i sinpπtq, 1q P St˝2pLCq

78

we obtain pHpΦq “ pγ1, V1q, where

γ1ptq “1

πp0, cospπtq ´ 1, sinpπtqq, V1ptq “ p´ sinpπtq, cos2

pπtq, cospπtq sinpπtqq.

From Example 3.2.1, we see that Lk2pγ1, V1q “ 1.

3.2.3 The Grassmann Manifolds

From the Stiefel manifold St2pVq, V “ LC or AC, we obtain a Grassmannian in the usual

way. We define

Gr2pVq :“ St2pVq{Up2q,

where A P Up2q acts on St2pVq by pointwise multiplication from the right:

ppφ, ψq ¨ Aq ptq :“ pφptq, ψptqq ¨ A for A P Up2q.

The Up2q-orbit of a point Φ “ pφ, ψq P St2pVq is the complex span of tφ, ψu and will be

denoted rΦs “ rφ, ψs. Thus the Grassmannian Gr2pVq is the space of 2-dimensional complex

subspaces of V . It is a compex manifold and we sketch the construction of holomorphic

charts for Gr2pVq in Proposition 4.1.3.

We define the stable Grassmannian of V to be the open submanifold

Gr˝2pVq :“ St˝2pVq{Up2q Ă Gr2pVq.

Recall from Lemma 3.1.8 that FrameHopf restricts to a homomorphism

FrameHopf : SUp2q Ñ SOp3q.

79

From this fact and the definition of H, we deduce that

Hppφ, ψq ¨ Aq “ Hpφ, ψq ¨ FrameHopfpAq for A P SUp2q, pφ, ψq P St˝2pVq. (3.15)

Therefore modding out by the action of the SUp2q term of the product

SUp2q ˆ Up1q “ Up2q

has the effect of modding out by the action of SOp3q by rotations on framed loop space. The

Up1q term of Up2q acts on framed loop space by global frame twisting—i.e. twisting each

frame vector V ptq around γ by the same angle. Explicitly, let eiθ P Up1q, let Hpφ, ψq “ pγ, V q

and let Hppφ, ψq ¨ eiθq “ pγ, V q. Then γ “ γ and

V “ cosp2θqV ` sinp2θqpT ˆ V q.

Recall from Section 2.1.2 that an equivalence class of a framed loop with respect to this

Up1q-action is called a relatively framed loop.

Thus the map pH induces a map on Gr˝2pLCq\Gr˝2pACq with image the moduli spaceM,

where we recall that

M “ trelatively framed loopsu{ttranslation, scaling, rotationu.

The induced map will still be denoted pH, and by further abuse of notation we will denote

pHprΦsq by pHpΦq.

The Grassmannians inherit natural Riemannian metrics from the Stiefel manifolds. This

follows immediately from checking that the restricted metric gL2

on St2pVq is invariant with

80

respect to the Up2q-action. Moreover, the restricted metric gS on

tframed loopsu{ttranslation, scalingu Ă S

is invariant with respect to rotations. Thus gS induces a well-defined Riemannian metric on

M. By abuse of notation, we will continue to denote these induced metrics by gL2

and gS ,

respectively. By the equivariance property (3.15) of H and by Proposition 3.2.7, we conclude

that the induced map pH on the Grassmannians is a local isometry with respect to gL2

and

gS .

Finally, we note that since Lk2 is invariant with respect to translation, scaling, and

rotation,M also has two connected components, indexed by Lk2. Thus we have proved our

second main result for this chapter.

Theorem 3.2.8. The induced map

pH : Gr˝2pLCq \Gr˝2pACq ÑM

is an isometry with respect to the induced metrics gL2

and gS . The Lk2 “ 1 component M

is isometric to Gr˝2pLCq and the Lk2 “ 0 component is isometric to Gr˝2pACq.

Proof. This follows from the discussion above. We should point out that this map is an

honest isometry, as opposed to the local isometry of Proposition 3.2.7. This follows since

Φ,´Φ P St2pVq differ by right multiplication by the negative 2ˆ2 identity matrix, an element

of Up2q. Thus rΦs “ r´Φs as elements of Gr2pVq.

81

Chapter 4

The Geometry of Spaces of Framed

Paths and Loops

The Grassmannian Gr2pVq (for V “ LC or AC) has a variety of natural geometric structures

which are inherited by the moduli space of framed loops M. In particular, Gr2pVq is a

complex Kahler manifold. It is complex in the strong sense that it admits holomorphic

charts (recall that there are weaker notions of complex manifolds in the Frechet category

[43]). The goal of this chapter is to describe the geometric structures on Gr2pVq and the

corresponding structures on M.

4.1 The Symplectic Structure of Framed Loop Space

4.1.1 The Kahler Structure of the Grassmannian

In this section, we describe the natural symplectic structure of the Grassmannians and the

induced symplectic structure of the moduli space of framed loopsM. We begin by describing

the tangent spaces of the Stiefel manifolds.

82

Lemma 4.1.1. The tangent space to St2pVq at Φ “ pφ, ψq is given by the following codimension-

4 subspace of V2:

TΦSt2pVq “"

δΦ “ pδφ, δψq P V2| Re

ż 2

0

φδφ dt “ Re

ż 2

0

ψδψ dt “

ż 2

0

φδψ ` δφψ dt “ 0

*

.

Proof. To derive this description of the tangent space, one simply notes that this vector space

is the kernel of the derivative at Φ of the map (3.13) used to define the manifold structure

of St2pVq, and the description follows by Corollary 2.2.10.

Recall that we denote the Up2q-orbit of Φ “ pφ, ψq P St2pVq by rΦs “ rφ, ψs P Gr2pVq

and that

rΦs “ rφ, ψs “ spanCtφ, ψu Ă V .

From Lemma 4.1.1, we will obtain a description of the tangent spaces to the Grassmannians.

We define the vertical tangent space to St2pVq at Φ to be the space of directions along

Up2q-orbit of Φ; that is,

T vertΦ St2pVq :“ tΦ ¨ ξ | ξ P up2qu.

The horizontal tangent space to St2pVq at Φ is then the orthogonal complement with respect

to the L2-metric

T horΦ St2pVq :“ tδΦ P TΦSt2pVq | gL2

Φ pδΦ, δΨq “ 0 @ δΨ P T vertΦ St2pVqu.

By the construction of Gr2pVq as the image of the submersion

St2pVq Ñ St2pVq{Up2q “ Gr2pVq,

83

the tangent space to Gr2pVq at rΦs can be identified with the horizontal tangent space to

St2pVq at Φ. Moreover, since the vertical space is finite-dimensional, this horizontal space

can be explicitly described.

Corollary 4.1.2. The tangent space to Gr2pVq at rΦs may be identified with the codimension-

4 Up2q-horizontal subspace of TΦSt2pVq given by

TrΦsGr2pVq « T horΦ St2pVq

“

"

δΦ P V2|

ż 2

0

φδφ dt “

ż 2

0

ψδψ dt “

ż 2

0

φδψ dt “

ż 2

0

ψδφ dt “ 0

*

. (4.1)

Note that if Φ1 and Φ2 lie in the same Up2q-orbit, then

T horΦ1St2pVq “ T horΦ2

St2pVq,

so that this description of the tangent spaces of Gr2pVq does not depend on the choice of

representative of rΦs.

Proof. Elements of up2q can be written in the form

A “

¨

˚

˝

ai z

´z bi

˛

‹

‚

,

where a, b P R and z P C. Thus the defining condition of the horizontal subspace can be

expressed as

gL2

pδΦ,Φ ¨ ξq “ gL2

ppδφ, δψq, pφai´ ψz, φz ` ψbiqq “ 0 @ a, b P R and z P C,

84

where δΦ “ pδφ, δψq and Φ “ pφ, ψq. Simplifying, we have

gL2

ppδφ, δψq, pφai´ ψz, φz ` ψbiqq “ Re 〈pδφ, δψq, pφai´ ψz, φz ` ψbiq〉L2

“ Re

ż

δφpφai´ ψzq ` δψpφz ` ψbiq dt,

and we conclude

a ¨ Im

ż

φδφ dt` b ¨ Im

ż

ψδψ dt` Re

ˆ

z

ż

φδψ ` δφψ dt

˙

“ 0

must hold for all a, b P R and z P C. By taking various choices of a, b and z and combining

with conditions which are already assumed on δΦ P TΦSt2pVq, we are able to derive the

conditions in (4.1). For example, by taking a “ 1 and b “ z “ 0 we obtain

Im

ż

φδφ dt “ 0.

Since

Re

ż

φδφ dt “ 0

was already a condition required to be tangent to St2pVq, we conclude

ż

φδφ dt “ 0.

85

The remaining conditions can be derived by other choices of parameters. We leave it to the

reader to check that the following choices work:

b “ 1, a “ z “ 0 ñ

ż

ψδψ dt “ 0

z “ 1, a “ b “ 0 ñ

ż

φδψ dt “ 0

z “ i, a “ b “ 0 ñ

ż

ψδφ dt “ 0.

We note that the tangent spaces to the Grassmannian are complex subspaces of V2, so

each tangent space inherits the complex structure

TrΦsGr2pVq Ñ TrΦsGr2pVq

δΦ ÞÑ iδΦ. (4.2)

Thus Gr2pVq has an almost complex structure, and we show it is in fact an honest complex

structure on Gr2pVq, as described in Section 2.3.1. To do so, we explicitly construct holomor-

phic charts on Gr2pVq. This construction uses the usual charts for a real finite-dimensional

Grassmannian, adapted to the complex infinite-dimensional setting (e.g., [56], Section 5).

We outline the proof here for the convenience of the reader.

Proposition 4.1.3. The Grassmannian Gr2pVq is a complex manifold with complex structure

(4.2).

Proof. Fix rΦ0s “ rφ0, ψ0s P Gr2pVq and let rΦ0sK denote the codimension-2 subspace of V

consisting of elements which are orthogonal to the 2-dimensional subspace rΦ0s Ă V with

respect to gL2.

86

We will model Gr2pVq near rΦ0s on the complex vector space TrΦ0sGr2pVq, described in

(4.1). We define a map

frΦ0s : TrΦ0sGr2pVq Ñ Gr2pVq

pδφ, δψq ÞÑ spanCtφ0 ` δφ, ψ0 ` δψu.

The image of frΦ0s is the open subset U Ă Gr2pVq consisting of 2-dimensional complex

subspaces of V which intersect rΦ0sK in exactly 0. Moreover, frΦ0s is a smooth bijection onto

its image with smooth inverse defined on S P U by

f´1rΦ0spSq :“

´

p´1S,rΦ0s

pφ0q ´ φ0, p´1S,rΦ0s

pψ0q ´ ψ0

¯

,

where pS,rΦ0s is orthogonal projection from S to rΦ0s—this is a well-defined isomorphism,

since S and rΦ0s are both 2-dimensional and S P U .

We leave it to the reader to check that transition maps take the form

f´1rΦ1s

˝ frΦ0spδφ, δψq “ p〈φ1, φ0〉L2 pφ0 ` δφq ` 〈φ1, ψ0〉L2 pψ0 ` δψq ´ φ1,

〈ψ1, φ0〉L2 pψ0 ` δφq ` 〈φ1, ψ0〉L2 pψ0 ` δψq ´ ψ1q

and that the derivative of this transition map at pδφ, δψq in the direction´

Ăδφ,Ăδψ¯

is given

by

D´

f´1rΦ1s

˝ frΦ0s

¯

pδφ, δψq´

Ăδφ,Ăδψ¯

“

´

〈φ1, φ0〉L2Ăδφ` 〈φ1, ψ0〉L2

Ăδψ,

〈ψ1, φ0〉L2Ăδφ` 〈ψ1, ψ0〉L2

Ăδψ¯

.

Thus the transition maps are clearly holomorphic.

87

4.1.2 The Grassmannian as a Kahler Reduction

We explained in Section 3.2.3 that the metric gL2

on V2 obtained by taking the real part

of the natural L2 Hermitian metric restricts to a metric on St2pVq and descends to a well-

defined metric on Gr2pVq, each still denoted gL2. By the same reasoning, the entire Hermitian

structure descends to Gr2pVq as well. In particular,

ωL2

:“ Ím 〈¨, ¨〉L2

is a symplectic form on V2 which we claim descends to a well-defined symplectic form on

Gr2pVq—we will abuse notation and continue to denote the induced form by ωL2. Indeed,

the closure property dωL2“ 0 follows from the closure of ωL

2as a form on the vector space

V2, which is clear since ωL2

doesn’t depend on its basepoint. The nondegeneracy of the

induced form follows easily because Gr2pVq is a complex manifold. In particular, for any

nonzero δΦ P TrΦsGr2pVq, iδΦ P TrΦsGr2pVq, and

ωL2

rΦspδΦ, iδΦq “ Ím

ż

〈δΦ, iδΦ〉C2 dt “ Re

ż

}δΦ}2C2 dt ‰ 0.

This induced symplectic form can be seen as arising from a symplectic reduction, as

defined in Section 2.3.2. We claim that the map

V2Ñ R4

pφ, ψq ÞÑ

ˆż 2

0

|φ|2 dt,

ż 2

0

|ψ|2 dt,Re

ż 2

0

φψ dt, Im

ż 2

0

φψ dt

˙

“`

}φ}2L2 , }ψ}2L2 ,Re 〈φ, ψ〉L2 , Im 〈φ, ψ〉L2

˘

88

introduced in Section 3.2.2 to show that St2pVq is a Frechet manifold can be viewed as the

moment map for the Up2q-action on V2 by right multiplication. More precisely, we have the

following proposition.

Proposition 4.1.4. The map

µUp2q : V2Ñ up2q

pφ, ψq ÞÑ i

¨

˚

˝

〈φ, φ〉L2 ´ 〈φ, ψ〉L2

´ 〈ψ, φ〉L2 〈ψ, ψ〉L2

˛

‹

‚

is a moment map for the Up2q-action on V2 by pointwise right multiplication. We conclude

that

Gr2pVq “ V2 � Up2q.

Remark 4.1.5. The moment map is closely related to the frame-Hopf map (see Section

3.1.2). The difference of the diagonal coordinates of the moment map is

i p〈φ, φ〉L2 ´ 〈ψ, ψ〉L2q “

ż 2

0

i`

|φ|2 ´ |ψ|2˘

dt.

The integrand is i times the first entry of FrameHopf1. The upper right entry of the moment

map is

í 〈φ, ψ〉L2 “

ż 2

0

íφψ dt.

The real part of the integrad is the second entry of FrameHopf1, while the imaginary part is

the negation of the remaining entry of FrameHopf1. Thus the closed framed loops correspond

to the level set of¨

˚

˝

i 0

0 i

˛

‹

‚

P up2q.

89

Proof. Let 〈¨, ¨〉up2q denote the standard inner product on up2q (an extension of the standard

inner product on sup2q), and fix an arbitrary matrix

ξ “

¨

˚

˝

ia z

´z ib

˛

‹

‚

P up2q “ tskew-Hermitian matricesu.

For the remainder of this proof, let F denote the map

F : V2Ñ R

Φ ÞÑ⟨µUp2qpΦq, ξ

⟩up2q

.

More explicitly,

F pΦq “1

2trace

`

µUp2qpΦq ¨ ξ˚˘

“1

2trace

¨

˚

˝

i

¨

˚

˝

〈φ, φ〉L2 ´ 〈φ, ψ〉L2

´ 〈ψ, φ〉L2 〈ψ, ψ〉L2

˛

‹

‚

¨

˚

˝

ía ´z

z íb

˛

‹

‚

˛

‹

‚

“1

2

`

a}φ}2L2 ´ iz 〈φ, ψ〉L2 ` iz 〈ψ, φ〉L2 ` b}ψ}2L2

˘

“1

2

`

a}φ}2L2 ` b}ψ}2L2

˘

` Im pz 〈φ, ψ〉L2q

Our goal is to show that

DF pΦqpδΦq “ ωL2

pδΦ,Φ ¨ ξq, (4.3)

90

where Φ ¨ ξ is the infinitesimal vector field associated to ξ. This can be established directly:

DF pΦqpδΦq “d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

1

2

`

a}φ` εδφ}2L2 ` b}ψ ` εδψ}2L2

˘

` Im pz 〈φ` εδφ, ψ ` εδψ〉L2q

“ aRe 〈φ, δφ〉L2 ` bRe 〈ψ, δψ〉L2 ` Im pz p〈φ, δψ〉L2 ` 〈δφ, ψ〉L2qq

“ Ím 〈δφ, iaφ〉L2 ´ Im 〈δψ, ibψ〉L2 ´ Im 〈δφ,´zψ〉L2 ´ Im 〈δψ, zφ〉L2

“ Ím 〈pδφ, δψq, piaφ´ zψ, zφ` ibψq〉L2

“ ωL2

pδΦ,Φ ¨ ξq.

It is easy to check that

ξ0 :“

¨

˚

˝

i 0

0 i

˛

‹

‚

is a regular value of µUp2q—for example, the collection

tpφ, 0q, p0, ψq, pψ, φq, piψ,íφqu

maps to a spanning set for up2q under DµUp2qpφ, ψq. Recalling that St2pVq “ µ´1Up2qpξ0q and

Gr2pVq “ St2pVq{Up2q, we have shown that

Gr2pVq “ V2 � Up2q,

i.e., the Grassmannians are obtained as symplectic reductions.

A primary feature of the Marsden-Weinstein reduction process is that the restriction of

the symplectic form to the level set submanifold degenerates exactly along the group orbit

fibers—i.e. the vertical space of the submersion. We record a useful version of this fact in

the special case of interest in the following lemma.

91

Lemma 4.1.6. Let rΦs P Gr2pVq, δΦ1 P ThorΦ St2pVq « TrΦsGr2pVq and δΦ2 P TΦSt2pVq and

let proj : TΦSt2pVq Ñ T horΦ St2pVq denote orthogonal projection. Then

ωL2

Φ pδΦ1, δΦ2q “ ωL2

Φ pδΦ1, projpδΦ2qq,

where ωL2

is the symplectic form of V2 restricted to St2pVq.

Proof. This statement is equivalent to the statement that

ωL2

Φ pδΦ1, δΦvert2 q “ 0

for all δΦ1 P ThorΦ St2pVq and δΦ2 P TΦSt2pVq, where Φvert

2 is the vertical component of the

tangent vector. Indeed, the vertical component can be written as δΦvert2 “ Φ ¨ ξ for some

ξ P up2q, and

ωL2

Φ pδΦ1,Φ ¨ ξq “⟨DµUp2qpΦqpδΦq, ξ

⟩up2q

“ 〈0, ξ〉up2q “ 0.

By Theorem 3.2.8, the various geometric structures of Gr2pVq are passed on to the moduli

space of framed loopsM. We have already seen (as part of the theorem) that pGr2pVq, gL2q

is locally isometric to pM, gSq, where gS is the metric induced from the natural metric on

framed path space constructed in Section 3.1.1. We now describe the induced complex and

symplectic structures on M.

4.1.3 The Induced Kahler Structure of M

To describe the complex structure of M, we note that it is an easy corollary of Proposition

3.1.10 that the map pH restricts to a locally isometric double-cover from

W :“ LpC2zt0uq \ApC2

zt0uq Ă PC2,

92

a disjoint union of open subsets of complex vector spaces, onto the space of frame-periodic

paths

rSper “ tframe-periodic framed pathsu{ttranslationu Ă rS.

An element of rSper is a framed path pγ, V q with γp0q “ 0 such that V is periodic and γ is

almost periodic in the sense that

γpkqp0q “ γpkqp2q @ k “ 1, 2, . . . ,

but γp0q is not necessarily equal to γp2q. Thus such a pγ, V q determines a loop in SOp3qˆR`.

By the previous discussion, M can be viewed as a symplectic reduction of rSper by the

action of SOp3qˆS1, where SOp3q acts by rigid rotations and S1 acts by global frame twists.

This is summarized by the commutative diagram:

W Gr˝2pLCq \Gr˝2pACq

rSper M

�Up2q

pHpH

�pSOp3q ˆ S1q

Therefore, it suffices to describe the complex structure on the simpler space rSper, as this

structure descends to M.

Let pγ, V q P rSper and let T “ γ1{}γ1} and W “ T ˆ V . Any variation pδγ, δV q can be

recovered from pδγ1, δV q, and this can be expressed uniquely as

pδγ1, δV q “ λstretchXstretch ` λtwistXtwist ` λbend1Xbend1 ` λbend2Xbend2 , (4.4)

93

T

V

WJ

-‐Jγ

J

-‐J

Xstretch Xtwist Xbend1 Xbend2

Figure 4.1: The complex structure ofM.

where the λ’s are R-valued functions and the X’s are the following four basic variations :

Xstretch “ p}γ1}T, 0q “ stretch tangent vector, leave frame vector unchanged

Xtwist “ p0,´W q “ twist the frame in the negative direction around γ

Xbend1 “ p}γ1}W, 0q “ bend the curve in the negative direction around V

Xbend2 “ p´}γ1}V, T q “ bend the curve in the negative direction around W.

Then we define an almost complex structure, denoted J , by extending the rule

J : Xstretch ÞÑ Xtwist, J : Xbend1 ÞÑ Xbend2 , and J2“ Íd. (4.5)

over linear combinations. The complex structure’s pointwise action is depicted in Figure 4.1.

We claim that this almost complex structure is, in fact, an honest complex structure and

that it is the induced structure coming from our identification with rS. To show this, we first

express the basic variations in quaternionic coordinates.

94

Lemma 4.1.7. Let pγ, V q P rSper. Using quaternionic coordinates, let q PW satisfy

pHpqq “ pγ, V q.

Then, under the map DpHpqq, we have the correspondences

q

2Ø Xstretch

iq

2Ø Xtwist

jq

2Ø Xbend1

kq

2Ø Xbend2 .

Proof. In quaternionic coordinates, pH takes the form

pHpqq “

ˆ

t ÞÑ

ż t

0

Impqiqq dt, Im

ˆ

qjq

}q}2H

˙˙

,

so for any variation δq, we have

DpHpqqpδqq “d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

˜

t ÞÑ

ż t

0

Imppq ` εδqqipq ` εδqqq dt, Im

˜

pq ` εδqqjpq ` εδqq

}q ` εδq}2H

¸¸

“

ˆ

t ÞÑ

ż t

0

2Impqiδqq dt,2

}q}2H

`

Impqjδqq ´ RepqδqqV˘

˙

“: pδγ, δV q,

where we once again use Im and Re in this context to denote the maps

Impq0 ` iq1 ` jq2 ` kq3q “ pq1, q2, q3q P R3, Repq0 ` iq1 ` jq2 ` kq3q “ q0 P R.

Thus

δγ1 “ 2Impqiδqq.

95

Now the claim follows by simply calculating explicitly. For δq “ q{2 we have

δγ1 “ 2Im´

qiq

2

¯

“ Impqiqq “ γ1

and

δV “2

}q}2H

ˆ

Im´

qjq

2

¯

´ 2Re

ˆ

q´q

2

¯

˙

V

˙

“1

}q}2H

`

Impqjqq ´ Re`

}q}2H˘

V˘

“ V ´ V “ 0.

Thus δq “ q{2 corresponds to Xstretch, as claimed.

Similarly, for δq “ iq{2,

δγ1 “ 2Im

ˆ

qiiq

2

˙

“ Imp´}q}2Hq “ 0

and

δV “2

}q}2H

˜

Im

ˆ

qjiq

2

˙

´ Re

˜

q

ˆ

iq

2

˙

¸

V

¸

“1

}q}2HpImp´qkqq ´ Repí}q}2HqV q

“ Ímpqkqq

}q}2H“ ´W,

so δq “ iq{2 corresponds to Xtwist.

The remaining computations are similar. For δq “ jq{2,

δγ1 “ 2Im´

qkq

2

¯

“ }γ1}W and δV “2

}q}2H

ˆ

Im´

´qq

2

¯

´ Re

ˆ

´qq

2j

˙

V

˙

“ 0.

96

and for δq “ kq{2,

δγ1 “ 2Im´

qp´jqq

2

¯

“ ´}γ1}V and δV “2

}q}2H

ˆ

Im´

qiq

2

¯

´ Re

ˆ

qq

2p´kq

˙˙

“ T.

Then we have the immediate corollaries:

Corollary 4.1.8. The complex structure defined in (4.5) on rSper is induced from the complex

structure of W via the map pH

Proof. In quaternionic coordinates, the usual complex structure is replaced by multiplication

by the quaternionic i. Thus Lemma 4.1.7 gives correspondences

i ¨q

2“

iq

2Ø J ¨Xstretch “ Xtwist

i ¨jq

2“

kq

2Ø J ¨Xbend1 “ Xbend2

i2 “ Íd Ø J2“ Íd.

Remark 4.1.9. It follows from Lemma 4.1.7 that rSper actually has a quaternionic structure.

Corollary 4.1.10. Let Φ “ pφ, ψq P W correspond to pγ, V q P rSper under pH. The basic

variations are represented in complex coordinates as

1

2Φ Ø Xstretch

i

2Φ Ø Xtwist

1

2p´ψ, φq Ø Xbend1

1

2píψ, iφq Ø Xbend2 .

Proof. Recall that the identification of C2 with H is given by pz, wq Ø z`wj. The corollary

follows by rewriting each quaternionic variation using this identification.

97

This complex structure transfers from rSper to M, although this requires some clarifica-

tion. Any functional linear combination of the form

λstretch1

2Φ` λtwist

i

2Φ` λbend1

1

2p´ψ, φq ` λbend2

1

2píψ, iφq (4.6)

determines an admissible variation of Φ “ pφ, ψq PW , since W is a union of open subsets of

vector spaces. Thus any variation of the form (4.4) of pγ, V q P rSper is admissible. However,

when we restrict to Gr2pVq, we impose conditions on which variations of rΦs P Gr2pVq of

the form (4.6) are admissible—this is because the tangent spaces to the Grassmannians have

positive codimension in W . Nonetheless, we have shown in the previous section that the

complex structure of Gr2pVq is well-defined. Similarly, when we restrict to M we impose

conditions on admissible variations of rγ, V s P M of the form (4.4). The fact that the

complex structure of M described in this section is well-defined is a corollary of Theorem

3.2.8, Proposition 4.1.4 and the discussion above.

ThusM has a well-defined complex structure. Readers familiar with the work of Millson

and Zombro will note that it is no coincidence that J closely resembles the complex structure

constructed on a related moduli space of loops in [55]. This will be treated later in Section

4.3.

Finally, the symplectic form on rSper is defined by

ωSp¨, ¨q :“ gSpJ ¨, ¨q.

As usual, we will not use special notation to distinguish this form from the induced form on

M.

98

4.2 Principal Bundle Structure

4.2.1 Statement of the Result

As we have so far identified S1 with r0, 2s{p0 „ 2q, we introduce the special notation S1

for the length-2π circle, considered as a subset of C in the natural way. The space LS1 is

a tame Frechet Lie group with the obvious pointwise multiplication. It will frequently be

convenient to denote points of LS1 by eiα, where α P PR is 2π ¨ k-periodic for some k P Z.

By identifying S1 with the set of constant loops in LS1, S1 embeds as a Lie subgroup. The

quotient LS1{S1 is once again a tame Frechet Lie group and it has a natural free action on

Gr2pVq by pointwise multiplication. Explicitly, the action is

preiαs ¨ rφ, ψsqptq :“ reiαptq ¨ φptq, eiαptq ¨ ψptqs,

where reiαs P LS1{S1 denotes the equivalence class of eiα P LS1.

According to Theorem 3.2.8, this action corresponds to an action of LS1{S1 on the moduli

space of relatively framed loops, M. Indeed, the action is by adjusting the framing: an

element eiα P LS1 acts on a framed loop pγ, V q by fixing γ and adjusting V according to the

formula

V ÞÑ cosp2αqV ` sinp2αq

ˆ

γ1

}γ1}ˆ V

˙

.

This action induces a well-defined action on equivalence classes of framed loops up to trans-

lation, scaling and rotation, and descends to a well-defined action of LS1{S1 on equivalence

classes up to relative framing. From this interpretation, we see that the action of LS1{S1 is

free—if

reiαs ¨ rγ, V s “ rγ, V s,

99

then the framed loopˆ

γ, cosp2αqV ` sinp2αq

ˆ

γ1

}γ1}ˆ V

˙˙

differs from pγ, V q by a rigid rotation. This is only possible if α is constant, and thus reiαs

is the identity class in LS1{S1.

The main result of this section is that this action gives M the structure of a principal

bundle over the moduli space of unframed loops. To be precise, let B denote the moduli

space of unframed loops

B :“ timmersed loops in R3u{ttranslation, scaling, rotationu.

Lemma 4.2.1. The moduli space B is a Frechet manifold.

Proof. We model B as

B « tγ P Imm0pS1,R3

q | lengthpγq “ 2u{SOp3q,

where we let Imm0pS1,R3q denote the space of immersed loops based at zero—an open

submanifold of a tame Frechet vector space.

The space

tγ P Imm0pS1,R3

q | lengthpγq “ 2u

is a tame Frechet manifold as it is the level set of a regular value of the length functional and

we once again invoke Corollary 2.2.10. We obtain smooth cross-sections for the SOp3q-action

by following an argument of Millson and Zombro ([55], Lemma 1.6). Let Ut0 denote the open

set of length-2 based immersed curves containing the curves with nonvanishing curvature at

t0 P r0, 2s. Every such curve γ has a principal normal vector Nγpt0q at γpt0q, and we take

the SOp3q-cross-section through Ut0 to be the inverse image of the identity matrix under the

100

map

Ut0 Ñ SOp3q

γ ÞÑ

ˆ

γ1pt0q

}γ1pt0q}, Nγpt0q,

γ1pt0q

}γ1pt0q}ˆNγpt0q

˙

.

It is easy to see that these sets do define cross-sections for the SOp3q-action and that

transition maps are smooth, thus proving the claim (see [55] for more details).

Thus our goal for this section is to show that LS1{S1 ãÑM Ñ B is a Frechet principal

bundle.

4.2.2 The Failure to Close Map

The obvious projection M Ñ B is the “forget framing” map rγ, V s ÞÑ rγs. It will be useful

to express this projection map in terms of the Grassmannian formalism. Let pHp1q denote the

first coordinate of pH; that is,

pHp1qprΦsq “ rγs P B ô pHprΦsq “ rγ, V s PM.

Then, by Theorem 3.2.8, the forget framing map can be expressed as

pHp1q : Gr2pLCq \Gr2pACq Ñ B

In this form it is easy to check from the explicit formula for pH that each fiber of the projection

is diffeomorphic to LS1{S1.

101

Thus it remains to show that M is locally diffeomorphic to B ˆ LS1{S1. We begin by

recalling that the twist rate of a framed path pγ, V q is defined to be

twpγ, V q :“

⟨d

dsV,W

⟩,

where d{ds is derivative with respect to arclength of γ and W “ γ1{}γ1}. The twist rate

is well-defined for relatively framed curves and is invariant under translation, scaling, and

rotation. We also recall that the total twist of pγ, V q is the integral of the twist rate with

respect to arclength; i.e.,

Twpγ, V q “1

2π

ż 2

0

twpγ, V q }γ1ptq}dt.

It will be useful to express the twist rate in pφ, ψq-coordinates. We first prove an auxilliary

lemma.

Lemma 4.2.2. Let q P PpHzt0uq correspond to the framed path pγ, V q in quaternionic

coordinates. Then the variation δq “ q1{}q}2H of q corresponds to the variation

pδγ1, δV q “

ˆ

d

dsγ1,

d

dsV

˙

of pγ, V q under DpHpqq.

Proof. Recall from the proof of Lemma 4.1.7 that the variation corresponding to δq is

pδγ, δV q “

ˆ

t ÞÑ

ż t

0

2Impqiδqq dt,2

}q}2H

`

Impqjδqq ´ RepqδqqV˘

˙

102

Thus for δq “ q1{}q}2H, we have

δγ1 “ 2Im

ˆ

qiq1

}q}2H

˙

“1

}q}2H2Impqiq1q

“1

}q}2H

d

dtImpqiqq

“1

}γ1}

d

dtγ1 “

d

dsγ1.

Moreover,

δV “2

}q}2H

˜

Im

ˆ

qjq1

}q}2H

˙

´ Re

˜

q

ˆ

q1

}q}2H

˙

¸

V

¸

“1

}q}4H

ˆ

d

dtImpqjqq ´ V

d

dt}q}2H

˙

“1

}q}2H

}q}2Hddt

Impqjqq ´ Impqjqq ddt}q}2H

}q}4H

“1

}γ1}

d

dtV “

d

dsV.

Lemma 4.2.3. Let Φ “ pφ, ψq P St˝2pVq correspond to a framed loop pγ, V q. Then

twpγ, V q “ ´2Im

`

φ1φ` ψ1ψ˘

p|φ|2 ` |ψ|2q2. (4.7)

Proof. Rearranging the expression on the right, we obtain

´2Im

`

φ1φ` ψ1ψ˘

p|φ|2 ` |ψ|2q2“ ´

2

}Φ}4C2

Im 〈Φ1,Φ〉C2

“ ´4

}Φ}2C2

Re

⟨Φ1

}Φ}2C2

,iΦ

2

⟩C2

.

103

In quaternionic coordinates, the last expression is equal to

´4

}q}2HRe

⟨q1

}q}2H,iq

2

⟩H.

Applying Lemma 4.2.2, Lemma 4.1.7 and Corollary 3.1.11, this is equal to

´4

}γ1}¨

1

4

ˆ⟨d

dsδγ, 0

⟩`

⟨d

dsV,W

⟩〈´W,W 〉

˙

}γ1} “

⟨d

dsV,W

⟩“ twpγ, V q,

where W “ γ1{}γ1} and δγ satisfies δγ1 “ ddsγ1.

The quantity on the right hand side of (4.7) will also be denoted by twpΦq. We may also

express the total twist in pφ, ψq-coordinates as

TwpΦq “1

2π

ż 2

0

twpΦq`

|φ|2 ` |ψ|2˘

dt.

Moreover, these maps induce well-defined maps on equivalence classes rΦs P Gr˝2pVq.

We now introduce the map

ĆFTC :MÑ S1Ă C

rγ, V s ÞÑ exppiπTwpγ, V qq

(“FTC” stands for “failure to close”—this will be explained momentarily). It is clear that

this map is well-defined, as Tw is well-defined for relative framings and is invariant under

translation, scaling and rotation. Perhaps less obvious is the fact that the value of ĆFTC does

not depend on the framing.

Lemma 4.2.4. The map ĆFTC :MÑ S1 induces a well-defined map on B.

104

Proof. We need to show that ĆFTC is LS1{S1-invariant. Let rγ, V s PM have Grassmannian

representation rΦs “ rφ, ψs. For reiαs P LS1{S1. It follows easily from Lemma 4.2.3 that

twpeiαΦq “ ´2α1

|φ|2 ` |ψ|2` twpΦq.

Thus

TwpeiαΦq “1

2π

ż 2

0

ˆ

´2α1

|φ|2 ` |ψ|2` twpΦq

˙

p|φ|2 ` |ψ|2qdt

“1

πpαp0q ´ αp2qq ` TwpΦq.

Since eiα is a loop, αp0q ´ αp2q is a multiple of 2π, and we have

exppiπTwpeiαΦqq “ exppiπTwpΦqq,

as claimed.

We denote the induced map by FTC : B Ñ S1. This is the failure to close map; it

measures the angle by which the Bishop frame [4] of a given loop fails to close. Recall from

Example 2.1.2 that the Bishop frame of a space curve γ is the relative framing obtained by

choosing a normal vector V p0q and parallel transporting it in the normal bundle of γ, and

that this framing does not necessarily close up even if the base curve γ is closed. With FTC

defined, we are equipped to prove the main theorem of this section.

4.2.3 The Principal Bundle Theorem

We are prepared to prove the main result of this section, which is stated formally as follows.

105

Theorem 4.2.5. The “forget framing” map

MÑ B

rγ, V s ÞÑ rγs

gives M the structure of a principal bundle over B with structure group LS1{S1.

The proof strategy is stated informally as follows. We divide the space of unframed loops

into two open sets using the map FTC. Roughly, there is an open set containing loops

whose Bishop frames fail to close by a small amount and an open set containing loops whose

Bishop frames fail to close by a large amount. For each open set, we assign a preferred

relative framing to each loop by “smoothly closing the Bishop framing”—more precisely, we

increase or decrease the twist rate by the smallest amount possible so that the resulting frame

smoothly closes. These frames are essentially the rotation minimizing frames of, e.g., [74].

The amount by which to increase the twist rate is not canonical over the space of all loops,

which is why the principal bundle is nontrivial. To show that M is locally diffeomorphic to

B ˆ LS1{S1, we define a map which takes rγ, V s to prγs, reiαsq, where reiαs is the loop class

measuring the difference between the relative framing rV s and the preferred relative framing.

We now make this precise.

Proof. We will construct explicit local trivializations ofM. First define local charts pfj, Ujq

for j “ ˘1 on S1 Ă C by

Uj “ S1ztju, j “ ˘1

and

fj : Uj Ñ p0, 2πq

f´1peiθq “ θ ` π for θ P p´π, πq and f1pe

iθq “ θ for θ P p0, 2πq.

106

Let

Uj :“ FTC´1pUjq and rUj :“ ĆFTC

´1pUjq.

Then Uj is an open subset of B, rUj is an open subset of Gr˝2pLCq \ Gr˝2pACq « M, and

pHp1qp rUjq “ Uj. Referring back to the informal discussion above, the open subset U´1 corre-

sponds to those loops whose Bishop framing “fails to close by a small amount”.

We claim that

rUj « Uj ˆ LS1{S1, j “ ˘1.

We will define an explicit diffeomorphism by first defining maps

Gj : rUj Ñ PR j “ ˘1,

where Gj is defined on Φ “ pφ, ψq by the formula

GjpΦqptq :“1

2π

ż t

0

twpΦqp|φ|2 ` |ψ|2q dt´t

2πfj ˝ĆFTCpΦq

and we note that this descends to a well defined map on equivalence classes rΦs. We also

note that

GjpΦqp0q “ 0,

and

GjpΦqp2q “ TwpΦq ´ pTwpΦq mod 2πq “ 2πk for some k P Z.

Therefore

rexppiGjpΦqqs P LS1{S1,

107

and we define our local trivializations by

rUj Ñ Uj ˆ LS1{S1

rΦs ÞÑ´

pHp1qpΦq, rexppiGjpΦqqs¯

j “ ˘1. (4.8)

Informally, the equivalence class of loops rexppiGjpΦqqs records the difference between the

original framing of the framed loop corresponding to Φ and the modified Bishop framing of

its base loop.

To show that these maps are diffeomorphisms, we construct their inverses. Let rΦs be

any element of the fiber of rγs P Uj. We claim that rexppiGjpΦqqΦs only depends on rγs and

not on the choice of rΦs; i.e., we wish to show that for any reiαs P LS1{S1,

rexppiGjpeiαΦqqeiαΦs “ rexppiGjpΦqqΦs.

From the definition of Gj, we easily deduce that

GjpeiαΦqptq “ αp0q ´ αptq `GjpΦqptq.

Up to S1-equivalence, we have

rexppipαp0q ´ αptq `GjpΦqptqqqs “ rexppiGjpΦptqqq exppíαptqqs,

We conclude

rexppiGjpeiαΦqqeiαΦs “ rexppiGjpΦqqe

íαeiαΦs “ rexppiGjpΦqqΦs.

108

Now we define the inverse map of (4.8) by

Uj ˆ LS1{S1

Ñ rUj

prγs, reiθsq ÞÑ rexppipGjpΦq ´ θqqΦs j “ ˘1,

where rΦs is any element of the fiber of rγs. Informally, this map produces of a framing of

a loop γ by taking its modified (closed) Bishop framing and adjusting it according to reiθs.

We leave it to the reader to check that these are our inverse maps and this completes the

proof.

4.2.4 Non-Existence of a Global Framing Algorithm

This theorem has a corollary which follows trivially but is of practical interest. A well-

studied problem in applied differential geometry is to algorithmically assign a framing or

relative framing to a given parameterized space curve [9, 14, 26, 74]. This has applications

to computer graphics, where one uses the framing to construct a tube around a given curve

for visual clarity [26], as well as animation, motion planning and camera tracking.

To make the problem precise, we define a curve framing algorithm to be a continuous

map from any subset of the space of immersed paths, Immpr0, 2s,R3q, to the space of framed

paths that is of the form γ ÞÑ pγ, V q. Similarly, we define a relative framing algorithm to

be a continuous map which assigns a relative framing to each immersed path lying in some

subset of Immpr0, 2s,R3q. If γ is a loop, then we require a (relative) framing algorithm to

produce a smoothly closed (relative) framing.

For example, the most classical curve framing algorithm assigns to an immersed path

γ its Frenet framing, where the frame vector is the principal normal vector to γ. The

principal normal vector is not well-defined any point where the curvature of γ vanishes. In

109

our language, the curve framing algorithm is defined on the subset

tγ P Immpr0, 2s,R3q | κpγq ą 0u.

A classical relative framing algorithm assigns to each path its Bishop framing. This

algorithm is defined on the full space of immersed paths. However, if γ is a loop then the

Bishop framing may fail to close and this is a problem in many applications (e.g., for the

application of providing a tubing of the curve, the smooth closure of the framing is required

if the tube is to be textured). Therefore, the restriction of this curve framing algorithm to

loops is only defined on the subset

tγ P ImmpS1,R3q | Wrpγq P Zu,

(by the Calugareanu-White-Fuller Theorem [20]) . In fact, every framing algorithm that has

been introduced fails on some subset of loop space. From our perspective, the reason for this

phenomenon is obvious: there is no continuous global section B ÑM.

Corollary 4.2.6. There is no Euclidean similarity-invariant curve framing algorithm or

relative framing algorithm with domain ImmpS1,R3q.

Proof. Any such framing algorithm or relative framing algorithm would induce a continuous

global section of the principal bundle MÑ B. This would imply that M is homeomorphic

to B ˆ LS1{S1, but M has two components and B ˆ LS1{S1 has countably infinitely many

componenents, so this is impossible.

Remark 4.2.7. Using similar methods to this section, one is able to show that

PS1{S1 ãÑ trel. framed pathsu {ttransl., rot., scal.u Ñ Immpr0, 2s,R3

q{ttransl., rot., scal.u

110

is a principal bundle. This bundle does admit a global section—e.g., the Bishop framing.

There is no contradiction here, as PS1{S1 is contractible.

4.3 Symplectic Reduction by the action of LS1{S1

Our goal in this section is to show that the symplectic reduction of the moduli space of

framed loopsM by the action of LS1{S1 described above is isomorphic to the moduli space

of Millson-Zombro, denoted MMZ . We recall that the Millson-Zombro moduli space is

MMZ :“ tarclength-parameterized (unframed) loopsu{ttranslation, scaling, rotationu.

In [55] it was shown that MMZ is a Frechet manifold. Moreover, it was shown to admit

a complex Kahler structure. This was accompished by showing that MMZ is obtained as

the symplectic reduction of LS2 by the action of SOp3q by rigid rotations. The work in

this section provides a new interpretation of the Kahler structure of MMZ and moreover

demonstrates an infinite-dimensional Gel’fand-Macphereson Correspondence.

4.3.1 The Momentum Map

We note thatMMZ is connected, whileM has two connected components. Indeed, perform-

ing the Kahler reduction process on M will produce two copies of MMZ . For this reason,

we will actually work with only one of the components of M—say the component contain-

ing framed loops with odd linking number. For this reason, we will work specifically with

St2pLCq and Gr2pLCq in this section (ignoring their anti-loop counterparts).

111

We begin by identifying the relevant pieces in the construction. The Lie algebra of LS1{S1

is LR{R. We define an inner product on LR{R by the formula

〈rαs, rβs〉LR :“1

2

ż 2

0

αβ dt´1

4

ż 2

0

α dt

ż 2

0

β dt, α, β P LR.

This is (up to a constant multiple) the inner product induced by the standard L2 inner

product on LR.

Now we note that the LS1{S1-action on Gr2pLCq is induced from the action of LS1 on

St2pLCq, defined by

peiα ¨ Φqptq :“ eiαptqΦptq.

Let α P LR, the Lie algebra of LS1. The exponential map LR Ñ LS1 is α ÞÑ eiα, so the

infinitesimal vector field associated to α is given by

Xα|Φ “d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

eiεαΦ “ iαΦ.

Thus a point rαs P LR{R induces a vector field Xrαs on Gr2pLCq defined by orthogonally

projecting; i.e.,

Xrαs|rΦs “ projpiα ¨ Φq,

where proj is orthogonal projection

proj : TΦSt2pLCq Ñ T horΦ St2pLCq « TrΦsGr2pLCq

with respect to gL2. This is well-defined, since the horizontal space is finite codimension in

the full tangent space.

112

We define our candidate for the moment map for the LS1{S1-action to be

µ “ µLS1{S1 : Gr2pLCq Ñ LR{R

rφ, ψs ÞÑ r|φ|2 ` |ψ|2s.

The rest of this section is devoted to proving that the Kahler reduction process is well-

defined in this setup and that it produces the space MMZ .

Lemma 4.3.1. The map µ is a moment map for the action of LS1{S1.

Proof. We first compute the derivative of the map

rΦs “ rφ, ψs ÞÑ 〈µprΦsq, rαs〉LR “1

2

ż 2

0

p|φ|2 ` |ψ|2qα dt´1

2

ż 2

0

α dt

at the point rΦs in the direction δΦ “ pδφ, δψq P T horΦ St2pLCq for fixed rαs P LR{R. This is

given by

d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

1

2

ż 2

0

p|φ` εδφ|2 ` |ψ ` εδψ|2qα dt´1

2

ż 2

0

α dt “

ż 2

0

Repφδφ` ψδψqα dt.

On the other hand, we need check that this is equal to

ωL2

rΦspδΦ, Xrαs|rΦsq “ Ím 〈δΦ, projpiαΦq〉L2 .

To see this, we write

Xrαs|rΦs “ projpiαΦq “ iαΦ´ piαΦqK,

113

where piαΦqK is perpendicular to T horΦ St2pLCq with respect to gL2

(i.e., piαΦqK is tangent to

the Up2q-orbit of Φ). Then

Ím 〈δΦ, projpiαΦq〉L2 “ Ím⟨δΦ, iαΦ´ piαΦqK

⟩L2

“ Ím 〈δΦ, iαΦ〉L2 ` Im⟨δΦ, piαΦqK

⟩L2

“ Re 〈αΦ, δΦ〉L2 ` Re⟨íδΦ, piαΦqK

⟩L2

“

ż 2

0

Re`

φδφ` ψδψ˘

α dt` 0,

and this proves the claim.

4.3.2 Manifold Structure of the Space of Arclength-Parameterized

Framed Loops

Next we choose a suitable level set of the moment map to construct an LS1{S1-invariant

submanifold of Gr2pLCq. Let 1 P LR{R denote equivalence class of the loop that takes

the value 1 identically. We wish to show that µ´1p1q is a submanifold with the desired

properties. From Lemma 3.1.7 we deduce that µ´1p1q corresponds under pH to the subset of

M consisting of relatively framed loops with arclength parameterized base curve:

µ´1p1q «Marc :“ trγ, V s PM | }γ1} ” 1u.

Proposition 4.3.2. The space µ´1p1q is an LS1{S1-invariant smooth Frechet manifold with

tangent space

TrΦsµ´1p1q «

δΦ P T horΦ St2pLCq | Re 〈Φ, δΦ〉C2 ” 0(

.

114

The strategy we will use to prove this theorem is to note that there is a natural identifi-

cation

µ´1p1q « pLS3

X St2pLCqq{Up2q,

where S3 Ă C2 denotes the standard 3-sphere and we consider LS3 as a submanifold of LC2.

Thus we first aim to show that LS3X St2pLCq is a Frechet manifold. This follows by noting

that it is a level set of the Frechet manifold LS3 ˆ R` with respect to the map

µUp2q : LS3ˆ R` Ñ up2q

pφ, ψq ÞÑ i

¨

˚

˝

〈φ, φ〉L2 ´ 〈φ, ψ〉L2

´ 〈ψ, φ〉L2 〈ψ, ψ〉L2

˛

‹

‚

Note that this is a restriction of the momentum map introduced in Proposition 4.1.4 for the

Up2q-action on LpC2zt0uq. We are considering LS3 ˆR` as embedded in LpC2zt0uq via the

embedding

LS3ˆ R` ãÑ LpC2

zt0uq

pΦ, rq ÞÑ rΦ.

Thus our goal is to show that elements of

LS3X St2pLCq « pLS3

ˆ R`q X St2pLCq

are always regular points for µUp2q, and then the proposition follows by Corollary 2.2.10. It

is not so easy to show explicitly that elements of LS3 X St2pLCq are regular points, so we

introduce the following technical lemmas. Their purpose is to allow us to produce enough

115

Figure 4.2: Regions described in the proof of Lemma 4.3.3.

tangent vectors δΦ P TΦLS3 to map to a spanning set for up2q under DµUp2qpΦq for any

Φ P LS3 X St2pLCq.

Lemma 4.3.3. Let eiα P LS1 such that

ż

S1

eiα dt “ 0.

Then there exist functions B1, B2 : S1 Ñ Rě0 such that

ż

S1

B1 ¨ eiα dt and

ż

S1

B2 ¨ eiαdt

are linearly independent over R as elements of R2 “ C.

Proof. The assumption that

0 “

ż

S1

eiαdt “

ż

S1

cosα dt` i

ż

S1

sinα dt

116

says that the average value of the loop eiα lies at the origin in C “ R2. We claim that there

exists an open subarc I1 Ă S1 such that | cosα| ą | sinα| on I1. This claim means that the

image of eiα on I1 lies in one of the highlighted regions in Figure 4.3.2 intersected with S1.

From this interpretation, it is clear that such a subarc must exist due to the average value

assumption. Similarly, there must exist some open subarc I2 Ă S1 such that | cosα| ă | sinα|

on I2.

Now we define Bj : S1 Ñ Rě0 to be a nonzero bump function supported on Ij. Then

the image of B1eiα lies entirely in one of the shaded regions of Figure 4.3.2 intersected with

S1, and this implies that the average value of B1eiα is nonzero and lies in the same shaded

region (though not necessarily on S1). Similarly, the average value of the image of B2eiα

is nonzero and lies in an unshaded region. Multiplication by a real number will not take a

point in a shaded region to a point in an unshaded region, so the proof is complete.

Before moving on to the next lemma, we note that the tangent space to LS3 at Φ is

concretely realized as

TΦLS3“ tδΦ P LC2

| Re 〈Φ, δΦ〉C2 ” 0u.

Lemma 4.3.4. Let Φ “ pφ, ψq P LS3 X St2pLCq. Then there exists pδφ, δψq P TΦLS3 such

that

Im⟨pφ, ψq, pδψ, δφq

⟩L2 ‰ 0.

Proof. Consider the tangent vector

pδφ, δψq “ píψ, iφq ´ Re 〈píψ, iφq, pφ, ψq〉C2 pφ, ψq P Tpφ,ψqLS3.

We claim that

Im⟨pφ, ψq, pδψ, δφq

⟩C2 ı 0

117

for this choice of δΦ. Indeed, if this is not the case, then

0 ” Im⟨pφ, ψq, piφ´ Re 〈píψ, iφq, pφ, ψq〉C2 ψ, ψ ´ Re 〈pψ, φq, pφ, ψq〉C2 φq

⟩C2

“ Im`

í|φ|2 ´ i|ψ|2 ´ 2Re 〈píψ, iφq, pφ, ψq〉C2 φψ˘

“ ´1´ 2Impψφ` φψqImpφψq

“ ´1` 4Impψφq2.

This implies that

Impψφq ”1

2or Impψφq ” ´

1

2,

contradicting pφ, ψq P St2pLCq as it implies

ż

Impψφq dt ‰ 0.

Now assume without loss of generality that Im⟨pφ, ψq, pδψ, δφq

⟩C2 takes some positive

value. Then there exists an open subarc I Ă S1 so that Im⟨pφ, ψq, pδψ, δφq

⟩C2 ą 0 on I.

Let I 1 Ă I be a proper open subset, and let B : S1 Ñ Rě0 be a smooth bump function with

B|I 1 ” 1, and supportpBq Ă I. Then

Im⟨pφ, ψq, pB ¨ δψ,B ¨ δφq

⟩L2 “

ż

S1

B ¨ Im⟨pφ, ψq, pδψ, δφq

⟩C2 dt

ě

ż

I 1Im⟨pφ, ψq, pδψ, δφq

⟩C2 dt

ą 0.

Since pB ¨ δφ,B ¨ δψq P TΦLS3, this proves the claim.

118

Proof of Proposition 4.3.2. Fix Φ “ pφ, ψq P LS3 X St2pLCq. Our goal is to show that Φ is

a regular point of the restricted map µUp2q. That is, we wish to find four tangent vectors in

TΦpLS3ˆ R`q “ tδΦ “ pδφ, δψq P LC2

| Re 〈Φ, δΦ〉C2 ” 0u ‘ tλΦ | λ P Ru

whose images are linearly independent under

DµUp2qpΦqpδΦq “ i

¨

˚

˝

2Re 〈φ, δφ〉L2 ´ 〈φ, δψ〉L2 ´ 〈δφ, ψ〉L2

´ 〈ψ, δφ〉L2 ´ 〈δψ, φ〉L2 2Re 〈ψ, δψ〉L2

˛

‹

‚

.

The first obvious choice is δΦ “ Φ, so that

DµUp2qpΦqpΦq “ i

¨

˚

˝

2 0

0 2

˛

‹

‚

“: A1.

To find the second tangent vector, we first observe that either

ż

Repφψq2 dt or

ż

Impφψq2 dt

must be positive. This follows since the assumptions |φ|2 ` |ψ|2 ” 1 and }φ}L2 “ }ψ}L2 “ 1

imply that |φ|2|ψ|2 ı 0, which in turn implies

0 ă

ż

|φ|2|ψ|2 dt

“

ż

|φψ|2 dt

“

ż

Repφψq2 `

ż

Impφψq2 dt.

119

Assume without loss of generality thatş

Repφψq2 dt ą 0 and consider the tangent vector

pδφ, δψq “ p´Repφψqψ,Repφψqφq P TΦLS3.

We have

Dµpφ, ψqp´Repφψqψ,Repφψqφq “ i

¨

˚

˝

´2ş

Repφψq2 dt 0

0 2ş

Repφψq2 dt

˛

‹

‚

“: A2.

Then A2 is linearly independent from A1, as the diagonal entries of A2 differ by a sign while

the diagonal entries of A1 are the same. In the case thatş

Impφψq2 dt ą 0, a similar choice

of tangent vector could have been made.

Thus it remains to find two more tangent vectors which map to complete a spanning set

for up2q. Since A1 and A2 are diagonal matrices, it suffices to show the set

t´ 〈φ, δψ〉L2 ´ 〈δφ, ψ〉L2 | pδφ, δψq P TΦLS3u.

of upper-right entries of DµUp2qpΦqpδΦq is two-dimensional (over R).

We proceed by checking two cases: either |ϕ|2 ” |ψ|2, or not. In the case that the

functions are not identically equal,

ż

|φ|2 ´ |ψ|2 dt “

ż

|φ|2 dt´

ż

|ψ|2 dt “ 0

implies that the function |φ|2´|ψ|2 must take both positive and negative values. Let I Ă S1

be an open subarc such that |ϕ0|2 ´ |ψ0|

2 ą 0 on I and let I 1 Ă I be a proper open subarc.

Let B : S1 Ñ Rě0 be a smooth bump function with supportpBq Ă I and B ” 1 on I 1. Now

we consider the tangent vector δΦ “ pB ¨ψ,´B ¨φq. The upper-right entry of DµUp2qpΦqpδΦq

120

is

〈φ,B ¨ φ〉L2 ´ 〈B ¨ ψ, ψ〉L2 “

ż

B ¨ p|φ|2 ´ |ψ|2q dt.

This is a positive real number; i.e.,

ż

B ¨ p|φ|2 ´ |ψ|2q dt ě

ż

I 1|φ|2 ´ |ψ|2 dt ą 0.

Finally we note that Lemma 4.3.4, implies that there exists a tangent vector δΦ such that

the upper-right entry of DµUp2qpΦqpδΦq has non-zero imaginary part, and this completes the

proof of the claim for this case.

Now assume that |φ|2 ” |ψ|2. As |φ|2 ` |ψ|2 ” 1, this implies that |φ|2 ” |ψ|2 ”?

2{2

and we write

φ “

?2

2eiθ1 and ψ “

?2

2eiθ2

for some smooth eiθ1 , eiθ2 P LS1. Consider the tangent vector

δΦ “ pireiθ1 ,íreiθ2q P TΦLS3,

where r P LRě0 is any smooth function. We have that the upper-right entry ofDµUp2qpΦqpδΦq

is?

2i

ż

reipθ1´θ2q dt.

Since

0 “

ż

φψ dt “1

2

ż

eipθ1´θ2q dt,

Lemma 4.3.3 implies that there exist functions B1, B2 : S1 Ñ Rě0 such that choosing r “ B1

and r “ B2 for the tangent vector above will yield linearly independent elements of R2 after

applying DµUp2qpΦq. This completes the proof that LS3 X St2pLCq is a Frechet manifold.

121

We immediately have a description of the tangent spaces of LS3 X St2pLCq. These are

given by

TΦLS3X St2pLCq “ tδΦ P TΦLS3

| δΦ P kerDµUp2qpΦqu

“ tδΦ P TΦSt2pLCq | Re 〈Φ, δΦ〉C2 ” 0u.

It remains to show that

µ´1p1q “ pLS3

X St2pLCqq{Up2q

is a Frechet manifold and to describe its tangent spaces. This follows easily from the principal

bundle structure

St2pLCq Ñ St2pLCq{Up2q « Gr2pLCq.

Indeed, any choices of slice charts for this principal bundle will restrict to slice charts for

LS3X St2pLCq Ñ pLS3

X St2pLCqq{Up2q « µ´1p1q.

Going through the proof of Corollary 4.1.2, one sees that the tangent spaces to µ´1p1q are

described by

TrΦsµ´1p1q « tδΦ P T horΦ St2pLCq | Re 〈Φ, δΦ〉C2 ” 0u.

Finally, the fact that µ´1p1q is LS1{S1-invariant is obvious from the definition of the

action on Gr2pLCq.

122

4.3.3 Proof of the Main Result

Now our goal is to show that µ´1p1q{pLS1{S1q may be identified with MMZ , that it has

a natural symplectic structure, and that this structure agrees with the one constructed by

Millson and Zombro. We first give a more careful description of the tangent spaces to µ´1p1q.

Throughout the following lemmas, we consider µ´1p1q as a Riemannian manifold with metric

defined by restricting the metric gL2

of Gr2pLCq. Similarly, the intersection LS3 X St2pLCq

inherits a Riemannian metric, and with these metrics

LS3X St2pLCq Ñ pLS3

X St2pLCqq{Up2q “ µ´1p1q

is a Riemannian submersion.

Lemma 4.3.5. The subspace of the tangent space to µ´1p1q that is horizontal with respect

to the LS1{S1-action is

T horrΦs µ´1p1q “

δΦ P T horΦ St2pLCq | 〈Φ, δΦ〉C2 ” 0(

.

The tangent space splits orthogonally (with respect to Re 〈¨, ¨〉PC2) as

TrΦsµ´1p1q “ T horrΦs µ

´1p1q ‘ T vertrΦs µ

´1p1q,

where T vertrΦs µ

´1p1q consists of tangent vectors along the LS1{S1-orbits through rΦs.

Proof. We begin by working in LS3XSt2pLCq, which we will denote by F for the remainder

of this proof. Fix Φ P F . The vertical space T vertΦ F consists of tangent vectors of the form

iαΦ P TΦF , where α P LR. Thus the horizontal subspace with respect to the LS1{S1-action

is the set

T horΦ F :“ tδΦ P TΦF | Re 〈iαΦ, δΦ〉L2 “ 0 for all α P LRu.

123

Examining the defining condition more closely, we see

Re 〈iαΦ, δΦ〉L2 “ Ím 〈αΦ, δΦ〉L2

“ Ím

ż 2

0

α 〈Φ, δΦ〉C2 dt

“ ´

ż 2

0

αIm 〈Φ, δΦ〉C2 dt

is required to be zero for all α P LR. This is only possible if Im 〈Φ, δΦ〉C2 ” 0. On the other

hand, combining this with the description of TΦP obtained in the proof of Proposition 4.3.2,

we obtain

T horΦ P “ tδΦ P TΦSt2pLCq | 〈Φ, δΦ〉C2 ” 0u.

Projecting via the submersion P Ñ P{Up2q “ µ´1p1q, we have

T horrΦs µ´1p1q “ tδΦ P T horΦ St2pLCq | 〈Φ, δΦ〉C2 ” 0u.

Thus we have proved the first claim.

We now explicitly show that the tangent space splits orthogonally. We once again begin

by working with P . For an arbitrary δΦ P TΦP , we write

δΦ “ pδΦ´ 〈Φ, δΦ〉C2 Φq ` p〈Φ, δΦ〉C2 Φq .

Then the second term in this decomposition is an element of the vertical space. Indeed,

〈Φ, δΦ〉C2 is purely imaginary by the assumption that δΦ P TΦP , thus

〈Φ, δΦ〉Φ “ iαΦ

124

for α “ í 〈Φ, δΦ〉C2 P LR. It remains to show that the first term lies in the horizontal

space. Indeed,

〈Φ, δΦ´ 〈Φ, δΦ〉C2 Φ〉C2 “ 〈Φ, δΦ〉C2 ´ 〈Φ, δΦ〉C2 }Φ}2C2 “ 0.

Finally, we note that this property is preserved by the Riemannian submersion P Ñ

µ´1p1q.

Lemma 4.3.6. The restriction of the symplectic form ωL2

to µ´1p1q degenerates exactly

along directions tangent to LS1{S1-orbits.

Proof. Let Xrαs|rΦs denote an element of T vertrΦs µ

´1p1q. Then for any δΦ P TrΦsµ´1p1q, Lemma

4.3.1 implies

ωL2

pδΦ, Xrαs|rΦsq “

ż

Re 〈Φ, δΦ〉C2 dt “ 0.

Thus the symplectic form degenerates along LS1{S1-orbits (i.e. the vertical tangent spaces).

By the orthogonal splitting given in Lemma 4.3.5, it suffices to show that the symplectic

form is non-degenerate when restricted to T horΦ µ´1p1q. Indeed, this follows immediately since

the horizontal subspace is closed under complex multiplication.

Finally we come to the main result of this section.

Theorem 4.3.7. The “forget framing map”

µ´1p1q ÑMMZ

rγ, V s ÞÑ rγs

gives µ´1p1q the structure of a principal bundle over MMZ with fibers diffeomorphic to

LS1{S1. Thus

Gr2pLCq � pLS1{S1q “ µ´1

p1q{pLS1{S1q «MMZ

125

inherits a symplectic structure, and this structure is isomorphic up to a constant factor to

the one constructed by Millson and Zombro.

Proof. Going through the proof of the first principal bundle theorem, Theorem 4.2.5, we see

that it can be directly adapted to treat the first claim. Thus Gr2pLCq� pLS1{S1q andMMZ

are diffeomorphic as Frechet manifolds.

We endow Gr2pLCq � pLS1{S1q with a symplectic structure by restricting ωL2

to the

horizontal tangent spaces of µ´1p1q, where it was shown in Lemma 4.3.6 to be nondegenerate.

We likewise endow Gr2pLCq � pLS1{S1q with a complex structure.

To see that the induced structures agree with those of Millson-Zombro, we recall from

Section 4.1.3 that the admissible variations of rγ, V s PM can be written pointwise as linear

combinations of the basic variations Xstretch, Xtwist, Xbend1 and Xbend2 . From Lemma 4.1.7,

we see that variations which are tangent to M � pLS1{S1q « Gr2pLCq � pLS1{S1q can no

longer have any Xstretch or Xtwist component. Thus elements pδγ, δV q of the tangent space

toM� pLS1{S1q at rγ, V s must satisfy 〈δγ, γ1〉 ” 0 and 〈δV,W 〉 ” 0. Moreover, the induced

complex structure takes Xbend1 to Xbend2 . This can be succinctly rewritten as

J ¨ δγ “ T ˆ δγ,

and this agrees with the complex structure constructed in [55]. Moreover, the induced

Riemannian metric reduces to

g1,1,1,1ppδγ1, δV1q, pδγ2, δV2qq “

1

4

ż 2

0

⟨d

dsδγ1,

d

dsδγ2

⟩ds “

1

4

ż 2

0

〈δγ11, δγ12〉 dt

since 〈δV,W 〉 ” 0 for any admissible variation. This metric agrees with the Millson-Zombro

metric up to a constant. Thus the symplectic structure agrees up to a constant as well.

126

Remark 4.3.8. The complexification of LS1{S1 is LC˚{C˚. This space is a Frechet Lie

group with an obvious action on Gr2pLCq that extends the action of LS1{S1. Using the

terminology of GIT quotients [73], the stable submanifold of Gr2pLCq with respect to the

action of LS1{S1 is the set of points rΦs such that there exists an element rreiαs P LC˚{C˚

with

rreiαs ¨ rΦs P µ´1´”

~1ı¯

.

It is easy to see that the stable submanifold is Gr˝2pVq, justifying our choice of terminology.

4.3.4 Discussion of the Main Result

It is interesting to more thoroughly compare the approach given here with the original

approach of Millson and Zombro. They showed that MMZ is realized as the symplectic

reduction of LS2 by the rotation action of SOp3q. Note that LS2 is the base space of the

principal bundle

LS1 ãÑ LS3Ñ LS2

obtained by applying the loop functor to the Hopf fibration. The action of LS1 on LS3 which

appears in this fibration is a restriction of the same LS1-action on LC2 that we have been

considering in this section. Moreover, LS3 can be seen as a level set of the momentum map

of LS1.

Proposition 4.3.9. The momentum map for the natural action of LS1 on LC2 is

LC2Ñ LR

Φ “ pφ, ψq ÞÑ |φ|2 ` |ψ|2.

127

The loop space LS3 is the level set of this map for the loop constantly taking the value 1. We

conclude that

LS2“ LC2 � LS1.

Proof. The first statement can be proved by following the proof of Lemma 4.3.1 almost

exactly. The second and third statements are obvious.

Thus we have the following commutative diagram which summarizes the two approaches

to obtaining the symplectic structure of MMZ .

LC2

St2pLCq LS3

Gr2pLCq LS2

µ´1Up2qp1q µ´1

SOp3qp0q

MMZ

�LS1�Up2q{Up2q {LS1

�pLS1{S1q �SOp3q

{pLS1{S1q {SOp3q

We have thus demonstrated an infinite-dimensional Gel’fand-Macphereson correpson-

dence: the space LC2 has Hamiltonian actions by the groups Up2q and LS1, and symplecti-

cally reducing in either order produces the same space.

The content of this section is expressed informally in terms of framed paths and loops in

the next diagram. Every space in the diagram is also up to translations and the left side is

modulo scaling.

128

frame-periodicframed paths

framed loopsarclength param.f-p framed paths

relativelyframed loopsmod rotations

arclength param.tangent-periodicunframed paths

arclength param.rel. framed loops

mod rotations

arclength param.unframed loops

arclength param.unframed loopsmod rotations

arclength param.

�LS1�Up2q

closure

mod rotations& rel. framing mod framings

�pLS1{S1q �SOp3qarclength param.

mod framings

closure

mod rotations

Finally, we remark here that a similarly pleasing picture holds in the finite-dimensional

world of moduli spaces of linkages. For fixed ~r “ pr1, . . . , rnq P Rně0, let

Pol~r :“ tpolygons in R3 with edgelengths determined by ~ru{ttransl., rot., scal.u.

Kapovich and Millson showed that (for generic ~r) this space is a symplectic manifold, ob-

tained as a symplectic reduction of pS2qn by the diagonal action of SOp3q [37]. Hausmann

and Knutson showed that Pol~r is obtained as a symplectic reduction of Gr2pCnq [30]—this

viewpoint has subsequently been used quite successfully in the theory of sampling off-lattice

random walks (e.g., [10, 11]). It was shown by Howard, Manon and Millson that the full

129

Grassmannian Gr2pCnq also has a polygon-theoretic interpretation, similar to the interpre-

tations of the Grassmannians Gr2pLCq and Gr2pACq given here [32]. It will be interesting to

study whether the bending flows introduced in [37] for polygon spaces have a smooth analog.

This will be the subject of future work.

4.4 The actions of Diff`pr0, 2sq and Diff`pS1q

Let Diff`pΣq denote the group of orientation-preserving diffeomorphisms of Σ, for Σ “ r0, 2s

or S1. This is a Frechet Lie group which acts by reparameterization on the space of framed

paths when Σ “ r0, 2s and the space of framed loops when Σ “ S1. More precisely, for

ρ P Diff`pΣq, the action on a framed curve pγ, V q is defined by

pρ ¨ pγ, V qqptq “ pγpρptqq, V pρptqqq.

The action of Diff`pr0, 2sq descends to a well-defined action on S, and the action of a subgroup

of Diff`pS1q descends to M (see Section 4.4.1 for details).

We are interested in the quotient spaces with respect to these group actions. For example,

the quotient space S{Diff`pr0, 2sq can be considered as the space of unparameterized framed

paths, or the space of shapes of framed paths. For some shape recognition applications, one

wants to match unparameterized framed paths and this can be formulated as determining

geodesic distance in the quotient space. See Chapter 6 for a more thorough explanation of

the use of these spaces in the shape recognition setting.

130

4.4.1 Manifold Structures of the Quotient Spaces

Our first goal for this section is to show that the spaces

S{Diff`pr0, 2sq and M{Diff`pS1q

are smooth Frechet manifolds. We will see below that there is a technical issue in the case

of the moduli space of framed loops. However, the claim does hold for framed path space.

Proposition 4.4.1. The space S{Diff`pr0, 2sq admits the structure of a Frechet manifold.

Proof. The action of Diff`pr0, 2sq is free on S. We can take a global slice of the action by

identifying

Sarc :“ tpγ, V q P S | }γ1} ” 1u « S{Diff`pr0, 2sq

via the bijection which takes pγ, V q to its Diff`pr0, 2sq-equivalence class. This map has

an inverse obtained by taking the Diff`pr0, 2sq-equivalence class of a framed loop pγ, V q

to pγ ˝ ρ, V ˝ ρq where ρ P Diff`pr0, 2sq is the unique reparameterization so that γ ˝ ρ is

arclength-parameterized.

It follows from Lemma 3.1.7 that Sarc is double-covered via pH by PS3, which is a Frechet

manifold. Thus

Sarc « PS3{pΦ „ ´Φq,

and we claim that the latter space has a manifold structure. Indeed, this amounts to choosing

smooth cross-sections to the Z2-action Φ ÞÑ ´Φ and this can be accomplished near Φ by using

the usual manifold charts for PS3 chosen small enough to not contain antipodal points.

We now turn to the Diff`pS1q action onM, where we immediately run into two problems.

The first is that with our usual identification,

M « tframed loops pγ, V q | γp0q “ 0, lengthpγq “ 2u{trelative framing, rotationu, (4.9)

131

the natural action is not well-defined. That is, the reparameterization γ ˝ ρ does not neces-

sarily satisfy γ ˝ρp0q “ 0. This problem may be remedied by instead considering the induced

action on the image of M under the map

MÑ LpSOp3q ˆ R`q{pSOp3q ˆ R` ˆ S1q

rγ, V s ÞÑ rpT, V, T ˆW q, }γ1}s .

In the above, the circle factor in the quotient on the right corresponds to relative framing.

The induced action is

ρ ¨ rpT, V, T ˆW q, }γ1}s “ rρ1pT ˝ ρ, V ˝ ρ, T ˝ ρˆW ˝ ρq, ρ1}γ1 ˝ ρ}s .

The above approach leads to a much more serious problem since the action is not free.

As a simple example, take

pγ, V q “ ppcospπtq, sinpπtq, 0q, p0, 0, 1qq

and

ρptq “ t` θ mod 2

for any θ P p0, 2q. Then

rpT, V, T ˆW q, }γ1}s “

»

—

—

—

—

–

¨

˚

˚

˚

˚

˝

´ sinpπtq 0 cospπtq

cospπtq 0 sinpπtq

0 1 0

˛

‹

‹

‹

‹

‚

, 1

fi

ffi

ffi

ffi

ffi

fl

(4.10)

132

and

ρ ¨ rpT, V, T ˆW q, }γ1}s “

»

—

—

—

—

–

¨

˚

˚

˚

˚

˝

´ sinpπpt` θqq 0 cospπpt` θqq

cospπpt` θqq 0 sinpπpt` θqq

0 1 0

˛

‹

‹

‹

‹

‚

, 1

fi

ffi

ffi

ffi

ffi

fl

. (4.11)

After applying trigonometric identities, we see that the matrix in (4.11) differs from the one

in (4.10) by right multiplication by

¨

˚

˚

˚

˚

˝

cospπθq 0 sinpπθq

0 1 0

´ sinpπθq 0 cospπθq

˛

‹

‹

‹

‹

‚

P SOp3q.

Thus our example is stablilized in (the image of) M by the Diff`pS1q-action.

To circumvent each of these problems, we restrict the action to the reparameterization

action by the group of based diffeomorphisms

Diff`0 pS1q :“ tρ P Diff`pS1

q | ρp0q “ 0u.

Remark 4.4.2. Diffeomorphisms of S1 of the form

t ÞÑ t` θ mod 2 θ P r0, 2q

form a subgroup of Diff`pS1q which is obviously diffeomorphic to S1 and we can identify the

based diffeomorphism group with the quotient

Diff`0 pS1q « Diff`pS1

q{S1.

133

Clearly the action by the subgroup S1 on M is not free. We will see later in Theorem 5.2.8

that the points stabilized by this S1 subgroup action are exactly the critical points of a certain

natural functional on M.

The natural action of Diff`0 pS1q is well-defined on the usual model ofM (4.9). Moreover,

the action is now clearly free. We are easily able to show that the quotient space is a manifold

based on previous work.

Proposition 4.4.3. The spaceM{Diff`0 pS1q has the structure of a smooth Frechet manifold.

Proof. We concretely represent M as

M « tpγ, V q | γp0q “ 0, lengthpγq “ 2u{trelative framing, rotationu.

By Proposition 4.3.2, the space

Marc :“ tframed loops pγ, V q | γp0q “ 0, }γ1} ” 1u{trelative framing, rotationu

is a smooth Frechet manifold. We claim that M{Diff`0 pS1q can be identified with Marc.

Indeed, we define a bijection from Marc to M{Diff`0 pS1q by

rγ, V s ÞÑ rrγ, V ss,

where the double brackets on the right denote the Diff`0 pS1q orbit of rγ, V s. The inverse of

this map is

rrγ, V ss ÞÑ rγ ˝ ρ, V ˝ ρs,

where ρ is the unique based reparameterization such that γ˝ρ is arclength parameterized.

134

We note that all Riemannian metrics defined in the previous chapter are invariant un-

der the Diff`-actions—indeed, this was by construction! Thus the metrics descend to the

quotient spaces.

4.4.2 Splitting the Tangent Space

It will be useful to decompose the tangent space ofM into the subspace tangent to Diff`0 pS1q-

orbits (the vertical tangent space) and its orthogonal complement (the horizontal tangent

space). For example, we will use this decomposition in Section 4.5.4 to study the sectional

curvatures of the quotient M{Diff`0 pS1q. This task is nontrivial since M and Diff`0 pS

1q are

both infinite-dimensional. We will first decompose the tangent spaces of the simpler space

ĂM :“ tframed loops pγ, V q | γp0q “ 0u.

Since the moduli space of interest is obtained from ĂM by quotienting by a finite-dimensional

group action, a decomposition of the tangent spaces to ĂM will easily yield a decomposition

for M.

By arguments similar to those of the previous chapter, ĂM is a tame Frechet manifold

(this can be seen by, e.g., viewing ĂM as a codimension-3 submanifold of LpSOp3q ˆ R`q)

with tangent spaces

Tpγ,V q ĂM “ tpδγ, δV q P LpR3ˆ R3

q | 〈δV, V 〉 “ 〈δγ1, V 〉` 〈γ1, δV 〉 ” 0, δγp0q “ 0u.

The diffeomorphism group Diff`0 pS1q acts on ĂM by reparameterization. The framed curve

elastic metric g1,1,1,1 is well-defined on ĂM, and going through the proof of Lemma 3.1.4, one

135

sees that it reduces to

g1,1,1,1pγ,V q ppδγ1, δV1q, pδγ2, δV2qq “

1

2`pγq

ż 2

0

⟨d

dsδγ1,

d

dsδγ2

⟩` 〈δV1,W 〉〈δV2,W 〉 ds,

where `pγq is the length of γ.

We are now prepared to state the splitting result for ĂM. We use the notation

L0R :“ tξ P LR | ξp0q “ 0u

for the based loop space of R.

Proposition 4.4.4. The subspace of Tpγ,V q ĂM which is tangent to the Diff`0 pS1q-orbits is

T vertpγ,V qĂM :“ tpξT, ξp´κ1T ` twW qq | ξ P L0Ru,

where T is the unit tangent to γ, W “ T ˆ V , κ1 is the first Darboux curvature of pγ, V q

and tw is the twist rate of pγ, V q.

The orthogonal complement with respect to g1,1,1,1pγ,V q is

T horpγ,V qĂM :“

"

pδγ, δV q P Tpγ,V q ĂM |

⟨d2

ds2δγ, T

⟩´ 〈δV,W 〉 tw ” 0

*

.

The tangent space Tpγ,V q ĂM splits as

Tpγ,V q ĂM “ T vertpγ,V qĂM‘ T horpγ,V q

ĂM.

The proof of the proposition requires a technical lemma.

Lemma 4.4.5. . Let pγ, V q be a framed loop, let d{ds denote derivative with respect to

arclength of γ, let κ and tw denote the curvature and twist rate of γ, respectively. The

136

operator L defined by

L : L0RÑ LR

ξ ÞÑd2

ds2ξ ´ pκ2

` tw2q ¨ ξ

is invertible.

Proof. We first show that L is injective. This follows because the unique solution to the

second-order linear ODE

d2

ds2ξ ´ pκ2

` tw2q ¨ ξ “ 0

with boundary conditions ξp0q “ ξp2q “ 0 is ξ “ 0.

Now we wish to show that L is onto. We first note that L is a degree-2 elliptic operator.

Indeed,

Lpξq “d2

ds2ξ ´ pκ2

` tw2qξ

“1

}γ1}

d

dt

1

}γ1}

d

dtξ ´ pκ2

` tw2qξ

“1

}γ1}2ξ2 ´

〈γ2, γ1〉}γ1}4

ξ1 ´ pκ2` tw2

qξ.

Following an argument in the proof of Lemma 4.5 in [53], we claim that the index of L,

indexpLq “ dim kerL´ dim cokerL,

is zero. Since L is injective, this implies that its cokernel is zero, hence L is onto. This

follows because L is homotopic to the invertible operator

ξ ÞÑ ξ2 ´ ξ, (4.12)

137

through perturbations by elliptic operators and index is invariant under such homotopies.

The index of (4.12) is zero (see [53]), thus the proof is complete.

We now proceed with the proof of Proposition 4.4.4.

Proof of Proposition 4.4.4. Let ρε be a path in Diff`0 pS1q with parameter ε such that ρ0 is

the identity and let ξ “ ddε

ˇ

ˇ

ε“0ρε so that

ξ P L0R,

which is the Lie algebra of Diff`0 pS1q (strictly speaking, the Lie algebra of Diff`0 pS

1q is the

space of vector fields on S1 with the zero vector at 0, but this is identified with LR by

comparing to the unit vector field). Then tangent vectors along the Diff`0 pS1q-orbit of pγ, V q

take the form

d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

pγpρεq, V pρεqq “ pγ1pρ0qξ, V

1pρ0qξq

“

ˆ

ξ}γ1}d

dsγ, ξ}γ1}

d

dsV

˙

“ pξ}γ1}T, ξ}γ1}p´κ1T ` twW qq ,

thus

T vertpγ,V qĂM “ tpξ}γ1}T, ξ}γ1}p´κ1T ` twW qq | ξ P L0Ru “ tpξT, ξp´κ1T ` twW qq | ξ P L0Ru.

To determine the horizontal subspace, we note that pδγ, δV q is horizontal if and only if

g1,1,1,1pγ,V q ppδγ, δV q, pξT, ξp´κ1T ` twW qqq “ 0

138

for all ξ P L0R. Writing this out, this means that for all ξ we have

0 “1

2`pγq

ż 2

0

⟨d

dsδγ,

d

dsξT

⟩` 〈δV,W 〉〈ξp´κ1T ` twW q,W 〉 ds

“1

2`pγq

ż 2

0

⟨d

dsδγ,

ˆ

d

dsξ

˙

T ` ξpκ1V ` κ2W q

⟩` 〈δV,W 〉 ξtw ds

“1

2`pγq

ż 2

0

d

dsξ

⟨d

dsδγ, T

⟩` ξ

ˆ⟨d

dsδγ, κ1V ` κ2W

⟩` 〈δV,W 〉 tw

˙

ds

“1

2`pγq

ż 2

0

ξ

ˆ

´

⟨d2

ds2δγ, T

⟩` 〈δV,W 〉 tw

˙

ds,

where we have integrated by parts to obtain the last line. This condition holds for every

ξ P L0R if and only if ⟨d2

ds2δγ, T

⟩´ 〈δV,W 〉 tw ” 0,

and we have established the claimed characterization of the horizontal tangent space.

Finally, we wish to show that the tangent space splits. Assuming that we have such a

projection, we can express an arbitrary tangent vector as

pδγ, δV q “ pξT, ξp´κ1V ` twW qq ` pδγhor, δV horq (4.13)

where

pδγhor, δV horq P T horpγ,V q

ĂM

and ξ P L0R. From (4.13) we conclude

⟨d2

ds2δγ, T

⟩“

d2

ds2ξ ´ ξκ2

`⟨δγhor, T

⟩, (4.14)

where κ2 “ κ21 ` κ

22 is the squared curvature of γ. On the other hand,

〈δV,W 〉 “ ξtw `⟨δV hor,W

⟩.

139

Multiplying this expression by tw and subtracting the result from (4.14) yields

⟨d2

ds2δγ, T

⟩´ 〈δV,W 〉 tw “ d2

ds2ξ ´ ξκ2

`⟨δγhor, T

⟩´ ξtw2

´⟨δV hor,W

⟩tw.

Utilizing the horizontality of pδγhor, δV horq, this reduces to

⟨d2

ds2δγ, T

⟩´ 〈δV,W 〉 tw “ d2

ds2ξ ´ pκ2

` tw2qξ “ Lpξq,

where L is the invertible linear operator from Lemma 4.4.5. Therefore we define

ξ :“ L´1

ˆ⟨d2

ds2δγ, T

⟩´ 〈δV,W 〉 tw

˙

.

This gives us a well-defined projection onto T vertpγ,V q

ĂM, thus proving that the tangent spaces

split.

Corollary 4.4.6. There is a well-defined projection

Trγ,V sMÑ T vertrγ,V sM,

where T vertrγ,V sM is the space of tangent vectors to Diff`0 pS

1q-orbits.

Proof. Since M “ ĂM{pSOp3q ˆ R`q is the image of a submersion with finite-dimensional

fibers, we can identify Trγ,V sM with a finite codimension subspace of Tpγ,V q ĂM. Thus we can

first project onto T vertpγ,V q

ĂM using Proposition 4.4.4, then project onto the finite codimension

subspace.

140

4.4.3 Diff`-Actions in Complex Coordinates

We conclude this section with a description of the Diff`-actions in complex coordinates. We

will once again focus on the case of loops, so that we will work in the Stiefel and Grassmann

manifolds. The case of framed paths is similar.

Lemma 4.4.7. The action of Diff`0 pS1q on St2pVq corresponding to the one described above

is

ρ ¨ Φ :“a

ρ1Φpρq.

This induces a well-defined action on Gr2pVq.

Proof. Let pHpΦq “ pγ, V q. We wish to show that?ρ1Φpρq maps under pH to pγpρq, V pρqq.

Since we are working up to translations, it suffices to show that this holds at the level of

LpSOp3q ˆ R`q; i.e. that

Hpa

ρ1Φpρqq “ pρ1γ1pρq, V pρqq

This is easy to see using quaternionic coordinates Φ Ø q. We have

Hpa

ρ1qpρqq “

˜

Impa

ρ1qpρqia

ρ1qpρqq,Imp

?ρ1qpρqj

?ρ1qpρqq

}?ρ1qpρq}2H

¸

“

˜

Impqpρqiqpρqqρ1,Impqpρqjqpρqq

}qpρq}2H

¸

“ pρ1γ1pρq, V pρqq.

The second claim follows by noting that if Φ0 and Φ1 differ by right multiplication by an

element of Up2q then ρ ¨ Φ0 and ρ ¨ Φ1 do as well, so that rρ ¨ Φs is well-defined.

We now describe the horizontal and vertical tangent spaces in these coordinates. Since

we have already used T horSt2pVq to denote the horizontal space with respect to the Up2q-

141

action, we will use T hor1

St2pVq and T vert1

St2pVq to denote the horizontal and vertical spaces

with respect to the Diff`0 pS1q-action.

Corollary 4.4.8. The tangent space of St2pVq decomposes as

TΦSt2pVq “ T hor1

Φ St2pVq ‘ T vert1

Φ St2pVq,

where T vert1

Φ St2pVq is the space of tangents to Diff`0 pS1q orbits, described explicitly as

T vert1

Φ St2pVq :“ tp1{2qξ1Φ` ξΦ1 | ξ P LRu,

and T hor1

Φ St2pVq is its orthogonal complement with respect to gL2, described explicitly as

T hor1

Φ St2pVq :“ tδΦ P TΦSt2pVq | Rep〈δΦ,Φ1〉C2 ´ 〈δΦ1,Φ〉C2q ” 0u.

Similarly,

TrΦsGr2pVq “ T hor1

rΦs Gr2pVq ‘ T vert1

rΦs Gr2pVq,

where T hor1

rΦs Gr2pVq (respectively, T vert1

rΦs Gr2pVq) is the orthogonal projection of T hor1

Φ St2pVq

(respectively, T vert1

Φ St2pVq) to TrΦsGr2pVq.

Proof. The statements that the tangent spaces split follows from Proposition 4.4.4, thus we

only need to show that the vertical and horizontal spaces are as described. Let ρε be a path

in Diff`0 pS1q such that ρ0 is the identity, and let ξ denote d

dε

ˇ

ˇ

ε“0ρε. Then

d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

?ρεΦpρεq “

1

2?ρ0

ξΦpρ0q `?ρ0Φ1pρ0qξ “

1

2ξΦ` ξΦ1.

This proves that the vertical space is as described.

142

A tangent vector δΦ “ pδφ, δψq lies in the horizontal space if and only if

0 “ gL2

ˆ

δΦ,1

2ξ1Φ` ξΦ1

˙

“ Re

ż 2

0

⟨δΦ,

1

2ξ1Φ` ξΦ1

⟩C2

dt

“ Re

ż 2

0

´1

2ξ p〈δΦ1,Φ〉C2 ` 〈δΦ,Φ1〉C2q ` ξ 〈δΦ,Φ1〉C2 dt

“ Re

ż 2

0

1

2ξ p〈δΦ,Φ1〉C2 ´ 〈δΦ1,Φ〉C2q dt

holds for all ξ P LR. This is true if and only if

Re p〈δΦ,Φ1〉C2 ´ 〈δΦ1,Φ〉C2q ” 0.

The claims that the tangent spaces split follow as corollaries of Proposition 4.4.4.

4.5 Riemannian Geometry of Framed Path and Loop

Spaces

4.5.1 Explicit Geodesics in Framed Path Space

A major motivation for the framework developed in the previous chapter is for applications to

elastic shape matching. This is a vibrant subfield of computer vision where shapes are treated

as points in an infinite-dimensional “shape manifold”, and shape similarity is measured by

geodesic distance in the shape manifold. In the following sections, we describe the geodesics

of the shape spaces S andM. The details of their application to shape recognition are treated

later in Chapter 6. For a more general overview of the field of elastic shape matching, see,

e.g., the survey articles [69, 77] and the references therein.

143

The geodesics of

S “ tframed pathsu{ttranslation, scalingu

are particularly easy to describe by virtue of Theorem 3.1.12. We recall that the theorem

says that S is locally isometric to an L2 sphere—in particular, it is isometrically double

covered by

S˝pPC2q “ tΦ P PpC2

zt0uq | 〈Φ,Φ〉L2 “?

2u Ă SpPC2q,

which is an open subset of the radius-?

2 L2-sphere in PC2, denoted SpPC2q. It follows that

the geodesics of S are exactly the geodesics of the sphere SpPC2q. There is a technical issue

in that geodesics joining points of S˝pPC2q do not necessarily stay in S˝pPC2q. However,

geodesic distance in the full L2-sphere is still a well-defined, geometrically-motivated distance

measure for S. Moreover, it is the case that geodesics of SpPC2q between elements of S˝pPC2q

tend to stay in S˝pPC2q. Heuristically, this is because a geodesic of SpPC2q can be thought

of as special homotopies between paths in C2. Then the geodesic staying in S˝pPC2q means

that the special homotopy never passes through 0 P C2, and this is a highly generic condition.

As SpPC2q is a round sphere in a vector space, its geodesics are great circles. These are

parameterized explicitly as follows. Let Φ0,Φ1 P SpPC2q and define

θ “ θpΦ0,Φ1q “ arccos1

2

ż 2

0

Re 〈Φ0,Φ1〉C2 dt.

Then geodesic distance in SpPC2q is given by?

2θ and the geodesic segment connecting Φ0

to Φ1 in SpPC2q is

Φu “sinpp1´ uqθq

sinpθqΦ0 `

sinpuθq

sinpθqΦ1.

144

4.5.2 Explicit Geodesics in Framed Loop Space

Now we turn to the slightly more involved task of explicitly describing the geodesics of M

with respect to its natural metric. By Theorem 3.2.8, these are the geodesics of Gr2pVq for

V “ LC2 orAC2. We once again run into the technical issue here thatM is actually isometric

to the disjoint union of stable Grassmannians, so that geodesics in the Grassmanians can

potentially correspond to paths in framed loop space which pass through singular framed

loops. We once again note that geodesic distance in the full Grassmannian gives a well-

defined metric onM and, by the same heuristic argument of the previous section, geodesics

between points of the stable Grassmannian tend to stay in the stable Grassmannian.

The description of the geodesics in Gr2pVq given here can be traced back to Neretin [59],

who gave a description of the geodesics of the finite-dimensional Grassmannians GrkpRnq

which is amenable to adaptation to the infinite-dimensional setting. The Neretin geodesics

were already adapted to describe the geodesics in the space of immersed plane curves in [76].

Let rΦ0s, rΦ1s P Gr2pVq—recall that these equivalence classes represent complex 2-planes

in V , where we have chosen particular orthonormal bases Φj for each subspace. The basic

description of the geodesic rΦus joining these points is to find the optimal bases for each

2-plane and then to simply interpolate. This process is described completely explicitly by

the following steps.

1. Write the projection map π : rΦ0s Ñ rΦ1s as a 2 ˆ 2 matrix in terms of the chosen

orthonormal bases.

2. Compute the singular value decomposition of the matrix π. This produces new or-

thonormal bases rΦ0 “

´

φ0, ψ0

¯

for rΦ0s and rΦ1 “

´

φ1, ψ1

¯

for rΦ1s such that πpφ0q “

λφφ1 and πpψ0q “ λψψ1, where 0 ď λφ, λψ ď 1.

3. Let θφ “ arccospλφq and θψ “ arccospλψq. These are the Jordan angles of rΦ0s and

rΦ1s.

145

Figure 4.3: Geodesic in M joining a p2, 3q-torus knot to a p2, 5q-torus knot.

4. If θφ, θψ ‰ 0, then the geodesic joining the subspaces is given by rΦus, where Φu “

pφu, ψuq is described by the formulas

φuptq “sinpp1´ uqθφqφ0ptq ` sinpuθφqφ1ptq

sin θφ

ψuptq “sinpp1´ uqθψqψ0ptq ` sinpuθψqψ1ptq

sin θψ.

If θφ “ 0 (i.e., πpφ0q “ φ1, which implies φ0 “ φ1), then the geodesic is simply given

by φuptq “ φ0 and likewise for ψu.

We also note that the geodesic distance between rΦ0s and rΦ1s is given explicitly by

dist “b

θ2φ ` θ

2ψ.

An example geodesic is shown in Figure 4.5.2.

146

This description of the geodesics of Gr2pVq has a useful corollary. For any isometric

embedding L : Cn Ñ V (with respect to the standard Hermitian metric on Cn and the

Hermitian L2-metric 〈¨, ¨〉L2 on V), we obtain an induced map

L˚ : Gr2pCnq Ñ Gr2pVq

spantw1, w2u ÞÑ spantLpw1q, Lpw2qu

for any linearly independent pair w1, w2 P Cn. In other words, Gr2pCnq embeds as Gr2pimagepLqq.

The following corollary shows that this is a totally geodesic embedding and follows essen-

tially immediately from our description of geodesics above. We note that a similar result

was shown to hold for Stiefel manifolds of real Hilbert spaces in [29].

Corollary 4.5.1. Let L : Cn ãÑ V be an isometric embedding as described above. Then

the induced map L˚ : Gr2pCnq ãÑ Gr2pVq is an isometric embedding as a totally geodesic

submanifold with respect to the standard induced metric on Gr2pCnq and the induced metric

gL2

on Gr2pVq.

Proof. One can easily show that the induced map L˚ is an embedding by representing L

explicitly in terms of a choice of orthonormal basis. The embedding takes the horizontal

tangent spaces of St2pCnq to the tangent spaces to the embedded Grassmannian, so the

assumption that L was an isometry implies that that L˚ is an isometric embedding. Thus

it remains to show that Gr2pCnq embeds as a totally geodesic submanifold.

Let rΦ0s and rΦ1s be elements of the image of L˚ and let rΦus “ rφu, ψus denote the

geodesic joining them in Gr2pVq, as described above. With the interpretation that the

embedded Grassmannian is Gr2pLpCnqq with metric induced by Gr2pVq, we see that the

geodesic in L˚pGr2pCnqq is obtained by following the same procedure above. Thus the image

of Gr2pCnq is totally geodesic.

147

As a sanity check, we will explicitly show that φu and ψu each lie in the image of L for all

u. First note that φ0, ψ0, φ1 and ψ1 all lie in the image of L. Indeed, since rΦ0s P imagepL˚q,

φ0 and ψ0 lie in imagepLq, hence φ0 does as well, and the same argument holds for the other

vectors. Then φu lies in the image of L for all u, as φu P spantφ0, φ1u. Therefore rΦus lies in

the image of L˚ for all u.

We can take this line of thought further and show that every collection of planes is

contained in a totally geodesic copy of some finite-dimensional Grassmannian. This aligns

with our point of view that much of the geometry of Gr2pVq has a finite-dimensional flavor.

Proposition 4.5.2. Let rΦ1s, . . . , rΦks be distinct elements of Gr2pVq. There exists an em-

bedded finite-dimensional Grassmannian which contains every rΦjs.

Proof. Let V denote the linear subspace of LC of minimal dimension which contains all planes

rΦ1s, . . . , rΦks and let dimpV q “ n ď 2k. Choose an isometric embedding L : Cn Ñ LC such

that imagepLq “ V . Then L˚pGr2pCnqq is a totally geodesic Grassmannian which contains

every plane rΦjs.

In particular, it follows that every geodesic triangle of Gr2pVq is contained in a finite-

dimensional totally geodesic embedded Grassmannian. The heuristics of Toponogov’s The-

orem (see, e.g., [12] Theorem 2.2) suggest that the sectional curvatures of Gr2pCnq and

Gr2pVq should be comparable. We will use a similar idea to describe the curvature of Gr2pVq

in Section 4.5.4.

To implement this framework as an elastic shape matching algorithm, it remains to com-

pute distance in S{Diff`pr0, 2sq andM{Diff`pS1q—i.e., to optimize over reparameterization.

This is achieved via dynamic programming and is treated in detail in Chapter 6.

148

4.5.3 The Exponential Map

The aim of this section is to describe geodesics of St2pVq and Gr2pVq in terms of initial

position and velocity. We once again obtain explicit geodesics. This is accomplished by

determining the geodesic equation of St2pVq and showing that it is integrable with an explicit

solution. The derivation given here adapts the finite-dimensional real version found in [16].

The results at the end of the section are proved by methods similar to those in [29] for the

real Hilbert space case.

In this section, it will be useful to identify St2pVq with a space of linear maps C2ˆC2 Ñ

V2. We first consider Φ “ pφ, ψq P V2 as a linear map by using matrix multiplication on the

right:

pφ, ψq

¨

˚

˝

z1 w1

z2 w2

˛

‹

‚

“ pφz1 ` ψz2, φw1 ` ψw2q.

The formal adjoint of Φ to be the linear map Φ˚ : V2 Ñ C2 ˆ C2 defined by

Φ˚pφ1, ψ1q “

¨

˚

˝

〈φ1, φ〉L2 〈ψ1, φ〉L2

〈φ1, ψ〉L2 〈ψ1, ψ〉L2

˛

‹

‚

.

This adjoint is formal in the sense that we do not claim that any properties of the adjoint

in a Hilbert space carry over. However, it will be useful later to note that some of the usual

basic properties of an adjoint hold for this definition. In particular, for any

A “

¨

˚

˝

a b

c d

˛

‹

‚

P C2ˆ C2,

149

we have

pΦAq˚pφ1, ψ1q “ paφ` cψ, bφ` dψq˚pφ1, ψ1q

“

¨

˚

˝

〈φ1, aφ` cψ〉L2 〈ψ1, aφ` cψ〉L2

〈φ1, bφ` dψ〉L2 〈ψ1, bφ` dψ〉L2

˛

‹

‚

“

¨

˚

˝

a 〈φ1, φ〉L2 ` c 〈φ1, ψ〉L2 a 〈ψ1, φ〉L2 ` c 〈ψ1, ψ〉L2

b 〈φ1, φ〉L2 ` d 〈φ1, ψ〉L2 b 〈ψ1, φ〉L2 ` d 〈ψ1, ψ〉L2

˛

‹

‚

“ A˚Φ˚pφ1, ψ1q,

so that pΦAq˚ “ A˚Φ˚ as linear maps V2 Ñ C2 ˆ C2.

With this formalism, the Stiefel manifold can be expressed as

St2pVq “ tΦ P V2| Φ˚Φ “ Id2ˆ2u (4.15)

and the tangent space to St2pVq at Φ is

TΦSt2pVq “ tδΦ P V2| Φ˚δΦ P up2qu.

Indeed, this is a direct translation of the description of TΦSt2pVq given in Lemma 4.1.1. For

points Φ1 P St2pVq, Φ2 P V2 and A P C2 ˆ C2, it therefore holds that

Φ1A “ Φ2 ô A “ Φ˚1Φ2.

Now we note that since St2pVq is a Riemannian submanifold of V2 with a flat metric, the

geodesics of St2pVq are paths Φu such that B2

Bu2Φu lies in the normal direction to St2pVq for all

u. Thus it will be useful to describe the normal spaces of St2pVq explicitly. In the following

150

lemma, we also give a description of the normal space to the Up2q-horizontal tangent space

of St2pVq, as this will be useful later on.

Lemma 4.5.3. Let Φ P St2pVq. The normal space to TΦSt2pVq Ă TΦV2 has L2-orthonormal

basis

pφ, 0q, p0, ψq,1?

2pψ, φq,

1?

2píψ, iφq. (4.16)

Let rΦs P Gr2pVq. The normal space to TrΦsGr2pVq « T horΦ St2pVq Ă TΦSt2pVq has orthonor-

mal basis

piφ, 0q, p0, iψq,1?

2piψ, iφq,

1?

2pψ,´φq. (4.17)

Proof. One simply needs to check that the vectors in the first list are orthonormal and that

they are each orthogonal to TΦSt2pVq. This is sufficient, since St2pVq is codimension-4 inW .

We leave it to the reader to check the first point and note that the second point is also easy

by our characterization of the tangent spaces of the Stiefel manifolds given in Lemma 4.1.1.

For example, let pδφ, δψq P TΦSt2pVq. Then

Re 〈pφ, 0q, pδφ, δψq〉PC2 “ Re

ż 2

0

φδφ dt “ 0.

The remaining vectors are also easy to check.

To prove the second statement, we note that orthonormality follows by the first statement

as each vector in the second list is obtained by applying the complex structure to a vector in

the first list. Moreover, the fact that the Grassmannian is a Kahler reduction of the Stiefel

manifold implies that the vectors in the seond list are normal to the horizontal tangent space

of the Stiefel manifold. Alternatively, this can be proved by explicit compuatations using

the characterization of the tangent spaces to the Grassmannian given in Corollary 4.1.2.

151

It is straightforward to check that the normal space to TΦSt2pVq can then be written as

tδΦ P V2| Im 〈φ, δφ〉L2 “ Im 〈ψ, δψ〉L2 “ 〈φ, δψ〉L2 ´ 〈δφ, ψ〉L2 “ 0u. (4.18)

Indeed, this set is real codimension-4 in V2 and it contains each vector in the list (4.16).

Using the description (4.15) of St2pVq, the set (4.18) translates to

tδΦ P V2| Φ˚δΦ is Hermitianu.

We are now prepared to describe the geodesics of St2pVq. For a path Φu in St2pVq, we

will use dots to denote differentiation with respect to u; that is,

9Φu :“B

BuΦu.

Proposition 4.5.4. The geodesic equation for St2pVq with its L2 metric is

:Φu ` Φu9Φ˚u 9Φu “ 0.

Thus the geodesic Φu with Φ0 “ Φ and 9Φu “ δΦ, say with }δΦ}L2 “ 1, is given by

Φu “ pΦ, δΦq expu

¨

˚

˝

Φ˚δΦ ´δΦ˚δΦ

Id2ˆ2 Φ˚δΦ

˛

‹

‚

Id4ˆ2 exppúΦ˚δΦq, (4.19)

where pΦ, δΦq is treated as a map C4 ˆ C4 Ñ V2. Therefore the exponential map of St2pVq

is well-defined.

152

Proof. As stated above, the geodesics of St2pVq are those paths Φu such that :Φu lies in the

normal space to Φu for all u. Taking two u-derivatives of the defining of St2pVq yields

B2

Bu2Φ˚uΦu “

B2

Bu2Id2ˆ2,

so that

:Φ˚uΦ` 2 9Φ˚u 9Φu ` Φ˚u:Φ “ 0. (4.20)

If :Φu lies the normal space to Φu, then

Φ˚u:Φu “ A

for some Hermitian 2ˆ 2 matrix A. We claim that this can be rewritten as

:Φu “ ΦuA. (4.21)

This holds because ΦuΦ˚u can be viewed as the linear map V2 Ñ V2 which linearly projects

each coordinate of an element of V2 to the span of φu and ψu in V . Since :Φu is normal to

the Stiefel manifold, the explicit desription of the normal spaces in Lemma 4.5.3 shows that

each coordinate of :Φu already lies in the span of φu and ψu. Thus ΦuΦ˚u

:Φu “ :Φu.

Replacing (4.21) into (4.20) and multiplying on the left by Φu, we have

0 “ ΦupΦuAq˚Φu ` 2Φu

9Φ˚u 9Φu ` ΦuΦ˚uΦuA

“ ΦuA˚Φ˚uΦu ` 2Φu

9Φ˚u 9Φu ` ΦuA

“ ΦuA` 2Φu9Φ˚u 9Φu ` ΦuA

“ 2:Φu ` 2Φu9Φ˚u 9Φu.

153

Now we wish to show that (4.19) is the solution to the geodesic equation with the given

initial conditions. Let Su “ 9Φ˚u 9Φu, interpreted as a path of 2ˆ 2 complex matrices. Then

9Su “ :Φu9Φu ` 9Φ˚u:Φu

“

´

´Φu9Φ˚u 9Φu

¯˚9Φu ´ 9Φ˚u

´

Φu9Φ˚u 9Φu

¯

“ ´SuΦ˚u

9Φu ´ 9Φ˚uΦuSu.

Now define Tu “ Φ˚u 9Φu. Then Tu is a skew symmetric matrix, since 9Φu is a tangent vector

to St2pVq. Moreover, it is straightforward to check that

T ˚u “9Φ˚uΦu.

Thus we have

9Su “ TuSu ´ SuTu. (4.22)

Next we note that

9Tu “ 9Φ˚u 9Φu ` Φ˚u:Φu

“ 9Φ˚u 9Φu ´ Φ˚uΦu9Φ˚u 9Φu

“ 9Φ˚u 9Φu ´ 9Φ˚u 9Φu

“ 0.

Thus Tu is constant and we let T :“ T0. Then (4.22) becomes the ODE

9Su “ TSu ´ SuT,

154

which has the unique solution

Su “ exppuT qS0 exppúT q.

Therefore the geodesic equation becomes

:Φu “ ´Φu exppuT qS0 exppúT q.

This can be rewritten as the integrable equation

d

du

´

Φu exppuT q, 9Φu exppuT q¯

“

´

Φu exppuT q, 9Φu exppuT q¯

¨

˚

˝

T ´S0

Id2ˆ2 T

˛

‹

‚

.

It is easy to see that (4.19) is the unique solution with the given initial conditions.

Lemma 4.5.5. For any path Φu in St2pVq, there exists a path Au in Up2q so that ΦuAu is

horizontal.

Proof. From the description of TrΦsGr2pVq « T horΦ St2pVq in Corollary 4.1.2, we rewrite the

horizontal space condition in the language of this section as

T horΦ St2pVq “ tδΦ P V2| Φ˚δΦ “ 0u.

Thus ΦuAu is horizontal if and only if

0 “ Φ˚B

BuΦuAu “ Φ˚ 9ΦuAu ` Φ˚Φu

9Au.

We then solve the problem by taking Au as a solution of the ODE 9Au “ ´Φ˚u 9ΦuAu.

155

Now we note that if a geodesic in St2pVq has initial data pΦ, δΦq with δΦ P T horΦ St2pVq,

then the geodesic stays horizontal. Otherwise it could be shortened by applying the pro-

jection of Lemma 4.5.5. We conclude that a geodesic in St2pVq with horizontal initial data

represents a geodesic in Gr2pVq. The next proposition follows immediately.

Proposition 4.5.6. The exponential map exp : TrΦsGr2pVq Ñ Gr2pVq is well-defined.

4.5.4 Sectional Curvatures

To get an idea of how the geodesics inM andM{Diff`0 pS1q behave, it is useful to characterize

the sectional curvatures of each space with respect to our metric; this is the content of our

next main theorem, which states that each space is nonnegatively curved. This result is

similar to Theorem 5.3 in [76], where the sectional curvature for the shape space of planar

loops is shown to be nonnegative. The proof strategy used here is different than that of [76].

The proof relies on O’Neill’s formula, which requires some setup to state. Let M be a

Riemannian manifold with a smooth free action by a Lie group G, so that π : M Ñ M{G

is a submersion. Then we note that any tangent vector to M{G at rps can be expressed as

DπppXq for some X P TpM . The tangent space TpM decomposes as

TpM “ T horp M ‘ T vertp M,

where T vertp M is the kernel of Dπp and T horp M is its orthogonal complement. For X P TpM ,

we denote its decomposition as

X “ Xhor`Xvert.

We are now prepared to state O’Neill’s theorem.

156

Theorem 4.5.7 (O’Neill [63]). Let π : M Ñ M{G be a Riemannian submersion, as above.

Let DπppXq, DπppY q P TrpsM{G be orthonormal with respect to the induced metric, where

X, Y P TpM . Then the sectional curvature of M{G at rps for the plane spanned by X and

Y is given by

secpXhor, Y horq `

3

4

›

›

›

“

Xhor, Y hor‰vert

›

›

›

2

g,

where sec is sectional curvature computed in M and r¨, ¨s is the Lie bracket of M .

Going through the proof of O’Neill’s formula, we see that it extends to infinite-dimensional

Riemannian submersions, provided all terms are well-defined (also see [76]). We are in par-

ticular considering the submersion M ÑM{Diff`0 pS1q. By Corollary 4.4.8, the horizontal

and vertical projections are well-defined. Moreover, we can compute the Lie bracket in M

easily via its correspondence with the Grassmannian. Given vector fields X, Y on Gr2pVq,

we can locally think of these as horizontal (with respect to the Up2q-action) vector fields on

LC2. Thus we can consider X and Y as smooth functions from LC2 to itself. Thus their

bracket in LC2 is given by

rX, Y sΦ :“ DXpΦqpY |Φq ´DY pΦqpX|Φq.

This bracket can then be projected to a Up2q-horizontal vector field, thus giving us the

bracket on Gr2pVq.

Theorem 4.5.8. The spacesM andM{Diff`0 pS1q both have non-negative sectional curva-

ture.

Proof. By Theorem 3.2.8, M is locally isometric to Gr2pVq, and it suffices to compute the

curvature of the Grassmannian to prove the first part of the theorem. Assuming that M

has non-negative sectional curvature, it follows by O’Neill’s formula thatM{Diff`0 pS1q does

as well, since MÑM{Diff`0 pS1q is a Riemannian submersion by construction.

157

We proceed with proving that Gr2pVq is nonnegatively curved. Let rΦs P Gr2pVq and let

δΦ1, δΦ2 P TrΦsGr2pVq be linearly independent and assume without loss of generality that

each vector is L2-normalized. Using the results of the previous section, and in particular

Proposition 4.5.6, choose a point rΦjs along the geodesic through rΦs with velocity vector

δΦj for j “ 1, 2. These points can be chosen to be distinct, by the explicit description

of geodesics given in Proposition 4.5.4. Next we use Proposition 4.5.2 to choose a totally

geodesic isometrically embedded finite-dimensional Grassmannian containing rΦs, rΦ1s and

rΦ2s. In particular, this Grassmannian contains rΦs and contains each δΦj in its tangent

space. Thus the sectional curvatures of Gr2pVq and the finite dimensional Grassmannian

agree at that point and plane. Finite-dimensional Grassmannians are well known to be

nonnegatively curved as they are compact symmetric spaces, thus the theorem follows.

158

Chapter 5

Application: Critical Points of Energy

Functionals

5.1 Background

In this chapter we consider a pair of natural energy functionals on the moduli space M of

framed loops. Before introducing the functionals, we give some historical context for our

results.

5.1.1 Elastic Energies

The study of critical points of energy functionals on curve spaces dates back to James

Bernoulli in the 17th century. Bernoulli sought to describe the equillibrium shape taken

by a springy wire with fixed boundary conditions. The problem was later solved by Euler,

who, following a suggestion of David Bernoulli, reformulated the problem as determining the

critical points of the energy functional

γ ÞÑ

ż

Σ

κpγq2 ds,

159

where γ : Σ Ñ R2 is an arclength-parameterized plane curve with fixed boundary conditions

(Σ is S1 or a closed interval). Accordingly, critical points of this functional are referred to

as Euler elastica.

Euler utilized variational techniques to give a complete characterization of the elastica. Of

particular interest are the Euler elastica with periodic boundary conditions; i.e., γ is a smooth

closed curve. By the Whitney-Graustein theorem [75], the space of arclength-parameterized

closed planar curves has countably many components—immersed plane curves are homotopic

through immersions if and only if they have the same winding number. For each nonzero

winding number component, the total squared curvature functional has a multiply-covered

oriented round circle as its only critical point, up to rotations and translations. On the

other hand, the zero winding number component has countably many critical points up to

rotation and translation. These are the multiply-covered “Bournoulli 8-curves”. See [44]

for an in-depth historical account of the elasticae and [66] for a modern derivation of the

periodic critical points.

An obvious generalization of the Euler elastica is to consider critical points of the total

squared curvature functional on arclength-parameterized space curves, which we continue to

refer to as elastica. A characterization of the closed elastica in R3 is given by Langer and

Singer [42]. Surprisingly, they showed that the knotted elastica are exactly the ph, kq-torus

knots satisfying h ą 2k.

The Euler elastica problem treats the original physical problem by considering the springy

wire as a 1-dimensional object. A more natural physical model is to treat the wire as having

a thickness. In this case, the wire is more readily modelled by a framed curve pγ, V q, where

γ is an arclength-parameterized curve. A relevant energy functional in this setup is the

Kirchhoff elastic energy

pγ, V q ÞÑ

ż

Σ

κpγq2 ` twpγ, V q2 ds.

160

Once again, the aim is to find critical points of this functional amongst framed curves

with fixed boundary conditions and such critical points are referred to as elastic rods. The

Kirchhoff elastic energy functional penalizes both bending and twisting of the rod. The

study of elastic rods is a vast topic, frequently covering more general forms of Kirchhoff

elastic energy than the one given here (see e.g. [45] and references therein).

The characterization of closed elastic rods was achieved by Ivey and Singer in [35], where

they showed that every torus knot type is realized by the base curve of an elastic rod. More-

over, they showed that for any relatively prime h and k, there exists a homotopy of elastic

rods from the h-times-covered circle to the k-times-covered circle possessing certain inter-

esting features (e.g., there is exactly one self-intersecting elastic rod between the beginning

and end of the homotopy).

5.1.2 Vortex Filament Equation

Another interesting and relevant energy functional is

γ ÞÑ1

2

ż

Σ

}γ1}2 dt, (5.1)

where γ is an arclength-parameterized space curve. As first noticed by Marsden and We-

instein [49], this energy functional is related to the evolution of a vortex filament. We now

demonstrate this fact somewhat informally. Let δγ be a variation of γ. Then the derivative

of the functional at γ in the δγ-direction is

d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

1

2

ż

Σ

}γ1 ` εδγ1}2 dt “

ż

Σ

〈γ1, δγ1〉 dt “ ´

ż

Σ

〈γ2, δγ〉 dt.

Using the Millson-Zombro Kahler structure on the space of (isometry classes of) arclength-

parameterized loops, we see that if γ P LR3, then the gradient of this functional is ´γ2 “

161

´κN , where N is the principal normal to γ. Applying the negative almost complex structure

to the gradient, we see that the Hamiltonian vector field associated to the energy functional

is ´T ˆ ´κN “ κB, where B is the binormal vector field to γ. Therefore the Hamiltonian

flow of the energy functional, γpt, uq, satisfies the vortex filament equation:

B

Buγ “ κB.

Solutions of this PDE are well known to approximate the evolution of a vortex filament in

an incompressible fluid.

5.1.3 Description of Results

We now outline the results of this chapter. In Section 5.2, we study the weighted total twist

functional of a framed loop defined by

ĂTw :“1

4

ż

S1

tw}γ1} ds.

It turns out that the weighted total twist functional is the momentum map for the action

of a subgroup of Diff`pS1q which is diffeomorphic to S1 (the basepoint action). This is the

subgroup containing the stabilizers of the Diff`pS1q action, as described in Section 4.4.1. We

use this fact to give a characterization of the critical points of ĂTw.

In Section 5.3 we introduce the total elastic energy functional, denoted E, which is a

generalization of Kirchhoff elastic energy to not necessarily arclength-parameterized curves.

This functional takes the form

Epγ, V q :“1

4

ż

S1

pκ2` tw2

` st2q}γ1}2 ds,

162

where st is a term penalizing change in velocity per unit arclength, defined by

st :“1

}γ1}

d

ds}γ1}.

This closely resembles the strain energy of an extensible elastic rod, as in [6], Section 10.2.11.

The extra factor of 1{}γ1} must be included in the stretch term so that the functional scales

appropriately with length, and thus descends to a well-defined functional on our moduli

spaces. Restricting to arclength parameterized curves, the total elastic energy functional

reduces to exactly Kirchhoff elastic energy.

We give a description of the critical points to E. The critical points have very simple

descriptions, while the set of all critical points has a rich structure. The key to achieving

this characterization is to show that E is represented in complex coordinates as

EpΦq “

ż

S1

}Φ1}2C2 dt.

One immediately notices that this is strikingly similar to the energy functional 5.1. It turns

out that critical points of the complex energy functional which correspond to closed framed

curves satisfy a simple second-order ODE whose solutions can be written out explicitly, thus

the critical points of E have very simple representations. On the other hand, to compare to

the results of Ivey and Singer, we find a family of homotopies of critical points which take

an h-covered circle to a k-covered circle and which pass through torus knots in between.

163

5.2 Weighted Total Twist

5.2.1 The Basepoint Action

Recall from Remark 4.4.2 that there is a Lie subgroup of Diff`pS1q consisting of rigid ro-

tations of S1, and that this Lie subgroup is naturally diffeomorphic to S1. The reparame-

terization action of Diff`pS1q on M thus restricts to a (non-free) action of S1 on M. This

action is given explicitly for u P S1 « r0, 2s{p0 „ 2q by the forumula

u ¨ rγptq, V ptqs :“ rγpt´ uq, V pt´ uqs.

It follows from Lemma 4.4.7 that this action is given explicitly in Gr2pVq-coordinates by

u ¨ rΦptqs :“ rΦpt´ uqs,

and that this action is induced from the S1-action on St2pVq defined by

pu ¨ Φqptq :“ Φpt´ uq.

Each of these closely related actions of S1 will be referred to as the basepoint action. The

basepoint action can be interpreted as changing the basepoint of a based framed curve.

We claim that the basepoint action is Hamiltonian and that its momentum map has a

natural curve-theoretic interpretation. We will see that its momentum map is weighted twist

rate, defined in Section 5.1.3 by the formula

ĂTwpγ, V q “1

4

ż

S1

twpγ, V q}γ1} ds.

164

Note that this functional is invariant under translation and rotation. It could be made

invariant with respect to scaling, but we instead opt to restrict to framed curves with base

curves of length-2 (see the following remark). Thus ĂTw induces a well-defined functional on

M.

Remark 5.2.1. The factor of 1{4 in the definition of ĂTw is convenient for our purposes.

As defined, ĂTw is invariant under rotations, translations and global frame twists, but it is

not invariant under scaling of the base curve. Thus ĂTw is not well-defined on M. The most

natural remedy for this problem is to instead normalize by 1{p2lengthpγqq. Since we typically

realize M by fixing lengths of base curves at 2, this reduces to our choice of normalization.

Therefore we will take the convention in this section that M is always realized by restricting

to equivalence classes of length-2 curves, so that ĂTw is a well-defined energy functional on

M.

Another natural choice of normalization would be 1{p2πq. With this choice, ĂTw agrees

with the classical total twist functional Tw when restricted to the submanifold of framed loops

with arclength-parameterized base curve. We will stick with the 1{4 normalization only for

the sake of simplifying notation later on.

Proposition 5.2.2. The moment map for the basepoint action of S1 on M is ĂTw.

Before proving the proposition, we prove some preliminary lemmas.

Lemma 5.2.3. Let rΦs P Gr2pVq correspond to rγ, V s under pH. Then

ĂTwpγ, V q “ ´1

2Im 〈Φ,Φ1〉L2 .

Remark 5.2.4. The second statement of the lemma can be written as

ĂTwpγ, V q “1

2ωL

2

Φ pΦ,Φ1q,

165

where the right side is interpreted as the symplectic form on the complex vector space V2.

However, neither Φ nor Φ1 lie in T horΦ St2pVq « TrΦsGr2pVq, so this notation would be con-

fusing in the context of the lemma.

Proof. By Lemma 4.2.3, twist rate is given in complex coordinates by the formula

twpγ, V q “ ´2Im 〈Φ,Φ1〉C2

}Φ}4C2

.

Thus the claimed formula holds by the fact that }γ1} “ }Φ}2C2 (see Lemma 3.1.7), whence

ĂTwpγ, V q “1

4

ż 2

0

twpγ, V q}γ1} ds “1

4

ż 2

0

´2Im 〈Φ,Φ1〉C2

}Φ}4C2

}Φ}2C2 ¨ }Φ}2C2 dt “ ´1

2Im 〈Φ,Φ1〉L2 .

Lemma 5.2.5. For r P R « LiepS1q, the corresponding infinitesimal vector field for the

basepoint action on St2pVq is given by

Xr|Φ “ rΦ1.

Thus the infinitesimal vector field on Gr2pVq is the orthogonal projection of Xr onto the

horizontal tangent space of St2pVq, denoted projpXrq.

Proof. Let uε be a path in S1 « r0, 2s{p0 „ 2q with u0 “ 0 and let ddε

ˇ

ˇ

ε“0uε P LiepS1q « R

be denoted r. Then

d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

Φpt´ uεq “ rΦ1.

It is straightforward to check that this vector lies in TΦSt2pVq. Thus its projection to

T horΦ St2pVq gives the infinitesimal vector field on Gr2pVq.

166

We now proceed with the proof of Proposition 5.2.2.

Proof of Proposition 5.2.2. This follows by straightforward computations in complex co-

ordinates. We first calculate the derivative of ĂTw. Let Φ “ pφ, ψq P LpC2zt0uq and

δΦ “ pδφ, δψq P TΦLpC2zt0uq « LC2. The derivative of ĂTw at Φ in the direction δΦ is

given by

d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

ĂTwpΦ` εδΦq “d

dε

ˇ

ˇ

ˇ

ˇ

0

´1

2Im

ż 2

0

pφ1 ` εδφ1qpφ` εδφq ` pψ1 ` εδψ1qpψ ` εδψq dt

“ ´1

2Im

ż 2

0

δφ1φ` φ1δφ` δψ1ψ ` ψ1δψ dt

“ ´1

2Im

ż 2

0

´δφφ1 ` φ1δφ´ δψψ1 ` ψ1δψ dt (5.2)

“ Ím 〈δΦ,´Φ1〉L2 “ ωL2

Φ pδΦ,´Φ1q,

where (5.2) follows after integrating by parts and uses the assumption that all of the functions

involved are periodic. If we had instead taken Φ P ApC2zt0uq then it is easy to check that

the same calculation holds. Moreover, the same calculation holds in local coordinates on

Gr2pVq, where we apply Lemma 4.1.6 to obtain

DTwpΦqpδΦq “ ωL2

rΦspδΦ, projp´Φ1qq (5.3)

for δΦ P T horΦ St2pVq « TrΦsGr2pVq. An implicit requirement for this claim to be well-defined

is that ´Φ1 P TΦSt2pVq. Indeed, we can check that the constraint equations of TΦSt2pVq are

satisfied; e.g.,

Re

ż

φp´φ1q dt “ ´

ż

Repφφ1q dt “ ´1

2

ż

pφφq1 dt “ 0.

167

Let 〈¨, ¨〉R denote the standard inner product on R “ LiepS1q (i.e. multiplication). Then

for any r P R

〈DTwpΦqpδΦq, r〉R “ ωL2

rΦspδΦ, projp´rΦ1qq “ ωL2

rΦspδΦ, Xr|rΦsq.

From Equation (5.3) of the proof of the proposition, we immediately obtain the following

corollary.

Corollary 5.2.6. The Hamiltonian vector field of ĂTw : Gr2pVq Ñ R is given at rΦs by

projp´Φ1q, where proj is orthogonal projection from TΦSt2pVq to T horΦ St2pVq « TrΦsGr2pVq.

Thus the gradient of ĂTw at rΦs is projpíΦ1q, where proj is orthogonal projection from

TΦSt2pVq to T horΦ St2pVq « TrΦsGr2pVq.

Remark 5.2.7. The gradient vector field of ĂTw is a projection of the field íΦ1. We will

ignore the projection for a moment and consider the flow of íΦ1 in PC2. Denote this flow by

Φpu, tq “ pφpu, tq, ψpu, tqq, where u is the flow parameter. We can interpret each coordinate

function as a map from C to itself, where the domain has coordinates u ` it. It is not hard

to see that this implies that each coordinate function is holomorphic. It would be interesting

to study the implications of this observation for the gradient flow of ĂTw.

5.2.2 Critical Points of Total Twist

By the discussion above, ĂTw is a natural energy functional to study on M. We conclude

this section by characterizing its critical points.

Theorem 5.2.8. The critical points of ĂTw :MÑ R are equivalence classes rγ, V s such that

γ is an arclength parameterized, length-2, multiply covered round circle and V has constant

168

Figure 5.1: A torus knot realized as a pushoff of a critical point of ĂTw and the correspondingClifford torus knot as a stereographic projection of Φ2,3.

twist rate. Thus a small pushoff γ ` εV forms a multiply-covered torus knot and every torus

knot type is realized as such a pushoff for some critical point. All critical points are unstable.

Each critical framed loop has a complex-coordinate representative as a torus knot on the

standard Clifford torus in S3.

Proof. It is a general fact that if µ is the momentum map of a circle action on a finite-

dimensional Kahler manifold, then the points of the manifold where the gradient vanishes

are exactly the fixed points of the circle action. We see this by identifying LiepS1q with R

and considering the momentum map as a Hamiltonian. Then the associated Hamiltonian

vector field is tangent to the group orbits. Moreover, the gradient vanishes exactly when the

Hamiltonian vector field vanishes since the Riemannian metric and symplectic structure are

related by the complex structure. The points where the Hamiltonian vector field vanishes

are the fixed point of the circle action.

The reasoning of the previous paragraph applies to an infinite-dimensional Kahler man-

ifold, provided all of the pieces are well-defined. By Corollary 5.2.6, this is the case for the

169

situation at hand. Thus we see that the points of the Grassmannian where the gradient of

the twist functional is zero correspond exactly to the points fixed by the reparameterization

action of S1. A point rΦs is fixed in the Grassmannian under the S1 action if and only if

rΦpt´ uqs “ rΦptqs for all u P S1« r0, 2s{p0 „ 2q.

Since the equivalence class rΦs is up to Up2q-action, this means that there is an Apuq P Up2q

for each u P S1 such that

Φpt´ uq “ Φptq ¨ Apuq. (5.4)

Setting t “ 0 in (5.4), we obtain

Φpúq “ Φp0q ¨ Apuq for all u P S1. (5.5)

An element Φ P St2pVq satisfies 5.5 if and only if its corresponding framed loop has constant

parameterization speed as well as constant (classical) curvature, torsion and twist rate.

Indeed, multiplication by an element of Up2q in complex coordinates corresponds to a rotation

together with a global frame twist of the corresponding framed loop. These actions do not

effect any of the geometric invariants. Thus (5.5) implies that the values of the geometric

quantities of such a corresponding framed loop at any time must agree with the values at

time 0. The framed loops with constant curvature, torsion, parameterization speed and twist

rate are exactly those described by the theorem.

The fact that pushoffs produce torus knots for these types of framed loops is obvious, as

is the fact that all torus knot types are obtained.

We observe that the critical points are unstable because for any critical point rγ0, V0s,

there is a one parameter family rγτ , Vτ s such that Twpγτ , Vτ q “ Twpγ0, V0q for all τ . For

170

example, such a one-parameter family can be obtained by keeping γτ “ γ0 and altering Vτ

by increasing twist rate at one point and decreasing it elsewhere to keep total twist fixed.

Finally, we wish to show that a critical rγ, V s has a complex representation as a knot

on the Clifford torus in S3. We do so by providing such representations explicitly. We

assume that the (perhaps multiply-covered) torus knot corresponding to rγ, V s is of type

ph, kq P Z ˆ Z. In other words, γ is an h-times-covered round circle, and V has constant

twist rate π ¨ k. We claim that the Clifford torus knot

Φh,k “ pφh,k, ψh,kq :“1?

2

ˆ

exp

ˆ

i

2pk ` hqπt

˙

, exp

ˆ

i

2pk ´ hqπt

˙˙

corresponds to rγ, V s under pH.

Indeed, applying pH to Φh,k, we get the base curve

γ “1

πhp0,´ cosphπtq, sinphπtqq .

This curve has been translated to take its average value at 0 (rather than our usual convention

of taking its basepoint at 0). Then γ is clearly an h-times-covered round circle. Moreover,

we apply Lemma 4.2.3 to calculate the twist rate of Φh,k:

twpΦh,kq “ ´2Im

`

φh,k ¨ φh,k 1 ` ψh,k ¨ ψh,k 1˘

}Φh,k}4C2

“ ´2Im

ˆ

1

2exp

ˆ

i

2pk ` hqπt

˙

¨ exp

ˆ

í

2pk ` hqπt

˙

¨í

2pk ` hqπ

`1

2exp

ˆ

i

2pk ´ hqπt

˙

¨ exp

ˆ

í

2pk ´ hqπt

˙

¨í

2pk ´ hqπ

˙

“ ´2Im

ˆ

í

4pk ` hqπ `

í

4pk ´ hqπ

˙

“ kπ,

as claimed.

171

5.3 Total Elastic Energy

5.3.1 Framed Curve Invariants

Before moving on to study the total elastic energy functional defined in Section 5.1.3, we

derive the complex-coordinate representations of the remaining framed curve invariants.

Lemma 5.3.1. Let Φ be a complex-coordinate representation of a framed loop pγ, V q. Then

the Darboux curvatures of pγ, V q are given by

κ1 “2Impφ1ψ ´ ψ1φq

}Φ}4C2

and κ2 “ ´2Repφ1ψ ´ ψ1φq

}Φ}4C2

.

Proof. Let Φ “ pφ, ψq correspond to the quaternion q under the identification C2 Ø H and

to pγ, V q under pH. Recall from Lemma 4.2.2 that the quaternionic variation δq1 “ q1{}q}2H

corresponds to the framed curve variation

pδγ11, δV1q “

ˆ

d

dsγ1,

d

dsV

˙

under DpHpqq. Also recall from Lemma 4.1.7 that the quaternionic variation δq2 “ kq{2

corresponds to the framed curve variation

pδγ12, δV2q “ p´}γ1}V, T q.

172

Now we note that under the identification C2 Ø H, kq corresponds to píψ, iφq. Putting

this together, we conclude

2Impφ1ψ ´ ψ1φq

}Φ}4C2

“2

}Φ}4C2

Im⟨pφ1, ψ1q, pψ,´φq

⟩C2

“2

}Φ}4C2

Re⟨pφ1, ψ1q, piψ,´φq

⟩C2

“4

}q}2HRe

⟨q1

}q}2H,´

kq

2

⟩H

“4

}γ1}

1

4

ˆ⟨1

}γ1}

d

dsγ1, V

⟩`

⟨d

dsV,W

⟩〈T,W 〉

˙

}γ1} (5.6)

“

⟨1

}γ1}

d

dsγ1, V

⟩(5.7)

Line (5.6) follows from Corollary 3.1.11 and Lemma 3.1.7 and by noting that δγ11 “ddsγ1

implies that ddsδγ1 “ p1{}γ1}q d

dsγ1 and likewise for δγ12 “ ´}γ1}V . Now we notice that for

T “ γ1{}γ1}

⟨1

}γ1}

d

dsγ1, V

⟩“

1

}γ1}

⟨d

ds}γ1}T, V

⟩“

1

}γ1}

⟨ˆd

ds}γ1}

˙

T ` }γ1}d

dsT, V

⟩“

⟨d

dsT, V

⟩“ κ1.

This completes the proof for κ1. To derive the formula for κ2, one repeats the argument

above using the basic variation jq{2.

Recall that the stretch rate of a parameterized framed curve pγ, V q is given by

st “1

}γ1}

d

ds}γ1}

173

and represents the relative change per unit arclength of the parameterization speed of γ. We

now derive an expression for stretch rate in complex coordinates.

Lemma 5.3.2. Let Φ be a complex-coordinate representation of a framed loop pγ, V q. Then

the stretch rate of pγ, V q is given by

st “2Repφφ1 ` ψψ1q

}Φ}4C2

.

Proof. The derivation is easier in this case:

1

}γ1}

d

ds}γ1} “

1

}Φ}4C2

d

dt}Φ}2C2

“1

}Φ}4C2

d

dtpφφ` ψψq

“2Repφφ1 ` ψψ1q

}Φ}4C2

.

Remark 5.3.3. A simple calculation shows

st “1

}γ1}

d

ds}γ1} “

1

}γ1}2d

dt〈γ1, γ1〉1{2 “ 〈γ

2, γ1〉}γ1}3

.

Recall from Section 2.1.2 that the normal vector of an arbitrarily parameterized space curve

γ is given by the formula

κN “γ2

}γ1}2´〈γ2, γ1〉}γ1}3

T “γ2

}γ1}2´ st ¨ T,

and we see that stretch rate has already appeared naturally in the context of curve theory.

174

5.3.2 Total Elastic Energy

We now return to the total elastic energy functional on defined in Section 5.1.3. This

functional is defined on a framed path pγ, V q by the formula

Epγ, V q “1

4

ż 2

0

`

κ2` tw2

` st2˘

}γ1}2 ds,

We claim that the functional E is natural from a variety of perspectives. First, the total

energy functional is a straightforward generalization of the Kirchhoff elastic energy func-

tional described in Section 5.1—in particular, restricting to the submanifold of arclength-

parameterized framed curves produces Kirchhoff energy exactly. Secondly, applications

sometimes require elasticity in the stretch term, for example in the case of polymers where

the distance between the atoms is able to vary [38]. Another feature of the functional E

is that it has an incredibly nice representation in complex coordinates. Finally, E induces

a well-defined functional on the moduli space M, and the critical point set of the induced

functional on M is quite rich.

Remark 5.3.4. The statement that E is well-defined on M is still under the convention

that M is realized as a restriction to equivalence classes of length-2 curves. As in the case

of ĂTw, we could replace the 1{4 normalization factor by 1{p2lengthpγqq to achieve an energy

functional which is invariant under scaling of the base curve. The choice of 1{4 is convenient

for calculations to follow.

We begin our study of E by describing it in complex coordinates.

Proposition 5.3.5. In complex coordinates for framed paths, the total elastic energy func-

tional takes the form

EpΦq “

ż 2

0

}Φ1}2C2 dt.

175

Proof. Appealing to Lemmas 4.2.3, 5.3.1 and 5.3.2, we write

EpΦq “1

4

ż 2

0

4

}Φ}4C2

`

}φψ1 ´ φ1ψ}2C ` }φφ1 ` ψψ1}2C

˘

}Φ}2C2dt

“

ż 2

0

1

}Φ}2C2

`

}φψ1 ´ φ1ψ}2C ` }φφ1 ` ψψ1}2C

˘

dt. (5.8)

Simplifying }φψ1 ´ φ1ψ}2C ` }φφ1 ` ψψ1}2C yields

}φ1ψ ´ ψ1φ}2C ` }φφ1 ` ψψ1}2C “ }φ

1ψ}2C ` }ψ1φ}2C ` }φφ

1}

2C ` }ψψ

1}

2C

´ 2Re`

φ1ψψ1φ˘

` 2Re´

φφ1`

ψψ1˘

¯

“ p}φ}2C ` }ψ}2Cqp}φ

1}

2C ` }ψ

1}

2Cq “ }Φ}

2C2}Φ1}2C2 .

Replacing this into (5.8) yields the result.

For a moment, we consider E as a functional on the space W “ LpC2zt0uq \ApC2zt0uq,

which was previously introduced in Section 4.1.3. Recall that under pH, W double-covers the

space of frame-periodic framed paths rSper.

Proposition 5.3.6. The gradient of E :W Ñ R with respect to the natural L2 metric takes

the form

gradpEq|Φ “ ´2Φ2.

Proof. We calculate the derivative of E at Φ PW in the direction δΦ P TΦW «W :

DEpΦqpδΦq “d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

ż 2

0

}Φ1 ` εδΦ1}C2 dt

“

ż 2

0

φ1δφ1 ` δφ1φ` ψ1δψ1 ` δψ1ψ1 dt

“ ´2Re

ż 2

0

φ2δφ` ψ2δψ dt

“ Re 〈´2Φ2, δΦ〉L2 ,

176

where we have once again employed integration by parts using the assumption that either

all functions involved are periodic or all functions are antiperiodic.

5.3.3 Critical Points of E

Noting that E is invariant under rigid rotations, global frame twists, translations and scaling

(see Remark 5.3.4), we conclude that it induces a well-defined functional on M, which we

continue to denote by E. Our goal for this section is to determine the critcal points of

E on M. Locally, M can be considered as a codimension-8 submanifold of rSper. Indeed,

M is identified with the disjoint union of Grassmannians, each Grassmannian is locally a

codimension-4 submanifold of the corresponding Stiefel manifold (by virtue of a slice chart

to the Up2q-action), and each Stiefel manifold is an honest codimension-4 submanifold ofW .

Thus we wish to find the points where the gradient of E points in the 8-dimensional normal

direction toM, which we recall was described explicitly in Lemma 4.5.3. Before stating the

main theorem of this subsection (Theorem 5.3.10), we introduce notation and prove some

preliminary lemmas.

For the remainder of this section, we consider complex paths of the form

Φptq “

¨

˚

˝

φptq

ψptq

˛

‹

‚

“

¨

˚

˝

z1 exppic1πt{2q ` z2 exppíc1πt{2q

w1 exppic2πt{2q ` w2 exppíc2πt{2q

˛

‹

‚

, (5.9)

where c1, c2 P Z satisfy c1 “ c2 mod 2 and z1, z2, w1, w2 P C. The collection of all such Φ

will be denoted Sol. We will see in the following lemma that these are certain solutions to a

family of second-order ODEs, and we will see in the proof of Theorem 5.3.10 that this family

of ODEs is relevant to the critical point set of E.

177

Lemma 5.3.7. Consider the system of second order ODEs

¨

˚

˝

φ2

ψ2

˛

‹

‚

“

¨

˚

˝

λ1 0

0 λ2

˛

‹

‚

¨

˚

˝

φ

ψ

˛

‹

‚

(5.10)

where Φ “ pφ, ψq : RÑ C2 and λ1, λ2 P R. The set Sol consists of solutions lying in W.

Proof. General solutions φ : RÑ C of the equation

φ2 “ λφ, λ P R (5.11)

take the form

φptq “ z1e?λt` z2e

´?λt

for z1, z2 P C. Indeed, such functions satisfy (5.11) and solutions of (5.11) are unique up

to choice of initial conditions. There are four (real) degrees of freedom for choosing initial

conditions, hence the (complex) parameters z1 and z2 appearing in the solution. We conclude

that general solutions Φ : RÑ C of (5.10) take the form

Φptq “

¨

˚

˝

φptq

ψptq

˛

‹

‚

“

¨

˚

˝

z1 expp?λ1tq ` z2 expp´

?λ1tq

w1 expp?λ2tq ` w2 expp´

?λ2tq

˛

‹

‚

(5.12)

for z1, z2, w1, w2 P C.

Of course, not all paths Φ of the form (5.12) lie inW . The path Φ is a loop (respectively an

anti-loop) if and only if exppa

λjtq is a complex loop (anti-loop) for j “ 1, 2. We conclude

thata

λj is a complex integer multiple of π (respectively, half-integer multiple of π) for

j “ 1, 2. Thus we have shown that a necessary and sufficient condition for Φ to lie in W is

that it Φ P Sol.

178

Thus Sol is a subset of W—if the cj are even, then Φ P LC2 and if the cj are odd then

Φ P AC2. Next we wish to characterize which elements Φ P Sol lie in St2pVq.

Lemma 5.3.8. Let Φ P Sol. Then

Φ P St2pVq ô

$

’

&

’

%

|z1|2 ` |z2|

2 “ |w1|2 ` |w2|

2 “ 1 c1 ‰ c2

pz1, z2q and pw1, w2q are orthonormal in C2 c1 “ c2 ‰ 0

Proof. It is straightforward to conclude that Φ P Sol is never an element of St2pVq if c1 “

c2 “ 0. Indeed, if Φ “ pφ, ψq “ pz, wq for some z, w P C, then

ż

φψ dt “ 2zw,

which is zero if and only if z or w is zero, and this violates normality.

Next we note that the set texppcπt{2q | c P 2Zu is L2-orthonormal as a subset of LC.

Similarly, texppcπt{2q | c P Z is oddu is an L2-orthonormal subset of AC. Let Φ P Sol with

c1 ‰ c2. Then

〈φ, φ〉L2 “ |z1|2 〈exppic1πt{2q, exppic1πt{2q〉L2 ` |z2|

2 〈exppíc1πt{2q, exppíc1πt{2q〉L2

` Re pz1z2 〈exppic1πt{2q, exppíc1πt{2q〉L2q

“ |z1|2` |z2|

2.

Similarly 〈ψ, ψ〉 “ |w1|2 ` |w2|

2.

179

Finally,

〈φ, ψ〉L2 “ z1w1 〈exppic1πt{2q, exppic2πt{2q〉L2 ` z1w2 〈exppic1πt{2q, exppíc2πt{2q〉L2

` z2w1 〈exppíc1πt{2q, exppic2πt{2q〉L2 ` z2w2 〈exppíc1πt{2q, exppíc2πt{2q〉L2

“ 0

This establishes the first set of necessary and sufficient conditions.

In the case that Φ P Sol with c1 “ c2 “ c ‰ 0, the same calculations show 〈φ, φ〉L2 “

|z1|2 ` |z2|

2 and 〈ψ, ψ〉L2 “ |w1|2 ` |w2|

2. However, in this case

〈φ, ψ〉L2 “ z1w1 〈exppicπt{2q, exppicπt{2q〉L2 ` z1w2 〈exppicπt{2q, exppícπt{2q〉L2

` z2w1 〈exppícπt{2q, exppicπt{2q〉L2 ` z2w2 〈exppícπt{2q, exppícπt{2q〉L2

“ z1w1 ` z2w2,

and this completes the proof.

We will show in the following theorem that critical points of E : M Ñ R can be repre-

sented as elements of Sol X pSt˝2pLCq \ St˝2pACqq. These representations are not unique, so

we impose some normalizing conditions. We define the set

Crit :“ tΦc | c P Zu Y tΦ P SolX pSt˝2pLCq \ St˝2pACqq | c1 ą c2 ě 0, z1 P Rě0u

“: Crit1 Y Crit2,

where

Φcptq “

¨

˚

˝

φcptq

ψcptq

˛

‹

‚

:“

¨

˚

˝

exppicπt{2q

exppícπt{2q

˛

‹

‚

. (5.13)

We will see in the following lemma the reason for the distinction of the set Crit1

180

Lemma 5.3.9. The action of Up2q on St2pVq restricts to a free transitive action on each set

Solc :“ tΦ P SolX pSt˝2pLCq \ St˝2pACqq | c1 “ c2 “ c ‰ 0u.

Thus Crit1 is a Up2q-cross-section ofď

cPZ

Solc.

Proof. Let

A “ eiθ

¨

˚

˝

u v

´v u

˛

‹

‚

denote an arbitrary element of Up2q and let Φ P Solc. Then

Φ ¨ A “

¨

˚

˝

eiθpuz1 ´ vw1q exppicπt{2q ` eiθpuz2 ´ vw2q exppícπt{2q

eiθpvz1 ´ vw1q exppicπt{2q ` eiθpvz2 ´ vw2q exppícπt{2q

˛

‹

‚

.

By a straightforward calculation, we see that the assumption that the parameters zj, wj

satisfy the conditions of Lemma 5.3.8 implies that the new parameters do as well. We

conclude that Up2q acts on Solc, and the fact that the action is free follows from the freeness

of the action on St2pVq.

Then it remains to show that the action is transitive. Let

Φptq “

¨

˚

˝

φptq

ψptq

˛

‹

‚

“

¨

˚

˝

z1 exppicπt{2q ` z2 exppícπt{2q

w1 exppicπt{2q ` w2 exppícπt{2q

˛

‹

‚

P Solc.

The claim follows by showing that there exists A P Up2q such that Φc ¨ A “ Φ. Indeed, this

is satisfied by the matrix

A “

¨

˚

˝

z1 w1

z2 w2

˛

‹

‚

,

181

which is an element of Up2q since the parameters zj, wj must satisfy the conditions of Lemma

5.3.8 and this implies that the columns of A are Hermitian orthonormal.

Since the Up2q-action is free, transitivity implies that Crit1 is a Up2q-cross-section ofŤ

cPZ Solc.

We are now prepared to state our main result.

Theorem 5.3.10. The critical points of E :MÑ R are in one-to-one correspondence with

the complex parameterizations in Crit.

Proof. Locally we consider M as a codimension-8 submanifold of rSper. Critical points of E

restricted toM occur when gradpEq lies entirely in the normal space toM. We now work in

complex coordinates, where the previous statements are translated into the statement that

critical points of E restricted to Gr2pVq occur when gradpEq lies in the 8-dimensional normal

space to Gr2pVq, considered locally as a submanifold of W . For rΦs P Gr2pVq, the normal

space to rΦs is spanned by the eight vectors listed in Lemma 4.5.3. Easy calculations show

that the gradpEq|Φ is orthogonal to each of the four vectors in the second list (4.17); e.g.

integrating by parts shows

Rep〈pφ2, ψ2q, piφ, 0q〉L2 “ Re

ż

φ2piφq dt

“ Re

ż

i|φ1|2 dt “ 0.

Thus we are looking for Φ P St2pVq such that the gradpEq|Φ lies in the 4-dimensional

(over R) normal space to the Stiefel manifold at Φ. That is,

Φ2 “ λ1pφ, 0q ` λ2p0, ψq ` λ3pψ, φq ` λ4píψ, iφq, for some λj P R.

182

We can rewrite this as

¨

˚

˝

φ2

ψ2

˛

‹

‚

“

¨

˚

˝

λ1 λ3 ´ iλ4

λ3 ` iλ4 λ2

˛

‹

‚

¨

˚

˝

φ

ψ

˛

‹

‚

.

The matrix is Hermitian, hence diagonalizable by unitary matrices by the spectral theorem.

Since we only care about solutions up to multiplication by unitary matrices, we can rename

parameters and replace our system by

¨

˚

˝

φ2

ψ2

˛

‹

‚

“

¨

˚

˝

λ1 0

0 λ2

˛

‹

‚

¨

˚

˝

φ

ψ

˛

‹

‚

. (5.14)

From Lemmas 5.3.7 and 5.3.8 we see that the critical points of E can be written as elements of

SolXpSt2pLCq\St2pACqq, and that this is described by explicit conditions on the parameters.

We conclude the proof by showing that for any Φ P Sol X pSt2pLCq \ St2pACqq, there

exists A P Up2q such that Φ ¨ A P Crit, and then that such an A is unique. This will be

accomplished in two cases. We first assume that c1 “ c2 “ c, so that Φ P Solc. By Lemma

5.3.9, there is a unique A P Up2q such that Φ ¨ A “ Φc P Crit1. Moreover, Φ ¨ A R Crit2 for

any A P Up2q. Thus the proof is complete in this case.

We now move on to the second case and assume that Φ P SolXpSt2pLCq\St2pACqq with

c1 ‰ c2. Our goal is to find a matrix A P Up2q such that Φ P Crit2. We first wish to show

that it can be arranged that c1 ą c2 ě 0. Indeed, by renaming parameters we can ensure

that c1 ě ć1 and c2 ě ć2. Since c1 ‰ c2, we finish rearranging by multiplying by

¨

˚

˝

0 1

1 0

˛

‹

‚

P Up2q

183

if necessary, thus ensuring that c1 ą c2. We finish this part of the argument by writing

z1 “ r1eiθ1 and multiplying by the matrix

A “

¨

˚

˝

eíθ1 0

0 eiθ1

˛

‹

‚

P Up2q.

Clearly, Φ Ä P Crit2, and we have proved the existence part of our claim in the case c1 ‰ c2.

Now we prove that such an A is unique. We will prove the equivalent statement that for

Φ, rΦ P Crit2, if Φ Ä “ rΦ, then A is the identity matrix. Let Φ and rΦ be written in the form

(5.9), with the parameters of rΦ given by rzj, rwj, and rcj, and let

A “ eiθ

¨

˚

˝

u v

´v u

˛

‹

‚

be an arbitrary element of Up2q. Then ΦÄ “ rΦ implies that their first coordinates cooincide;

i.e.,

eiθ puz1 exppipc1qπt{2q ` uz2 exppípc1qπt{2q

´vw1 exppipc2qπt{2q ´ vw2 exppípc2qπt{2qq

“ rz1 exp pi prc1q πt{2q ` rz2 exp pí prc1q πt{2q .

From the L2-orthonormality of the exponential functions and the conditions c1 ą c2 ě 0 and

rc1 ą rc2 ě 0, we deduce that there are two cases: either rc1 “ c1, or rc1 “ c2. In the first case,

we once again appeal to L2-orthonormality to conclude that

eiθuz1 “ rz1 and v “ 0.

184

The first equality implies that eiθu is a positive real number, as z1, rz1 P Rě0. The second

equality implies that |eiθu| “ 1, therefore A is the identity matrix.

Now we assume that rc1 “ c2. This implies

rz1 “ éiθvw1, rz2 “ é

iθvw2, u “ 0.

Thus the equality of the second coordinates of Φ ¨ A and rΦ reads

eiθv pz1 exppipc1qπt{2q ` z2 exppípc1qπt{2qq

“ rw1 exp pi prc2q πt{2q ` rw2 exp pí prc2q πt{2q ,

whence we conclude that rc2 “ c1. However, this is a contradiction, as we have

rc2 “ c1 ą c2 “ rc1,

violating rΦ P Crit2.

Using our convenient complex form of E and the previous theorem, we are able to calculate

the possible energy levels that a framed curve can realize. Let Φ “ pφ, ψq P Crit. Then

φ1ptq “iπc1

2z1 exppiπc1t{2q `

íπc1

2z2 exppíπc1t{2q,

so

|φ1ptq|2 “

ˇ

ˇ

ˇ

ˇ

iπc1

2z1 exppiπc1t{2q

ˇ

ˇ

ˇ

ˇ

2

`

ˇ

ˇ

ˇ

ˇ

íπc1

2z2 exppíπc1t{2q

ˇ

ˇ

ˇ

ˇ

2

` 2Re

˜

iπc1

2z1 exppiπc1t{2q

ˆ

íπc1

2z2 exppíπc1t{2q

˙

¸

“

´πc1

2

¯2

´ 2´πc1

2

¯2

Re pz1z2 exppiπc1tqq .

185

Similarly,

|ψ1ptq|2 “´πc2

2

¯2

´ 2´πc2

2

¯2

Re pw1w2 exppiπc2tqq .

Thus

EpΦq “

ż 2

0

}Φ}2C2 dt

“

ż 2

0

´πc1

2

¯2

´ 2´πc1

2

¯2

Re pz1z2 exppiπc1tqq `´πc2

2

¯2

´ 2´πc2

2

¯2

Re pw1w2 exppiπc2tqq dt

“π2pc2

1 ` c22q

2´ 2

´πc1

2

¯2

Re

ˆ

z1z2

ż 2

0

exppiπc1tq dt

˙

´ 2´πc2

2

¯2

Re

ˆ

w1w2

ż 2

0

exppiπc2tq dt

˙

“π2pc2

1 ` c22q

2.

Therefore we have proved the following corollary.

Corollary 5.3.11. The critical energy levels of E :M Ñ R are quantized and the possible

critical energy levels are

π2pc21 ` c

22q

2

for any integers c1, c2 which are not both zero.

5.3.4 One-Parameter Families of Critical Points

We now move on to understanding these critical points of E in framed loop coordinates—e.g.,

what is the knot type of the base curve of a critical rγ, V s? Towards this goal, we examine

a particular collection of 1-parameter families of critical points.

186

Let c1 ą c2 ě 0 be integers. We will consider the simple 1-parameter family Φu “

pφu, ψuq, u P r0, 1s, where

φuptq :“ u exppic1πt{2q `?

1´ u2 exppíc1πt{2q, ψuptq :“ exppic2πt{2q.

Note that each Φu in this family is an element of the set Crit defined in the previous sub-

section. It will be convenient to introduce the change of variables

h :“c1 ` c2

2and k :“

c1 ´ c2

2.

Then, up to a translation, Φu maps under pH to the framed loop pγu, Vuq, where

γuptq “2

π

¨

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˝

u?

1´ u2

h` ksinpph` kqπtq

ú

kcospkπtq `

?1´ u2

hcosphπtq

u

ksinpkπtq `

?1´ u2

hsinphπtq

˛

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‚

,

and

Vuptq “2

}Φu}2C2

¨

˚

˚

˚

˚

˝

u sinphπtq ´?

1´ u2 sinpkπtq

cosphπtq cospkπtq ` u?

1´ u2

p1´ u2q sinphπtq cospkπtq ´ u2 cosphπtq sinpkπtq

˛

‹

‹

‹

‹

‚

,

with

}Φu}2C2 “ 2` 2u

?1´ u2 cospph` kqπtq.

187

Then Φ0 corresponds under pH to the framed loop pγ0, V0q, where

γ0ptq “2

πh

¨

˚

˚

˚

˚

˝

0

cosphπtq

sinphπtq

˛

‹

‹

‹

‹

‚

and

V0ptq “

¨

˚

˚

˚

˚

˝

´ sinpkπtq

cosphπtq cospkπtq

sinphπtq cospkπtq

˛

‹

‹

‹

‹

‚

“ cospkπtq

¨

˚

˚

˚

˚

˝

0

cosphπtq

sinphπtq

˛

‹

‹

‹

‹

‚

` sinpkπtq

¨

˚

˚

˚

˚

˝

1

0

0

˛

‹

‹

‹

‹

‚

Clearly γ0 is an arclength-parameterized h-times-covered round circle. Moreover, it is clear

from the second form of V0ptq that the linking number of V0 and the image of γ0 is k.

Similarly, Φ1 corresponds to

γ1ptq “2

πk

¨

˚

˚

˚

˚

˝

0

´ cospkπtq

sinpkπtq

˛

‹

‹

‹

‹

‚

and

V1ptq “

¨

˚

˚

˚

˚

˝

sinphπtq

cosphπtq cospkπtq

´ cosphπtq sinpkπtq

˛

‹

‹

‹

‹

‚

“ .´ cosphπtq

¨

˚

˚

˚

˚

˝

0

´ cospkπtq

sinpkπtq

˛

‹

‹

‹

‹

‚

` sinphπtq

¨

˚

˚

˚

˚

˝

1

0

0

˛

‹

‹

‹

‹

‚

which is an arclength parameterized k-covered round circle whose image is linked ´h times

by V1.

Thus it remains to determine what happens between the endpoints of the one-parameter

family. Our first goal is to show that the base curve γu is nonembedded for exactly one value

188

of u P p0, 1q. This can be achieved by directly analyzing the explicit formula for γu. The

proof follows slightly more easily by utilizing the following generalization of Lemma 3.2.1,

which has essentially the same proof.

Lemma 5.3.12. Let Φ “ pφ, ψq P St2pVq correspond to the framed loop pγ, V q under pH.

Then γ has a self-intersection γpt0q “ γpt1q for 0 ă a ă b ă 2 if and only if Φ satisfies

ż b

a

|φ|2 ´ |ψ|2 dt “

ż b

a

φψ dt “ 0.

Corollary 5.3.13. For the 1-parameter family of critical points described above, the only

u P p0, 1q for which the base curve γu is nonembedded is

u “

c

k2

h2 ` k2.

Proof. The 1-parameter family is given in complex coordinates by

pφuptq, ψptqq “ pu exppic1πt{2q `?

1´ u2 exppíc1πt{2q, exppic2πt{2qq,

thus the conditionż b

a

φuψu dt “ 0

reads

0 “

ż b

a

u exppic1πt{2q exppíc2πt{2q `?

1´ u2 exppíc1πt{2q exppíc2πt{2q dt

“u

ikπexppikπbq `

?1´ u2

íhπexppíhπbq ´

u

ikexppikπaq ´

?1´ u2

íhπexppíhπaq,

189

or, equivalently,

hupexppikπaq ´ exppikπbqq “ k?

1´ u2pexppíhaq ´ exppíhbqq.

Taking the squared absolute value of each side yields

h2u2p2´ 2Re exppikπpa´ bqqq “ k2

p1´ u2qp2´ 2Re exppíhpa´ bqqq.

Now we claim that Re exppikπpa ´ bqq “ Re exppíhpa ´ bqq. If this is the case then we

immediately obtain our result, as this implies

h2u2“ k2

p1´ u2q, or u “

c

k2

h2 ` k2.

Thus it remains to show that Re exppikπpa´bqq “ Re exppíhpa´bqq, which is equivalent

to cosppa ´ bqkπq “ cosppa ´ bqhπq. From the explicit description of γu, we see that for γu

to have a self-intersection at times a ă b it must be that sinpph ` kqbq “ sinpph ` kqaq.

Therefore it is necessary that

b “

$

’

&

’

%

a` 2jh`k

or

2j`1h`k

´ a,j “ 0, 1, . . . , h` k ´ 1.

Taking b “ a` 2j{ph` kq, we have

cosppa´ bqkπq “ cos

ˆ

2j

h` kkπ

˙

“ cos

ˆ

2j

h` kpph` kq ´ hqπ

˙

“ cos

ˆ

2jπ ´2j

h` khπ

˙

“ cos

ˆ

2j

h` khπ

˙

“ cosppa´ bqhπq.

190

The case b “ p2j ` 1q{ph` kq ´ a follows similarly.

We now describe the knot types which appear in the 1-parameter family γu.

Lemma 5.3.14. Assume that h and h ` k are relatively prime. Then for 0 ă u ă

a

k2{ph2 ` k2q, γu parameterizes an ph, h` kq-torus knot.

Proof. Since γu has the same knot type for 0 ă u ăa

k2{ph2 ` k2q, it suffices to prove that

γu is an ph, h` kq-torus knot for sufficiently small u. Consider the parameterized surface

pt, sq ÞÑ2

π

¨

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˝

u?

1´ u2

h` ksinpπsq

cospπtq

ˆ

ú

kcospπsq `

?1´ u2

h

˙

sinpπtq

ˆ

ú

kcospπsq `

?1´ u2

h

˙

˛

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‚

.

In this form, it is easy to see that for u ă 1{2, the surface is an embedded torus of revolution

with elliptical cross sections. Making the change of variables s ÞÑ ph ` kqt and t ÞÑ ht, we

obtain a loop on the torus

rγuptq :“2

π

¨

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˝

u?

1´ u2

h` ksinpph` kqπtq

cosphπtq

ˆ

ú

kcospph` kqπtq `

?1´ u2

h

˙

sinphπtq

ˆ

ú

kcospph` kqπtq `

?1´ u2

h

˙

˛

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‚

which is clearly an ph, h` kq-torus knot.

191

Noting that

cospph` kqπtq cosphπtq “ cospkπtq ´ sinphπtq sinpph` kqπtq

and

cospph` kqπtq sinphπtq “ ´ sinpkπtq ` cosphπtq sinpph` kqπtq,

we conclude that

rγuptq “ γuptq ù

k

¨

˚

˚

˚

˚

˝

0

sinphπtq sinpph` kqπtq

´ cosphπtq sinpph` kqπtq

˛

‹

‹

‹

‹

‚

Therefore γu and rγu are isotopic for sufficiently small u ą 0, and γu is an ph, h ` kq-torus

knot.

Lemma 5.3.15. Assume that k and h` k are relatively prime. Then fora

k2{ph2 ` k2q ă

u ă 1, γu parameterizes a p´k, h` kq-torus knot.

Proof. The proof is the same as that of Lemma 5.3.14, except we start with the parameterized

surface

pt, sq ÞÑ2

π

¨

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˝

u?

1´ u2

h` ksinpπsq

cospπtq

ˆ

ú

k`

?1´ u2

hcospπsq

˙

sinpπtq

ˆ

ú

k`

?1´ u2

hcospπsq

˙

˛

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‚

,

192

which is an embedded torus for u ą 1{2, and we make the change of variables s ÞÑ ph` kqt

and t ÞÑ ´kt. This produces a p´k, h` kq-torus knot rγu which can be written as

rγuptq “ γuptq ´

?1´ u2

h

¨

˚

˚

˚

˚

˝

0

sinpkπtq sinpph` kqπtq

cospkπtq sinpph` kqπtq

˛

‹

‹

‹

‹

‚

and we conclude that γu is isotopic to rγu for sufficiently large u ă 1, so γu is a p´k, h`kq-torus

knot.

The results of this subsection are summarized as:

Theorem 5.3.16. Let h, k be integers such that gcdph, h ` kq “ gcdpk, h ` kq “ 1. The

critical point set of E :M Ñ R contains a 1-parameter family rγu, Vus, u P r0, 1s, such that

there exists a unique u P p0, 1q where γu is nonembedded, and:

(i) pγ0, V0q is an arclength parameterized h-times-covered round circle linked k-times,

(ii) γε is a ph, h` kq-torus knot for sufficiently small ε ą 0,

(iii) γ1´ε is a p´k, h` kq torus knot for sufficiently small ε ą 0,

(iv) pγ1, V1q is an arclength parameterized k-times-covered round circle linked ´h-times.

An example is shown in Figure 5.3.4 of a 1-parameter family of critical points for the

h “ 2, k “ 1 case. Each shape in the figure is a ribbon parameterized according to pγu, Vuq

with centerline γu (in black). Note that the times are not equally separated and that the

curve is nonembedded ata

1{5 “a

k2{ph2 ` k2q. Also note that the u “ 0 ribbon appears

as a Mobius band because the centerline is actually double-covered.

193

0 0.1 0.25 0.4 1/5

0.5 0.75 0.99 1

Figure 5.2: The 1-parameter family of critical points with h “ 2, k “ 1.

Another example is shown in Figure 5.3.4 with h “ 3 and k “ 1. In this example, only

the base curve γu is shown for clarity. The starting and ending points are omitted from the

figure, as they are multiply covered circles.

194

0.1 0.4 4/13 0.6 0.99

Figure 5.3: The 1-parameter family of critical points with h “ 3, k “ 2.

195

Chapter 6

Application: Shape Recognition

6.1 Introduction

6.1.1 Background

The Riemannian geometry of infinite-dimensional shape manifolds of paths or loops in Rn has

become an active field of research over the last decade. Interest in these spaces is driven by

their applications to computer vision (see the survey papers [3, 69]). Such a shape manifold

is typically realized as a quotient of the space of immersions

ImmpΣ,Rnq “ tγ P C8pΣ,Rn

q | γ1ptq ‰ 0 for all t P Σu ,

where Σ is an interval or S1. To form the shape space, we quotient by “shape-preserving”

group actions—usually isometries and homotheties of the ambient Rn and diffeomorphisms

of Σ, which act by reparameterizing the immersed manifold. The points (equivalence classes

of curves) in the quotient manifold are “shapes” of immersed copies of Σ. By defining

a Riemannian metric on the shape space, we can (in theory) measure geodesic distance

196

between shapes, thereby defining a natural, geometrically-motivated distance function on

the space of shapes.

Example 6.1.1. In the case n “ 2 and Σ “ S1, the elements of the shape space

ImmpS1,R2q{ttranslation, rotation, scaling, reparameterizationu

are viewed as outlines of images of objects. Geodesic distance between points of the shape

space is a measure of distance between shapes. This yields an object classification algorithm.

Example 6.1.2. As a more relevant example, we consider facial recognition. The facial

recognition problem is mathematically realized as defining a distance metric on the space of

immersed surfaces in R3, up to natural equivalences. A computationally effective approach

to elastic shape recognition for surfaces is to compare collections of closed curves immersed

in the surfaces [65]. Curves on an immersed surface in R3 have a natural framing, thus

the surface shape matching problem can be formulated as shape matching of collections of

framed loops. Therefore shape similarity for faces can be measured by geodesic distance in

M or M{Diff`0 pS1q.

To study the Riemannian geometries of shape spaces of paths or loops in Rn, one begins by

choosing a convenient metric on the space of immersions ImmpΣ,Rnq, where Σ is an interval

or S1. Techniques from Riemannian geometry are then applied to study the geodesics in the

more complicated shape space.

The simplest choice of reparameterization-invariant metric on the space of immersions is

the L2 metric: for γ P ImmpΣ,Rnq, let δγ1 and δγ2 denote variations of γ and define the L2

metric by

gL2

γ pδγ1, δγ2q “

ż

Σ

〈δγ1, δγ2〉 ds,

197

where ds is measure with respect to arclength. It was shown in [52] that geodesic distance

with respect to the L2 metric vanishes, so this metric is not suitable for shape recognition

applications.

A wide variety of more complicated Riemannian metrics have been introduced in the

literature, largely produced by adjusting the L2 metric in some manner. The Sobolev-type

metrics [53, 71], which include inner products of higher-order derivatives of the variations

δγj in the integral, are particularly well-suited to shape recognition problems. For example,

it was shown in [7] that ImmpS1,R2q is geodesically complete with respect to a family of

Sobolev-type metrics.

The elastic metrics of [57], which we discussed previously in Section 3.1.1, are a partic-

ularly well-studied family of Sobolev-type metrics on ImmpΣ,R2q of the form

ga,bγ pδγ1, δγ2q “

ż

Σ

a

⟨d

dsδγ1, T

⟩⟨d

dsδγ2, T

⟩` b

⟨d

dsδγ1, N

⟩⟨d

dsδγ2, N

⟩ds, (6.1)

where a, b ą 0, d{ds is derivative with respect to arclength, N is the unit normal to γ and

T is the unit tangent to γ. The first term compares the bending deformation of γ induced

by the variations δγ1 and δγ2, while the second term compares the stretching deformation.

Remarkably, it was shown in [76] that geodesics may be described explicitly with respect to

this metric in the case a “ b. In Section 3.1.1 we introduced an analogous family of metrics

on the space of framed curves and we showed in Section 4.5.1 that geodesics in S and M

are explicitly computable with respect to this metric.

A common technique used to study geodesics in immersion space with respect to a chosen

metric is to flatten the metric by applying a transform. This entails isometrically mapping

the immersion space into some other Riemannian manifold with a metric which is simpler to

work with. Examples include the Square Root Velocity Function-transform [68], which maps

γ to γ1{a

}γ1}, the Q-transform [41], which maps γ toa

}γ1}γ, and the complex square-root

198

transform studied in [76] (it was recently shown in [2] that, in the planar case, these are all

specific examples of the so-called R-transform). The complex square-root transform is only

defined for planar curves γ, and the transform takes the curve to?γ1, where γ1 is treated as a

complex number. The proof of Theorem 3.2.8 presented in this work made use of a transform

which may be viewed as a quaternionic analogue of the complex square root transform.

6.1.2 Description of Results

In this chapter we are particularly interested in shape matching algorithms for the shape

spaces of open framed paths S{SOp3q and S{pSOp3q ˆ Diff`pr0, 2sqq. The reason for this is

that these shape spaces have the most obvious applications. We are particularly interested

in shape matching for protein backbones, as the shape of a protein backbone is well-known

to correlate to its function. Another potential application of this theory would be oriented

trajectory recognition for, e.g., automated aircraft training or gesture recognition.

We show that the geodesic distance problem for S{SOp3q has an exact closed-form solu-

tion. We then move on to treating the geodesic distance problem for S{pSOp3qˆDiff`pr0, 2sqq.

We will see that geodesic distance degerates on S{pSOp3qˆDiff`pr0, 2sqq and this makes the

space non-Hausdorff. This is remedied by identifying elements of S{SOp3q up to L2 con-

vergence, thereby producing a Hausdorff space. Geodesic distance in this space is treated

computationally using a dynamic programming algorithm.

6.2 Shape Matching for Framed Paths

6.2.1 Lifting a Framed Path

In order to perform shape analysis using our coordinate system, we need a concrete descrip-

tion of how to choose a representative Φ P S˝pPC2q of an arbitrary framed path pγ, V q P S.

199

In this chapter it will frequently be useful to work in quaternionic coordinates. Accordingly,

we define SpPHq to be the L2-sphere in PH (with L2 metric induced by the identification

C2 « H) and

S˝pPHq :“ tq P SpPHq | qptq ‰ 0 @ tu.

In order to lift to the L2-sphere we are assuming that lengthpγq “ 2, which can be easily

arranged as a preprocessing step.

Our approach to lifting a framed path is to use axis-angle coordinates. Any element

of SOp3q besides ˘Id3ˆ3 has a unique representation as p~n, βq up to multiplication by ´1,

where ~n P R3 is a unit vector along the 1-dimensional subspace fixed by the rotation, and

β is the angle rotated around this axis. Since the identity corresponds to a rotation by

β “ 0, there is no well-defined choice of ~n to represent the identity. Note that p~n, βq and

p´~n,´βq clearly correspond to the same rotation. There are easy and numerically stable

algorithms for determining the axis-angle coordinates of rotation matrix—see, e.g., Chapter

16 of [27]. Moreover, it is well-known that the unit quaternion corresponding to SOp3q under

the quaternionic version of FrameHopf is cospβ{2q ` sinpβ{2q~n. In this representation, we

are thinking of ~n as an element of the purely imaginary quaternions under their obvious

identification with R3.

Thus we have a lifting procedure taking a framed path pγ, V q P S to an element of

S˝pPC2q, modulo a technical issue which we will address momentarily. The procedure first

represents pγ, V q as ppT, V,W q, }γ1}q P PpSOp3q ˆ R`q. We then convert to axis-angle co-

ordinates pp~nptq, βptqq, }γ1ptq}q, choosing p~nptq, βptqq to be a smooth path in S2 ˆ R. By

continuity, there is a unique way to make this choice up to global multiplication by ´1.

Finally, we lift to

qptq “a

}γ1} pcospβ{2q ` ~n sinpβ{2qq .

200

The remaining technical issue that this procedure is not well-defined for a framed curve

pγ, V q such that pT ptq, V ptq,W ptqq “ Id3ˆ3 for some t P r0, 2s. We get around this issue

by noting that we are really interested in equivalence classes up to rotations. Thus we can

preprocess by applying some rotation to pγ, V q so that its image curve in SOp3q does not hit

the identity. A generic framed curve has the property that pT ptq, V ptq,W ptqq ‰ Id3ˆ3 for all

t, so such rotations are plentiful.

6.2.2 Optimizing Geodesic Distance Over Rotations

The moduli space of framed paths S admits a free action of SOp3q by rotations. It is easy

to see that the natural metric gS on S is invariant under the SOp3q action, thus gS descends

to a well-defined metric on the quotient space S{SOp3q. We described explicit geodesics in

S in Section 4.5.1 and our goal in this section is to describe geodesics in S{SOp3q. This can

be restated as finding a horizontal geodesic in S with respect to SOp3q. That is, let pγ0, V0q,

pγ1, V1q P S. Then the horizontal geodesic joining the equivalence classes in S{SOp3q is

realized as the geodesic in S between pγ0, V0q and A ¨ pγ1, V1q, where A P SOp3q is given by

A “ argmintdistSppγ0, V0q, B ¨ pγ1, V1qq | B P SOp3qu

and distS is geodesic distance in S. Thus geodesic distance in S{SOp3q between the equiva-

lence classes of framed paths is

mintdistSppγ0, V0q, A ¨ pγ1, V1qq | A P SOp3qu. (6.2)

The goal of this section is to give explicit descriptions of the minimizing rotation A and

geodesic distance distS .

201

We remark here that algorithms for optimizing geodesic distance over rotations are abun-

dant in the elastic shape recognition literature for a variety of shape space regimes. These

algorithms typically involve matrix decompositions. Such decompositions are numerically

efficient, but it is nonetheless useful and aesthetically pleasing to have a closed-form solution

for the rotation registration problem. Our closed-form solution in quaternionic coordinates

also allows us to write out solutions in complex and axis-angle coordinates. This allows us

to compute rotation-optimized geodesic distance without the need to lift data points from

S to S˝pPHq.

Recall from Theorem 3.1.12 that S is isometrically double-covered by the open subset of

the L2 sphere in PC2

S˝pPC2q “ tΦ P PpC2

zt0uq | 〈Φ,Φ〉L2 “?

2u.

We saw in Equation 3.15 that the right action of SUp2q on S˝pPC2q corresponds under pH

to the rotation action of SOp3q on S. The SUp2q-action in the covering space turns out

to be simpler to work with than the SOp3q-action on S. If Φj P S˝pPC2q corresponds to

pγj, Vjq P S for j “ 0, 1, then we can replace the computation of (6.2) with the computation

min tdistpΦ0,Φ1 ¨ Aq | A P SUp2qu , (6.3)

where dist is geodesic distance in SpPHq—for the remainder of this chapter, we slightly abuse

notation and take the convention that dist always refers to geodesic distance in whichever

space makes sense in context. The computation (6.3) can in turn be simplified by applying

the identifications of Section 2.4. Namely, let qj P PH correspond to Φj for j “ 0, 1 under the

identification induced by C2 « H. We also identify SUp2q « S3. The next lemma translates

the SUp2q-action into quaternionic coordinates.

202

Lemma 6.2.1. Let Φ “ pφ, ψq P PC2 correspond to the quaternionic path q P PC2 and let

A “

¨

˚

˝

u v

´v u

˛

‹

‚

P SUp2q

correspond to p “ u` vj P S3. Then the right action of A on Φ corresponds to quaternionic

right multiplication of q by p, q ¨ p.

Proof. This is a straightforward calculation. The group action in complex coordinates is

Φ ¨ A “ puφ´ vψ, vφ` uψq.

On the other hand, p acts on q “ φ` ψj by

q ¨ p “ pφ` ψjq ¨ pu` vjq “ pφu` ψjvjq ` pφvj` ψjuq “ puφ´ vψq ` pvφ` uwqj, (6.4)

where the second equality follows by noting that for any complex number x` iy,

jpx` iyq “ xj´ yk “ px´ iyqj. (6.5)

The right hand side of (6.4) is identified with puφ´ vψ, vφ` uψq.

Then (6.2) becomes

min

distpq0, q1 ¨ pq | p P S3(

. (6.6)

From the discussion in Section 4.5.1 and the fact that

Re 〈Φ0,Φ1〉C2 “ Repq0 ¨ q1q

203

we see that dist is given in quaternionic coordinates by

distpq0, q1q “?

2 arccos1

2Re

ż 2

0

q0q1 dt.

The next theorem shows that (6.6) has a simple closed-form solution.

Theorem 6.2.2. Let q0 and q1 denote points of SpPC2q in quaternionic coordinates. Then

argmin


“pq

}pq}H,

where

pq “

ż 2

0

q1 ¨ q0 dt P H.

We conclude that geodesic distance in SpPC2q{SUp2q is given by

min


“?

2 arccos1

2

›

›

›

›

ż 2

0

q1 ¨ q0 dt

›

›

›

›

H.

Proof. Let q0, q1 P SpPHq. Our goal is to find a solution to

pp :“ argmin


“ argmin

"

?2 arccos

ˆ

1

2Re

ż 1

0

q0pq1 ¨ pq dt

˙

| p P S3

*

.

Any solution pp satisfies

pp “ argmax

"

Re

ż 2

0

q0pq1 ¨ pq dt | p P S3

*

“ argmax

"

Re

ż 2

0

q0 ¨ p ¨ q1 dt | p P S3

*

“ argmax

"

Re

ˆ

p

ż 2

0

q1 ¨ q0 dt

˙

| p P S3

*

.

204

The last line follows by noting that for any quaternions q “ Q0 ` iQ1 ` jQ2 ` kQ3 and

p “ P0 ` iP1 ` jP2 ` kP3,

Repqpq “ Q0P0 ´Q1P1 ´Q2P2 ´Q3P3 “ Reppqq,

and that p is a constant.

Let

pq :“

ż 2

0

q1 ¨ q0 dt P H.

We claim that pp “ pq{}pq}H. Indeed, under the identification H « R4,

Re

ˆ

p

ż 2

0

q1 ¨ q0 dt

˙

“ 〈p, pq〉R4 ,

and this claim is just an application of the obvious fact that for any w P Rn,

argmaxt〈v, w〉Rn | v P Sn´1

Ă Rnu “

w

}w}.

Therefore geodesic distance between the equivalence classes of q0 and q1 in SpPC2q{SUp2q is

given by

min


“?

2 arccos1

2Re

˜

pq

}pq}Hpq

¸

“?

2 arccos1

2}pq}H

“?

2 arccos1

2

›

›

›

›

ż 2

0

q1 ¨ q0 dt

›

›

›

›

H

We easily obtain the geodesic distance formula in complex and axis-angle coordinates.

205

Corollary 6.2.3. Geodesic distance between equivalence classes of points Φ0 and Φ1 in

SpPC2q{SUp2q is given by?

2 arccos 12χ, where

χ2“

›

›

›

›

ż 2

0

φ0φ1 ` ψ0ψ1 dt

›

›

›

›

2

C`

›

›

›

›

ż 2

0

ψ0φ1 ´ φ0ψ1 dt

›

›

›

›

2

C

If

pγ0, V0q “ pp~n0, β0q, }γ10}q, and pγ1, V1q “ pp~n1, β1q, }γ

11}q

are axis-angle representations of the framed curves associated to Φ0 and Φ1, then geodesic

distance between the equivalence classes of these points in S{SOp3q is given by?

2 arccos 12ζ,

where

ζ2“

›

›

›

›

ż 2

0

a

}γ10} ¨ }γ11}

ˆ

cos

ˆ

β0

2

˙

cos

ˆ

β1

2

˙

` sinpβ0{2q sinpβ1{2q 〈~n0, ~n1〉˙

dt

›

›

›

›

2

R

`

›

›

›

›

ż 2

0

a

}γ10} ¨ }γ11}

ˆ

sin

ˆ

β0

2

˙

cos

ˆ

β1

2

˙

~n0 ´ cos

ˆ

β0

2

˙

sin

ˆ

β1

2

˙

~n1

´ sin

ˆ

β0

2

˙

sin

ˆ

β1

2

˙

~n0 ˆ ~n1

˙

dt

›

›

›

›

2

R3

.

Proof. Let points Φj “ pφj, ψjq, for j “ 1, 2, be identified with the quaternionic paths

qj “ φj ` ψjj. Once again using the observation (6.5), we obtain

q0 ¨ q1 “ pφ0 ` ψ0jq ¨ pφ1 ´ ψ1jq

“ pφ0φ1 ` ψ0ψ1q ` pψ0φ1 ´ φ0ψ1qj.

Thus

›

›

›

›

ż 2

0

q0 ¨ q1 dt

›

›

›

›

2

H“

›

›

›

›

ż 2

0

pφ0φ1 ` ψ0ψ1q ` pψ0φ1 ´ φ0ψ1qj dt

›

›

›

›

2

H

“

›

›

›

›

ż 2

0

φ0φ1 ` ψ0ψ1 dt

›

›

›

›

2

C`

›

›

›

›

ż 2

0

ψ0φ1 ´ φ0ψ1 dt

›

›

›

›

2

C.

206

To prove the second claim, write qj “b

}γ1j}pcospβj{2q ` ~nj sinpβj{2qq for j “ 0, 2. Then

q0 ¨ q1 “a

}γ10} ¨ }γ11}pcospβ0{2q ` ~n0 sinpβ0{2qqpcospβ1{2q ´ ~n1 sinpβ1{2qq

“a

}γ10} ¨ }γ11}pcospβ0{2q cospβ1{2q ` sinpβ0{2q sinpβ1{2q 〈~n0, ~n1〉` sinpβ0{2q cospβ1{2q~n0

´ cospβ0{2q sinpβ1{2q~n1 ´ sinpβ0{2q sinpβ1{2q~n0 ˆ ~n1q.

The last line makes use of the following identity for multiplication of purely imaginary

quaternions q “ Q1i`Q2j`Q3k and p “ P1i` P2j` P3k:

q ¨ p “ ´pQ1P1 `Q2P2 `Q3P3q ` pQ2P3 ´Q3P2qi` p´Q1P3 `Q3P1qj` pQ1P2 ´Q2P1qk

“ ´ 〈q, p〉` q ˆ p,

where we are utilizing the identification of purely imaginary quaternions with R3 to write

the last line. Now the result follows by integrating, taking the quaternionic norm and

decomposing accordingly.

6.2.3 Modding out by the Diff`-Action

In this section we aim to describe geodesic distance in the shape space

S{pSOp3q ˆDiff`pr0, 2sqq,

where Diff`pr0, 2sq acts as usual by reparameterizations (see Section 4.4), but we will see

momentarily that there is a problem with this goal. We first record the following lemma,

whose proof is obvious by definition.

Lemma 6.2.4. The actions of SOp3q and Diff`pr0, 2sq on S commute.

207

Thus we can first consider only modding out by Diff`pr0, 2sq. We wish to define a dis-

tance metric on S{Diff`pr0, 2sq by optimizing geodesic distance between points in S over

reparameterizations. Recall that the geodesics of S are the geodesics of S˝pPC2q, but that a

geodesic between points of S˝pPC2q may cross through the set SpPC2qzS˝pPC2q. Nonethe-

less, geodesic distance in SpPC2q is still a well-defined, geometrically-motivated metric on

S. Thus we might try to shift the problem of geodesic distance in S{Diff`pr0, 2sq to that

of geodesic distance in SpPC2q{Diff`pr0, 2sq. Here we run into a major problem in that this

quotient is not Hausdorff. This can be demonstrated by adapting an example from Section

3.5 of [76] as follows.

Example 6.2.1. Let ρ : r0, 2s Ñ r0, 2s be a smooth bijective map which is non-decreasing

and constant on some subinterval I Ă r0, 2s. The sequence of diffeomorphisms tρnu defined

by ρnptq “ p1´ 1{nqρptq ` t{n converges in C8 to ρ. For any Φ P S˝pPC2q, this sequence of

diffeomorphisms acts on Φ to produce a sequence tΦnu in the Diff`pr0, 2sq-orbit of Φ given

by

Φnptq “a

p1´ 1{nqρ1ptqΦpp1´ 1{nqρptq ` t{nq.

This sequence converges in C8 to a smooth path Φ8 P SpPC2q. Moreover, we see that Φ8

lies in SpPC2qzS˝pPC2q, since Φ8 has norm zero on the interval I. Since the Diff`pr0, 2sq-

action fixes S˝pPC2q, it must be that Φ8 is not in the Diff`pr0, 2sq-orbit of Φ. We conclude

that the Diff`pr0, 2sq-orbits of Φ and Φ8 are distinct but arbitrarily close to one another,

thus SpPC2q is not Hausdorff.

The problem boils down to the fact that the Diff`pr0, 2sq-orbits are not closed. The most

obvious remedy for this issue is to partition SpPC2q into slightly larger equivalence classes.

One possibility is to consider elements of SpPC2q up to Frechet equivalence, where we enlarge

the quotient set from Diff`pr0, 2sq to the set of monotone relations (see [76], Section 3.5 for

208

a definition). Another approach, and the one we adopt here, is to partition SpPC2q into

closures of Diff`pr0, 2sq orbits (this is the approach taken in, e.g., [64, 68]).

Let us precisely define the space of interest. We define the shape space of framed paths

pS to be the collection of equivalences classes JΦK, where JΦK is the set

JΦK “ closureL2pta

ρ1Φpρq | ρ P Diff`pr0, 2squq X SpPC2q.

That is, Φ P JΦ0K if and only if Φ P SpPC2q and there exists a sequence of diffeomorphisms

ρn such that Φ0pρnq Ñ Φ in L2. We claim that optimized geodesic distance now gives a

well-defined distance metric on pS. Define

dpSpJΦ0K, JΦ1Kq “ inftdistpΦ0,

a

ρ1Φpρqq | ρ P Diff`pr0, 2squ,

where dist is geodesic distance in SpPC2q. This can be written in quaternionic coordinates

as

dpSpJq0K, Jq1Kq “ inf

"

?2 arccos

1

2Re

ż 2

0

a

ρ1 ¨ q0 ¨ q1pρq dt | ρ P Diff`pr0, 2sq

*

.

We now show that the above definitions are reasonable from two perspectives. First, it

is necessary to show that dpS is actually a distance metric.

Proposition 6.2.5. The map dpS is a distance metric on pS.

Proof. The most important point here is that

dpSpJq0K, Jq1Kq “ 0 ô Jq0K “ Jq1K.

209

The left implication is obvious, so we focus on the right implication. If the distance between

Jq0K and Jq1K is zero, then there exists a sequence of diffeomorphisms ρn such that

Re

ż 2

0

a

ρ1n ¨ q1pρnq ¨ q0 dtÑ 2 as nÑ 8,

which impliesż 2

0

a

ρ1n ¨ q0 ¨ q1pρq dtÑ 2 as nÑ 8.

In other words,?ρ1n ¨ q1pρnq converges weakly in L2 to q0. On the other hand,

?ρ1n ¨ q1pρnq

and q0 are elements of the L2-sphere SpPHq, so their L2-norms agree for all n. Together

these statements imply that?ρ1n ¨ q1pρnq Ñ q0 in L2. But then Jq1K “ Jq0K, and this proves

the claim.

The map dpS is obviously nonnegative-valued and symmetric, and the triangle inequality

follows because it is satisfied by geodesic distance, dist.

Next we wish to show that these equivalence classes are geometrically meaningful in some

sense; i.e., that they are not so large that they are no longer useful for shape recognition

applications.

Proposition 6.2.6. Let q P Jq0K and let pH1pqq “ γ and pH1pq0q “ γ0. Then the images of γ

and γ0 agree.

Proof. Let tρnu be a sequence of diffeomorphisms such that?ρ1nq0pρq Ñ q in L2. Then

there exists a subsequence tρnku such thata

ρ1nkq0pρnkq Ñ q in L2 and pointwise almost

everywhere. Therefore γ0pρnkq Ñ γ pointwise almost everywhere. Let E a subset of r0, 2s

such that γpρnkq converges pointwise on E and such that r0, 2szE has zero measure. We

first claim that the images of γ0 and γ agree on E. Indeed, for t P E, let t0 “ lim ρnkptq.

Then γ0pρnkptqq Ñ γ0pt0q “ γptq. Similarly, for fixed t0 P E, let t “ lim ρ´1nkpt0q. Then

210

γ0pρnkptqq Ñ γptq “ γpt0q. Therefore the images of γ0 and γ agree on E. Since each curve is

smooth, we conclude that the images of γ0 and γ agree on all of r0, 2s.

6.2.4 Computationally Optimizing Over Reparameterizations

We have shown that dpS is a well-defined metric. Now the question becomes whether it

can actually be computed. In this section, we adapt a well-known dynamic programming

algorithm in order to approximate dpS . The approach taken here follows the exposition in

[64], Section 3.4.2.

The task is accomplished by discretizing and approximating diffeomorphisms of r0, 2s

by piecewise linear homeomorphisms of r0, 2s with vertices on a fixed grid such that each

segment has positive slope. The general algorithm begins by choosing partitions 0 “ t11 ă

t12 ă ¨ ¨ ¨ ă t1n1“ 2 of the x-axis and 0 “ t21 ă t22 ă ¨ ¨ ¨ ă t2n2

“ 2 of the y-axis of r0, 2s ˆ r0, 2s.

For the sake of simplicity, let us take the partitions to be the same and evenly spaced, so

that we can use the notation 0 “ t1 ă t2 ă ¨ ¨ ¨ ă tn “ 2. Then gridpoints on r0, 2s ˆ r0, 2s

take the form

a “ aij “ pti, tjq “

ˆ

2pi´ 1q

n´ 1,2pj ´ 1q

n´ 1

˙

.

Each grid point has a collection of potential successors. These are points ak` on the grid such

that the segment connecting ak` to aij has positive slope and does not pass through any other

grid point. The first condition is expressed simply as k ă i and ` ă j. The second condition

is included so that paths can be unambiguously expressed as concatenations of segments

joining vertices, and is expressed as gcdpk ´ i, `´ jq “ 1 (making use of the even spacing of

the partition). Thus admissible paths are represented as sequences pa1, a2, . . . , amq, where

each ai is a gridpoint, and we interpret the path as the concatenation of the edges joining ai

to ai`1.

211

We now introduce a variety of relevant functionals. For fixed q0, q1 P SpPHq and an

increasing linear diffeomorphism ρ : ra, bs Ñ rc, ds between subintervals of r0, 2s, we define

F pρq :“1

2Re

ż b

a

a

ρ1 ¨ q0 ¨ q1pρq dt

“1

2

ˆ

d´ c

b´ a

˙1{2

Re

ż b

a

q0 ¨ q1pρq dt.

This is referred to as the energy of ρ. For a grid point aij and an admissible successor ak` we

will use the notation F pak`, aijq for the energy of the linear diffeomorphism taking rtk, tis to

rt`, tjs. This can also be interpreted as the energy of the edge joining the points ak` and aij.

Remark 6.2.7. Note that the energy functional is derived from the geodesic distance between

q0 and q1 in SpPHq. One might consider using a functional based on the distance formula in

SpPHq{Up2q given by Theorem 6.2.2. We will see that the norm appearing in that formula

would cause problems with the proposed algorithm.

By a slight abuse of notation, we define the energy of an admissible path pa1, . . . , amq to

be the sum

F pa1, . . . , amq :“m´1ÿ

i“1

F pai, ai`1q.

Thus the energy of a path starting at a1 “ pt1, t1q “ p0, 0q and ending at am “ ptn, tnq “ p2, 2q

is, upon applying arccos and multiplying by?

2, an approximation of the geodesic distance

in SpPCq between Φ0 and?ρ1Φ1pρq, where ρ is the diffeomorphism approximated by the PL

path pa1, . . . , amq.

Turning toward optimizing over such PL paths, we define Dpaq to be the energy of the

optimal path joining the origin to the gridpoint a. By optimal we mean of maximum energy.

Large energy will be converted to small geodesic distance after taking arccos. More precisely,

212

we define

Dpaq :“ maxtF pa1, . . . , amq | pa1, . . . , amq is admissible, a1“ p0, 0q, am “ au.

Thus the problem of minimizing geodesic distance over admissible paths is equivalent to

computing Dp2, 2q.

The following lemma is standard (see, e.g., [64]), but it is important in that it shows that

dynamic programming is an appropriate approach to computing Dpaq.

Lemma 6.2.8. If Dpaq is realized by pa1, . . . , am “ aq, then for any grid point ai in this

path, Dpaiq is realized by pa1, . . . , aiq.

Proof. Let pa1, . . . , am “ aq be such an optimal path. If Dpaiq is not realized by pa1, . . . , aiq,

then there exists an admissible path pb1, . . . , bj “ aiq with strictly higher energy. But then

the concatenation pb1, . . . , bj, ai`1, . . . , amq is an admissible path ending at a with higher

energy than the original minimizing path, and we have obtained a contradiction.

Lemma 6.2.8 means that the problem of finding the optimal PL ρ is amenable to the

dynamic programming approach, and we take our functional equation to be

Dpaq “ maxtDpa1q ` F pa1, aq | a1 is an admissible predecessor for au.

The dynamic programming algorithm calculates Dp2, 2q by calculating Dpaq for every grid-

point in row-by-row order. The procedure is outlined in Algorithm 1. The algorithm is

essentially the same as the one employed in, e.g., [57] and the pseudocode is essentially the

same as that of [64], Section 3.4.2. The difference here is that the energy functions used

are specific to our setup, and that the presence of arccos in the geodesic distance formula

causes us to find maximum energy paths (as opposed to minimum energy paths in other

algorithms).

213

Algorithm 1 PL Optimization

Input: Size of grid partition n, Edge energies F pa1, a2q for all grid points a1 and a2

Output: Energy minimizing path Π from p0, 0q to p2, 2q and its energy Dp2, 2qInitialize D: Set Dp0, 0q “ 0, Dpaq “ ´8 for all other gridpoints aInitialize Π: Set Π “ pp2, 2qq

1: for r “ 1 to n do2: for c “ 1 to n do3: for Each predecessor a of acr do4: if Dpaq ` F pa, acrq ą Dpacrq then5: Set Dpacrq “ Dpaq ` F pa, acrq6: Set P pacrq “ a7: end if8: end for9: end for10: end for11: Set a “ p2, 2q12: while a ‰ p0, 0q do13: Set a “ P paq14: Prepend a to Π15: end while

We explore a simple example in Figures 6.1, 6.2 and 6.3. Figures 6.1 and 6.2 show

geodesics between a helix and a semicircle. Each curve has its Frenet framing, which is not

pictured for visual clarity. Figure 6.1 shows the geodesic in SpPHq, where the beginning

and ending curves have arclength parameterizations. The next figure shows the geodesic

after the helix has been optimized over Diff`pr0, 2sq. We can see from the figures that the

geodesics look quite different. Figure 6.3 gives a visualization of the reparameterization of

the helix and shows the corresponding gedesic distances.

A major source of future work will be to rigorously apply this algorithm to perform shape

recognition and modelling for, e.g., protein backbones, immersed surfaces, camera tracking,

root systems, hand gestures and aircraft trajectories.

214

Figure 6.1: Geodesic before reparameterization.

Figure 6.2: Geodesic after reparameterization.

215

~1.55 ~0.98

Figure 6.3: Geodesic distances.

216

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