Contents Introduction - ETH · ETH Zurich in 2016. Disclaimer. This is a preliminary version. Please report any typos, mistakes, comments etc. to [email protected]. Acknowledgements

RIEMANNIAN GEOMETRY

MARC BURGERSTEPHAN TORNIER

Abstract. These are notes of the course Differential Geometry II held atETH Zurich in 2016.

Disclaimer. This is a preliminary version. Please report any typos, mistakes,comments etc. to [email protected].

Acknowledgements. Thanks to those who pointed out typos and mistakesas these notes were written, in particular S. Ammann, V. Junet, W. Liu, M.Wasem and J. Wettstein.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Review of Differential Geometry . . . . . . . . . . . . . . . . . . . . . 42. Riemannian Metrics, Covariant Derivative and Geodesics . . . . . . . 283. Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554. What’s Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Introduction

In this introduction, we outline how Bernhard Riemann resolved three importantproblems of his time with the definition of what we now call a Riemannian manifold

in his habilitation treatise [Rie54]. The three problems revolve around how to dealwith the geometry of curves and surfaces after Gauss, Euclid’s fifth postulate andmanifolds which “can’t be seen”. This is a largely informal section. Its notation isdeliberately old-fashioned and should not be taken too seriously.

Given a curve −→c : R → E2, t 7→ −→c (t) in Euclidean space and points a, b ∈ R

with a < b one can talk about the length of the segment [−→c (a),−→c (b)]:

b

b

E2

−→clength(−→c [a, b]) =

∫ b

a

‖−→c ′(t)‖ dt.

This defines the intrinsic metric of −→c . Also, a curve −→c with −→c ′(t) 6= 0 for allt ∈ R can be reparametrized which is really saying that −→c (R) with its intrinsicmetric is isomorphic to R. This already finishes the metric classification of curves.

However, there are interesting invariants describing how the curve sits in Eu-clidean space such as curvature: A circle of radius r is defined to have constantcurvature 1/r. Given a curve −→c : R → E2 and an orientation of −→c via normalvectors −→n : R → E2 the curvature of −→c at the point −→c (t) is defined as

K(−→c (t)) = ±1

radius of osculating circle at −→c (t)

Date: June 2, 2016.

1

2 MARC BURGER STEPHAN TORNIER

where the sign depends on whether −→n (t) points towards or away from the osculatingcircle which is the unique circle tangent of order three at −→c (t).

b

b

E2

+

−

In other words, let t1 ≤ t ≤ t2. Then if −→c (t1), −→c (t) and −→c (t2) are not collinear,there is a unique circle through the three of them. Take the limit as t1 and t2 tendto t from below and above respectively.

Given a surface S in E3 and points p and q on S the intrinsic distance of p andq is defined as

b

p

b

q

dS(p, q) = inf

length(−→c )

∣∣∣∣∣∣∣−→c : [0, 1] → E3 smooth :

−→c ([0, 1]) ⊆ S−→c (0) = p−→c (1) = q

One may then ask what the shortest path between p and q is. This questionwas studied at least since the work of the Bernoulli’s on variational questions. Thestudy of the distance on S and the determination of shortest paths – geodesics –amount to the intrinsic geometry of S.

A surface S as above, or rather a piece of it, can – being two-dimensional – beparametrized by two real parameters, say u and v. A curve −→c (t) lying on S canthen be given by −→c (t) = (x(u(t), v(t)), y(u(t), v(t)), z(u(t), v(t)))T and we compute

‖−→c ′(t)‖2 = E u′(t)2 + 2F u′(t)v′(t) +G v′(t)2.

The expression ds2 = E du2 + 2F dudv + G dv2 was used to refer to the first

fundamental form. This notation should not be confused with notation from thetheory of differential forms but rather be granted as moonshine from the 18thcentury. The point is that the inner distance on S determines and is determined bythe first fundamental form.

What can be said about curvature in the case of surfaces? The mathematician’sway to deal with it is to reduce the question to the case of curves discussed above.Let m ∈ S and let −→n denote a normal vector to S at p. Further, let −→u ∈ TmS bea unit tangent vector and let P denote the plane spanned by −→n and −→u . Then theintersection of S and P is a curve −→c and we define

Π : TmS → R, −→u 7→ K(−→c at m).

The map Π assumes both a minimum K1(m) and a maximum K2(m) which maytake arbitrary signs. Gauss defined curvature to be the product of these two prin-cipal curvatures: K(m) := K1(m)K2(m). This makes sense geometrically:

RIEMANNIAN GEOMETRY 3

b

m

K1(m),K2(m) > 0

bm

K1(m),K2(m) < 0

bm

K1(m) < 0, K2(m) > 0

He proved the following Theorema Egregium.

Theorem (Gauss). The curvature KS of a surface S only depends on the intrinsicgeometry of S. It admits a formula in terms of the coefficients E, F and G of thefirst fundamental form.

He was also conscious of the fact that this notion of curvature assumes that we

can see the surface under consideration inside Euclidean space E3 and hence doesnot lend itself to investigate e.g. projective space P2(R) which is defined as

P2 R := R3 \0/ ∼where x ∼ y if and only if there is λ ∈ R \0 such that x = λy. The Klein bottleis another example.

We now turn to Euclid’s fifth postulate, the parallel postulate. Euclid’s approachto the geometry was to define the objects under consideration, namely points andlines, and to postulate “self-evident” statements, such as that there should be aunique line going through two distinct points, with the help of which further objectscould be introduced and statements made. His fifth postulate stated that given aline l and a point P not on l there is a unique line through P not intersecting l.

In 1829, Lobatchevsky constructed a geometry that did not satisfy Euclid’s fifthpostulate. In fact Gauss had already known about this but had not publishedanything as he feared his reputation.

Finally, Bernhard Riemann introduced n-dimensional manifolds and their tan-gent spaces in his habilitation treatise in 1854. He suggested to smoothly put ascalar product 〈−,−〉m on each tangent space TmM of a manifold of M whichallows one to define the notion of length of a curve in M and geodesics. He alsointroduced what is now called the Riemann curvature tensor, a revolutionary wayto measure curvature. It is related to sectional curvature, Ricci curvature and scalarcurvature. Sectional curvature is a curvature notion that generalizes Gauss curva-ture: Let P ⊆ TmM be a two-dimensional vector subspace. Given ε > 0, considerthe “circle” CP (m, ε) of points in M that are reached from m by following the ge-odesic flow along unit tangent vectors in P . Then the length of the closed curveCP (m, ε) is given by

length(CP (m, ε)) = 2πε− π

3σm(P )ε3 +O(ε4)

where σm(P ) is the sectional curvature which, as a consequence, is an intrinsicobject. By the way, note that it is a big mystery why there never is a term in ε2.

Riemann ended his habilitation treatise with the example of the open unit diskx ∈ Rn |∑n

i=1 x2i < 1 in Rn on which he defines the metric

ds2 =dx21 + · · ·+ dx2n

(1−∑ni=1 x

2i )

2 .

This coincides exactly with Lobatchevsky’s non-Euclidean plane. Overall, Riemann’shabilitation treatise dealt with many problems in one stroke.


Course Outline. In this course, we will define Riemannian metrics on smoothmanifolds and use them to study geodesics. We also study derivates of vector fieldswith respect to each other, leading to the notion of connection. In general, thereare many possible connections. However, on a Riemannian manifold there is a pre-ferred one, the Levi-Civita connection. Using the framework of connections we willextremely efficiently identify geodesics as curves whose acceleration vanishes. Afterintroducing various notions of curvature we move on to the relation between localcurvature properties of a Riemannian manifold and its global properties, e.g. prop-erties of its fundamental group or de Rham cohomology spaces. For instance, thereis the following classical theorem.

Theorem (Berger-Klingenberg). LetM be a simply-connected smooth manifold andassume that 1/4 < σm(P ) ≤ 1 for all m ∈ M and P ≤ TmM be two-dimensional.Then M is homeomorphic to Sn where n = dimM .

In the context of dynamical systems and ergodic theory we shall see that thenegative curvature world is very different from the above.

1. Review of Differential Geometry

In this chapter, we recall some fundamentals from differential geometry. When-ever possible, we refer to the lecture notes [BT15] of the Differential Geometry Icourse of the fall semester 2015. The main new topics include a more thoroughtreatment of vector fields, basic concepts of vector bundles, a short section on Liegroups and one on covering maps and fibrations.

1.1. Smooth Manifolds and Smooth Maps. Recall that a topological manifold

of dimension n is a topological space M which is Hausdorff, second-countable andlocally homeomorphic to Rn. A smooth atlas A on M is a collection of charts A =(Uα, ϕα) | α ∈ A such that

⋃α∈A Uα = M and all coordinate transformations

θβα = ϕβ ϕ−1α : ϕα(Uα ∩ Uβ) → ϕβ(Uα ∩ Uβ) is smooth.

Uα Uβ

ϕα(Uα) ϕβ(Uβ)

ϕα ϕβ

θβα

θαβ

One shows that every smooth atlas is contained in a unique maximal one. Asmooth manifold of dimension n is a topological n-manifold together with a max-imal smooth atlas. Recall that an exponent of a manifold typically indicates itsdimension. Let Mm and Nn be smooth manifolds and let f : M → N be a map.We say that f is differentiable at p ∈ M if there are charts (U,ϕ) at p ∈ M and(V, ψ) at f(p) such that f(U) ⊆ V and ψfϕ−1 : ϕ(U) → ψ(V ) is differentiable atϕ(p). Further, f is smooth if it is continuous and if for all charts (U,ϕ) of M and(V, ψ) of N satisfying f(U) ⊆ V the map ϕ(U) → ψ(V ) is smooth. Note that thecontinuity assumption in the last definition is crucial as a map that is wild to theextent that there are no pairs of charts as above should not be termed smooth.Given a map f : M → N that is differentiable at p ∈ M the rank rankp(f) of f at


p is defined to be the rank of the linear map Dpf : TpM → Tf(p)N . We furtherrecall that a smooth map f : Mm → Nn is an immersion if rankp(f) = m for allp ∈ M , an submersion if rankp(f) = n for all p ∈ M and an embedding if it is animmersion and a homeomorphism onto its image. In the last situation, f(N) is aregular submanifold and f is a diffeomorphism onto its image.

An important tool to construct manifolds is the following.

Theorem 1.1. Let Mm and Nn be smooth manifolds and let f : Mm → Nn be asmooth map of constant rank k. Then for any y ∈ f(M) the subset f−1(y) ⊆M isa regular submanifold of dimension m− k.

By applying Theorem 1.1 in the case where M and N are vector spaces and f isa linear map, note that it is a non-linear version of the first isomorphism theoremfrom linear algebra.

Example 1.2. (Orthogonal Groups). Let q : Rn → R be a quadratic form. Then thereis a matrix A ∈ Sym(n,R) such that q(x) = xTAx. Recall Silvester’s classificationof quadratic forms via their signature and assume from now on that q is non-degenerate, equivalently, that A ∈ GL(n,R). For example, if n = 2 we may haveq(x1, x2)

T = x21 − x22. Notice that q producing vectors of length zero says nothingabout it being degenerate or non-degenerate. The orthogonal group of q is

O(q) := h ∈ GL(n,R) | q(hx) = q(x) ∀x ∈ Rn= h ∈ GL(n,R) | hTAh = A

Also recall that O(q) is compact if and only if q is either positive or negative definite.Regardless of the signature of q, as long as it is non-degenerate, O(q) has dimensionn(n− 1)/2 as can be seen using Theorem 1.1: Namely, consider the map

f : GL(n,R) → GL(n,R), g 7→ gTAg.

In order to compute the derivative of f we recall that GL(n,R) acquires its manifoldstructure as an open subset of Mn,n(R) ∼= Rn·n. As such we identify Tg GL(n,R)with Mn,n(R) for every g ∈ GL(n,R). We compute

(Dgf)(Y ) = Y TAg + gTAY

for all Y ∈ Tg GL(n,R). Observe, that the map Dgf takes values in Sym(n,R). Infact every S ∈ Sym(n,R) is in the image of Dgf as

Dgf

(1

2A−1(g−1)TS

)= S.

Consequently, rank(Dgf) = n(n+1)/2 and the constant rank theorem implies thatO(q) = f−1(A) is a submanifold of dimension n2 − n(n+ 1)/2 = n(n− 1)/2.

The case in which q is degenerate is left to the reader.

Example 1.3. (Symplectic Group). Let V be a finite-dimensional real vector spaceand let ω : V ×V → R be an alternating bilinear 2-form. If ω is non-degenerate, i.e.x ∈ V | ω(x, y) = 0 ∀y ∈ V = 0, it is called symplectic. In this case, dimV = 2nand there is a basis e1, . . . , en, f1, . . . , fn such that ω(x, y) = xT Jy where

J =

(0 Id

− Id 0

)

Note the sharp contrast to the variety of quadratic forms. The symplectic group isgiven by

Sp(2n,R) = g ∈ GL(2n,R) | ω(gx, gy) = ω(x, y) ∀x, y ∈ R2n= g ∈ GL(2n,R) | gTJg = J.


It is an exercise to show that Sp(2n,R) is a submanifold of GL(n,R) of dimensionn(2n+ 1).

We shall see later that admitting a a symplectic 2-form is an interesting invariantfor manifolds, in contrast to being Riemannian.

1.2. Tangent Bundle. Given a smooth manifold Mm and p ∈ M , the tangent

space TpM of M at p is defined as the quotient of

Ap := (U,ϕ, ξ) | (U,ϕ) is a chart at p, ξ ∈ Rn

by the equivalence relation

(U,ϕ, ξ) ∼p (V, ψ, η) ⇔ Dϕ(p)(ψϕ−1)ξ = η.

In this way, TpM := Ap / ∼p is anm-dimensional vector space canonically attachedto a point p ∈M .

U V

b

ξb

ηϕ(U) ψ(V )

ϕ ψ

ψ ϕ−1

Note that every chart (U,ϕ) of M at p gives rise to a vector space isomorphismθ(U,ϕ,p) : R

m → TpM by setting ξ 7→ [(U,ϕ, ξ)].If f :M → N is a map which is differentiable at p ∈M then there is a canonical

linear map Dpf : TpM → Tf(p)N such that whenever (U,ϕ) is a chart at p and(V, ψ) is a chart at f(p) with f(U) ⊆ V , the diagram

Rmθ(U,ϕ,p)

//

Dϕ(p)ψfϕ−1

TpM

Dpf

Rmθ(V,ψ,f(p))

// Tf(p)N

commutes. There is an alternative, more classical and convenient way to definetangent vectors. It fits into the paradigm of reducing new problems to the case ofcurves. Consider the set Cp of pairs (I, c) consisting of an open interval I ⊆ R

containing 0 ∈ R and smooth curve c : I → M with c(0) = p. We declare (I, c) ∼p(J, γ) if there is a chart (U,ϕ) at p such that (ϕ c)′(0) = (ϕ γ)′(0).

Lemma 1.4. Let M be a smooth manifold and let p ∈ M . Further, let (U,ϕ) be achart of M at p. Then the map

Cp → Ap, (I, c) 7→ (U,ϕ, (ϕ c)′(0))

induces a well-defined bijection Cp/ ∼p→ Ap / ∼p.

The verification of Lemma 1.4 is left to the reader.


1.2.1. Tangent bundle. Finally, we now turn to the tangent bundle, see [BT15, Sec.2.3]. As a set, the tangent bundle of a smooth manifold M is defined as

TM :=⋃

x∈M

x × TxM = (x, v) | x ∈M, v ∈ TxM.

It comes with the canonical projection map π : TM → M, (x, v) 7→ x. Given anopen set U ⊆M we set TU = π−1(U). Further, if (U,ϕ) is a chart of M , we definethe map Dϕ : TU → ϕ(U)× Rn by (x, v) 7→ (ϕ(x), (Dxϕ)v).

Lemma 1.5. Let Mm be a smooth manifold. Then TM admits a topological 2m-manifold structure for which (TU,Dϕ) | (U,ϕ) chart is a smooth atlas.

1.2.2. Vector bundles. The tangent bundle is an example of the more general notionof vector bundle which we introduce now.

Definition 1.6. A vector bundle is a triple (π,E,B) consisting of a smooth mapπ : E → B of smooth manifolds such that

(i) π is surjective,(ii) there is an open cover (Ui)i∈I of B and a collection of diffeomorphisms hi

(i ∈ I)

hi : π−1(Ui) → Ui × Rn

such that hi(π−1(x)) = x × Rn, and(iii) for all i, j ∈ I, the map hi(hj |Ui∩Uj )−1 : (Ui ∩ Uj) × Rn → (Ui ∩ Uj) × Rn

given by (x, v) 7→ (x, gij(x)v), where gij : Ui ∩ Uj → GL(n,R), is smooth.

It is an exercise to show that the tangent bundle of a manifold is in fact a vectorbundle in the sense of Definition 1.6. Part (iii) of said definition entails that eachfiber π−1(x) (x ∈ B) has a well-defined vector space structure.

Example 1.7. As another example of vector bundles, recall the GrassmannianG(k, n) := L ⊆ Rn | dimL = k and define

E(k, n) = (L, v) | L ∈ G(k, n), v ∈ LIn a sense to be made precise, these vector bundles are universal.

Definition 1.8. Let (π,E,B) be a vector bundle.

(i) A (smooth) section of (π,E,B) is a (smooth) map s : B → E such thatπ s = idB.

(ii) The vector bundle (π,E,B) is trivial if there is a diffeomorphism h : E →B×Rn such that h|π−1(x) : π

−1(x) → x×Rn is a vector space isomorphismfor every x ∈ B.

Note that for any vector bundle, the zero section is smooth.

1.3. Vector Fields.

Definition 1.9. Let M be a smooth manifold. A (smooth) vector field on M is a(smooth) section X :M → TM of the tangent bundle.

Let Γ(TM) denote the span of all smooth vector fields on M . This space is notonly a real vector space but in fact a module over C∞(M): Given f ∈ C∞(M) andX ∈ Γ(TM) we define (fX)(x) := f(x)X(x) for all x ∈M .

Proposition 1.10. Let M be a smooth manifold. Then TM is trivial if and only ifthere are smooth vector fields X1, . . . , Xm on M such that (X1(x), . . . , Xm(x)) is abasis of TxM for all x ∈M .


A manifold is parallelizable if its tangent bundle is trivial. When mathemati-cians first looked at this notion they were surprised to find that there are non-parallelizable manifolds, for instance almost all spheres.

Example 1.11.

(i) It is evident that Rn is parallelizable for all n.(ii) We now show that also all tori T n := (S1)n are parallelizable. To this end,

we view S1 as the abelian group S1 := z ∈ C | |z| = 1. Recall thatfor every ∈ T n the left multiplication map L : T n → T n, z 7→ z is adiffeomorphism. A trivialization of T(T n) is thus given by

T(T n) → T n × T1(Tn), (, v) 7→ (,DL−1(v))

This argument applies in the more general context of Lie groups.(iii) As to the spheres, it is a fact that Sn is parallelizable if and only if

n ∈ 1, 3, 7. In these cases, the respective sphere can be identified as theunit length elements in a real division algebra, namely either the complexnumbers, the quaternions or the octonions. The case of S1 is covered above.

The key parallelizability does not lie in a manifold’s fundamental group but inits higher-dimensional fundamental groups. Given Proposition 1.10, the followingtheorem shows that all even-dimensional spheres, except S0, are not parallelizable.The proof we give is due to Milnor and is not very telling but nicely uses the theorydeveloped in part one of this course.

Theorem 1.12. The sphere Sn admits an everywhere non-zero smooth vector fieldif and only if n is odd.

Proof. We first show that the condition is sufficient: If n = 2m−1 we may constructa nowhere vanishing vector field as follows: Exhibit

S2m−1 =

(x1, y1), . . . , (xm, ym)

∣∣∣∣∣m∑

i=1

(x2i + y2i ) = 1

⊆ R2m .

Now, given the rotation

r(t) =

(cos t sin t− sin t cos t

)∈ SO(2)

define R(t) := r(t)⊕ · · · ⊕ r(t) acting on R2m. Then R(t)(S2m−1) ⊆ S2m−1 and wemay set

X(z) :=d

dt

∣∣∣∣t=0

t 7→ R(t)z

for all z ∈ S2m−1. One verifies that 〈X(z), z〉 = 0 and ‖X(z)‖ = 1. Therefore, X isa unit vector field on S2m−1.

To show necessity we argue by contradiction. Assume that there is a smoothmap X : Sn → Rn+1 such that 〈X(x), x〉 = 0 for all x ∈ Sn. Multiplying with anappropriate smooth function we may assume that ‖X(x)‖ = 1 for all x ∈ Sn. Now,let ε > 0 and define fε : Sn → Rn+1 by fε(x) := x+ εX(x). Then fε(x) has norm√1 + ε2 for every x ∈ X since

〈fε(x), fε(x)〉 = 〈x+ εX(x), x+ εX(x)〉 = 〈x, x〉 + ε2〈X(x), X(x)〉 = 1 + ε2.

Overall, fε is a smooth map taking values in Sn(√1 + ε2). In order to be able to

talk about the degree of fε we endow Sn(r) (r > 0) with the orientation it obtainsas boundary of the regular domain Bn+1(r) = x ∈ Rn+1 | ‖x‖ ≤ r.

In fact, deg fε = 1 which can be seen as follows: Consider the map projectionMε : Sn(

√1 + ε2) → Sn given by y 7→ y/‖y‖ which is an orientation-preserving


diffeomorphism. Precomposing it with fε yields Mεfε(x) = (x+εX(x))/(√1 + ε2)

which is homotopic to M0f0 = IdSn . Hence degMε fε = 1 and thus deg fε = 1.To obtain the announced contradiction, consider the differential form

ω =

n+1∑

i=1

(−1)ixi x1 ∧ · · · ∧ dxi ∧ · · · ∧ dxn+1 ∈ Ωn(Rn+1) ∈ Ωn(Rn+1).

For instance, if n = 3, then ω = x1 dx2∧dx3−x2 dx1∧dx3+x3 dx1∧dx2. Observe

d(xi dx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxn+1) = dxi ∧ dx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxn+1

= (−1)i−1 dx1 ∧ · · · ∧ dxn+1.

Hence dω = (n+ 1)dx1 ∧ · · · ∧ dxn+1. Let ω|Sn(r) denote the pullback of ω via theinjection Sn(r) → Rn+1. On the one hand, Stokes’ Theorem yields

∫

Sn(r)

ω|Sn(r) =∫

Bn+1(r)

dω = (n+ 1) vol(Bn+1(r)) = (n+ 1)cnrn+1

where cn is a constant depending solely on n. In particular, for r =√1 + ε2 we get

∫

Sn(r)

ω|Sn(r) = (n+ 1)cn(1 + ε2)n+12 .

On the other hand, we have∫

Sn(1)

(fε)∗(ω|Sn(r)) = deg fε

∫

Sn(r)

ω|Sn(r) =∫

Sn(r)

ω|Sn(r) = (n+1)cn(1+ε2)

n+12 .

Regarding the leftmost expression, note that f∗ε (ω|Sn(r)) is a differential form on

Sn(1) which depends on ε. By definition

(f∗ε ω)x(v1, . . . , vn) = ωfε(x)(Dxfεv1, . . . , Dxfεvn)

which is a sum over an index i ranging between 1 and n + 1 in which the i-thsummand is of the form

(xi + εXi(x))(dx1 ∧ · · · dxi ∧ · · · ∧ dxn+1)((Id+εDxX)v1, . . . , (Id+εDxX)vn+1)

On this expression, one sees that (f∗ε ω|Sn(r)) =

∑nj=0 ε

jηj where ηj ∈ Ωn(Sn). Weobtain ∫

Sn(1)

(f∗ε )(ω|Sn(r)) =

n∑

j=0

εj∫

Sn(1)

ηj

which is a polynomial in ε. On the other hand, if n is even then (1 + ε2)(n+1)/2 isnot a polynomial.

There is a shorter proof of Theorem 1.12, see e.g. [Lee10, Thm. 13.32], using anowhere-vanishing vector field to build a homotopy between the identity and theantipodal map.

1.3.1. Vector fields and derivations. Given a manifold M and a point p ∈ M ,recall that a derivation at p is a linear map λ : C∞(M) → R which satisfiesλ(fg) = f(p)λ(g)+ g(p)λ(f). Let Derp(C∞(M)) denote the space of all derivationsat p ∈M . We have shown the following.

Proposition 1.13. Retain the above notation. The map

TpM → Derp(C∞(M)), v 7→ f 7→ (Dpf)(v)is a vector space isomorphism.


In what is to follow we will often abuse notation by identifying a tangent vectorwith a derivation and vice-versa.

In order to express vector fields in local coordinates, let (U,ϕ) be a chart of Mand p ∈ U . For f ∈ C∞(M), define

∂j(p)(f) :=∂(f ϕ−1)

∂xj(ϕ(p)).

Then ∂j(p) ∈ DerpC∞(M) and (∂1(p), . . . , ∂m(p)) is a basis of TpM ∼= DerpC∞(M).Therefore, any vector field X : M → TM can be expressed as

X(p) =

m∑

i=1

Xi(p)∂i(p)

on U with well-defined functions Xi : U → R. In this context, the following lemmais an important exercise.

Lemma 1.14. Retain the above notation. The vector field X is smooth if and onlyif its local expression in every chart is given by smooth functions.

We now exhibit a hidden algebra structure on Γ(TM) to which end we recall thefollowing definition: A derivation of C∞(M) is a linear map δ : C∞(M) → C∞(M)such that δ(fg) = δ(f)g + fδ(g) for all f, g ∈ C∞(M). Derivations can be definedfor any associative algebra over any field, they are purely algebraic objects.

Now, given a smooth vector field X ∈ Γ(TM) on M and f ∈ C∞(M) we defineLX(f)(p) = X(p)f . Then the map LX : C∞(M) → C∞(M) is a well-defined linearmap. Furthermore, we have the following.

Proposition 1.15. Retain the above notation. The map LX : C∞(M) → C∞(M) isa derivation and the map Γ(TM) → Der(C∞(M)), X 7→ LX is an isomorphism.

Proof. The fact that LX is a derivation follows from the fact that f 7→ (LXf)(p)is a derivation at p ∈M .

Injectivity of the map Γ(TM) → Der(C∞(M)), X 7→ LX follows from the factthat TpM and Derp(C∞(M)) are isomorphic.

We now turn to surjectivity: Let δ ∈ Der(C∞(M)). Then for every p ∈ M ,the map f 7→ δ(f)(p) is a derivation at p. Hence there is a well-defined vectorX(p) ∈ TpM such that δ(f)(p) = X(p)(f) for all f ∈ C∞(M). Together, thesevectors form a vector field X on M . We need to verify that X is smooth. To thisend, we argue in local coordinates: Let (U,ϕ) be a chart of M and let X(p) =∑m

i=1Xi(p) ∂i(p) (p ∈ U) be the associated representation of X . Smoothness of Xamounts to smoothness of the coefficient functions Xi (i ∈ 1, . . . ,m). For everyf ∈ C∞(M), the map p 7→ δ(f)(p) = X(p)f is smooth. In particular, the mapp 7→∑m

i=1Xi(p)∂i(p)(f) is smooth on U . Now recall that

∂i(p)(f) =∂(f ϕ−1)

∂xi(ϕ(p)).

Hence, if f ϕ−1 : ϕ(U) → R was given by x 7→ xl we would obtain ∂i(p)(fl) = δiland conclude that Xl is smooth. However, there may not be such a function definedon the whole of M . Nevertheless, we can remedy the argument by multiplying witha smooth bump function.

The interpretation of smooth vector fields on a manifold M as derivations ofC∞(M) is important because it uncovers an important structure on Γ(TM), namelya Lie algebra structure.

Definition 1.16. Let W be a vector space and A,B ∈ End(W ). The bracket of Aand B is given by [A,B] := AB −BA ∈ End(W ).


The relevance of this notion in our context is due to the following.

Lemma 1.17. Let M be a manifold and let δ, δ′ ∈ Der(C∞(M)). Then [δ, δ′] ∈Der(C∞(M)).

Note that one would not expect a composition of derivations to be a derivationin general as second-order derivatives might be introduced.

Proof. (Lemma 1.17). We compute

δδ′(fg) = δ(δ′(f)g + fδ′(g)) = δδ′(f)g + δ′(f)δ(g) + δ(f)δ′(g) + δδ′(g)f

as well as

δ′δ(fg) = δ′(δ(f)g + fδ(g)) = δ′δ(f)g + δ(f)δ′(g) + δ′(f)δ(g) + δ′δ(g)f

and therefore[δ, δ′](fg) = [δ, δ′](f)g + [δ, δ′](g)f.

Definition 1.18. Let M be a smooth manifold and let X,Y be smooth vector fieldson M . The bracket [X,Y ] of smooth vector fields X and Y is the smooth vectorfield corresponding to [LX , LY ] ∈ Der(C∞(M)) via the isomorphism Γ(TM) →Der(C∞(M)).

Symbolically, we have L[X,Y ] = [LX , LY ]. We now compute [X,Y ] in local coor-dinates. Let (U,ϕ) be a chart of M and write

X(p) =

m∑

i=1

Xi(p)∂i(p) and Y (q) =

m∑

j=1

Yj(q)∂j(q)

for p, q ∈ U . Then for every f ∈ C∞(M) we have

LX(LY (f))(p) =

m∑

i=1

Xi(p)∂i(p)(LY (f))

where

∂i(p)(LY (f)) =

m∑

j=1

∂i(p)(Yj · ∂jf)

=

m∑

j=1

∂i(p)Yj · ∂j(p)(f) +m∑

j=1

Yj(p)∂2(f ϕ−1)

∂xj∂xi(ϕ(p))

We therefore have

LX(LY (f))(p) =∑

i,j

Xi(p)∂i(p)(Yj)∂j(p)(f) +∑

i,j

Xi(p)Yj(p)∂2(f ϕ−1)

∂xj∂xi(ϕ(p))

and

LY (LX(f))(p) =∑

i,j

Yi(p)∂i(p)(Xj)∂j(p)(f) +∑

i,j

Yi(p)Xj(p)∂2(f ϕ−1)

∂xj∂xi(ϕ(p))

︸︷︷︸∑i,j Yj(p)Xi(p)

∂2(fϕ−1)∂xi∂xj

(ϕ(p))

By Schwarz’s theorem, we thereby conclude

(L[X,Y ]f)(p) =m∑

j=1

(m∑

i=1

Xi(p)∂i(p)(Yj)− Yi(p)∂i(p)(Xj)

)∂j(p)(f)


That is

[X,Y ](p) =

m∑

j=1

Zj(p)∂j(p)

where

Zj(p) =

m∑

i=1

(Xi(p)∂i(p)(Yj)− Yi(p)∂i(p)(Xj))

Note that the value of [X,Y ] at p ∈ M does not only depend on the values of Xand Y at p, which would make the bracket a linear algebra story, but on the valuesof X and Y on a neighbourhood of p.

Also observe that the bracket [X,Y ] is bilinear, i.e. for all smooth vector fieldsX1, X2, Y on M and scalars λ1, λ2 ∈ R we have

[λ1X1 + λ2X2, Y ] = λ1[X1, Y ] + λ2[X2, Y ]

and similarly for the Y -slot. It seems like this turns [−,−] into a “product”. However,it is not associative. Instead it satisfies Jacobi’s identity which is the content of thefollowing Proposition.

Proposition 1.19. Let M be a smooth vector field and let X , Y and Z be smoothvector fields on M . Then

[X, [Y, Z]] + [Y, [X,Z]] + [Z, [Y,X ]] = 0

A way to remember this identity is to note that in the second and third term,the entries X , Y and Z are cyclically permuted. It holds true more generally: Givena vector space W and endomorphisms A,B,C ∈ EndW it is an easy computation.

Another interpretation of the identity is the following: Given a smooth vectorfield X on M , define

ad(X) : Γ(TM) → Γ(TM), Y 7→ [X,Y ].

Then ad(X) ∈ End(Γ(TM)) and the Jacobi identity amounts to ad(X) preservingbrackets: For all X1, X2 ∈ Γ(TM) we have

ad([X1, X2]) = [adX1, adX2].

There is also a geometric interpretation of the Jacobi identity in terms of flows ofthe occurring vector fields which we exhibit later.

1.3.2. Vector Fields on Rn. Let Ω ⊆ Rn be an open set. A smooth vector field onΩ is a smooth map X : Ω → Rn. Note however, that the mental picture of a vectorfield in which the vectors are viewed as being attached to the point. We recall thefollowing existence and uniqueness theorem for integral curves of vector fields inRn or rather solutions of ordinary differential equations, see e.g. [Kön13, 4.2 II].

Theorem 1.20. Let Ω ⊆ Rn be open and let Y : Ω → Rn be a smooth vector field.

(i) For every x0 ∈ Ω there is an open interval Ix0 ⊆ R containing 0 ∈ R andan open set Vx0 ⊆ Ω containing x0 ∈ Ω such that for every x ∈ Vx0 thereexists a smooth curve cx : Ix0 → Ω such that

cx(0) = x

Y (cx(t)) = c′x(t) ∀t ∈ Ix0

.

(ii) For every x ∈ Vx0 , any smooth curve γ : I → Ω satisfyingγx(0) = x

Y (γx(t)) = γ′x(t) ∀t ∈ Ix0

coincides with cx on some neighbourhood of 0 ∈ R.(iii) The map Vx0 × Ix0 → Ω, (x, t) 7→ cx(t) is smooth.


Recall that the key to the proof of Theorem 1.20 consists in transforming anordinary differential equation into an integral equation, identifying contractivityand applying Banach’s fixed point theorem. An analogous statement holds in thecase of manifolds.

Corollary 1.21. Let M be a smooth manifold and let X : M → TM be a smoothvector field.

(i) For every x0 ∈ M there is an open interval Ix0 ⊆ R containing 0 ∈ R andan open set Vx0 ⊆M containing x0 ∈M such that for every x ∈ Vx0 thereexists a smooth curve cx : Ix0 → M such that

cx(0) = x

X(cx(t)) = c′x(t) ∀t ∈ Ix0

.

(ii) For every x ∈ Vx0 , any smooth curve γ : I →M satisfyingγx(0) = x

X(γx(t)) = γ′x(t) ∀t ∈ Ix0

coincides with cx on some neighbourhood of 0 ∈ R.(iii) The map Vx0 × Ix0 →M , (x, t) 7→ cx(t) is smooth.

In Corollary 1.21, note that c′x(t) := Dt(cx)(1). Its proof consists of expressingeverything in local coordinates, applying Theorem 1.20 and transforming back.Later on, we will globalize this local existence and uniqueness statement in the caseof compact manifolds.

Retain the notation of Corollary 1.21. For every t ∈ Ix0 we may define the mapθt : Vx0 →M , x 7→ cx(t). The following interesting local group property of the mapsθt is due to the uniqueness part of Corollary 1.21. Taking the necessary precautions,its proof is mostly formal.

Corollary 1.22. Retain the above notation. Let t1, t2 and t1+ t2 be in Ix0 . Further,let W ⊆ Vx0 be such that θt1(W ) ⊆ Vx0 . Then θt2 θt1 and θt1+t2 are defined andagree on W .

Proof. Without loss of generality, we assume t1, t2 ≥ 0. Given x ∈ W we considerthe curve γ(s) := cx(t1 + s) for s ∈ [0, t2]. Setting T : R → R to denote thetranslation s 7→ t1 + s we can rewrite γ(s) = cx T (s). We now compute

γ′(s) = (Dsγ)(1) = Ds(cx T )(1) = DT (s)cx DsT (1)

= DT (s)cx(1) = X(cx(T (s))) = X(γ(s)).

Now observe that γ(0) = cx(t1). Hence, by uniqueness, γ(s) = ccx(t1)(s). In partic-ular, for s = t2 we obtain for all x ∈W :

θt1+t2(x) = cx(t1 + t2) = γ(t2) = ccx(t2) = θt2(cx(t1)) = θt2(θt1(x)).

Hence the assertion.

Later on, we will revisit this result in the case of Lie groups where it takes amore global form.

Example 1.23. LetM = Rn and consider the radial vector fieldX(x) :=∑n

i=1 xi∂i(x).To determine its integral curves c, note that the condition X(c(t)) = c′(t) translatesto c′i(t) = ci(t) where c(t) = (c1(t), . . . , cn(t)). Consequently c(t) = kie

t. Taking intoaccount an initial condition cx(0) = x we get cx(t) = etx and therefore θt(x) = etx.


One checks that indeed θt1+t2 = θt2 θt1 in this case.

b

Example 1.24. To illustrate that the precautions taken in Corollary 1.22 are indeednecessary, consider the case where M = R and X(x) := x2∂x(x). The associatedinitial value problem is c′x(t) = c2x(t) with cx(0) = x. Taking the physicist’s approachto solving this, we obtain

−c′c2

=

(1

c

)′

= −1 ⇒ 1

c= −t+K

and hence c(t) = 1/(−t + K). Therefore, cx(t) = 1/(−t + 1/x). Assuming x > 0the maximum interval of definition of cx is (−∞, 1/x). In particular, there is nouniform interval (−ε, ε) of definition that works for all initial values.

A possible condition to ensure that the maps θt are defined on the whole manifoldfor all times is compact support of the underlying vector field as in the followingproposition.

Proposition 1.25. Let M be a smooth manifold and let X ∈ Γ(TM) be a smoothvector field with compact support. Then the map, R → Diff(M), t 7→ θt is ahomomorphism.

Homomorphisms as in Proposition 1.25 are termed one-parameter subgroups of

diffeomorphisms for obvious reasons.

Proof. Let K = supp(X) = x ∈M | X(x) 6= 0. Using Corollary 1.21 we maycoverK with open sets V1, . . . , Vl such that there are intervals I1, . . . , Il to the extentthat for every x ∈ Vi there is a solution cx : Ii → M of the initial value problemX(cx(t)) = c′x(t), cx(0) = x. Choose ε > 0 such that (−ε, ε) ⊆ ⋂l

i=1 Ii and setΩ := V1∪· · ·∪Vl. Then for every t ∈ (−ε, ε) the map θt : Ω →M is defined. OutsideK, nothing happens: If x ∈M\K then cx(t) = x is a solution to X(cx(t)) = c′x(t),cx(0) = x for all t ∈ R. In this way, we obtain a map θt : M → M defined on thewhole of M for all t ∈ (−ε, ε). We now use Corollary 1.22 to extend θt to all times:First of all, Id = θt−t = θ−t θt and hence θt ∈ Diff(M). Secondly, given t ∈ R, lett = k · (ε/2) + r where k ∈ Z and r ∈ (−ε, ε); then set θt := (θε/2)

k θr ∈ Diff(M).It remains to verify that θt1+t2 = θ2 θ1 which can be done using additivity forsmall values of t1, t2 and t1 + t2.

The flows associated to vector fields are very interesting. Although they distortthe area started with they sometimes act volume-preservingly. In this case the fol-lowing may hold: Given a subset A ⊆M , consider the set 0 ≤ t ≤ T | θt(x) ∈ A ⊆R. Taking its volume, normalizing by T and letting T tend to infinity, Birkhoff’sergodic theorem states that

limT→∞

1

TL(0 ≤ t ≤ T | θt(x) ∈ A) = vol(A)

Chaotic behaviour and return to the starting point is typical in particular in thecase of compact manifolds. For instance, given an irrationally oriented vector fieldon the two-torus, the flow always returns to any set with positive measure.


Corollary 1.26. Let M be a smooth compact manifold and let X ∈ Γ(TM) be asmooth vector field on M . Then there is a one-parameter group of diffeomorphismsR → Diff(M), t 7→ θt such that for every x ∈M , the map R →M , t 7→ θt(x) is theintegral curve of X passing through x at time t = 0.

We end the discussion of vector fields by introducing the push-forward of avector field by a diffeomorphism, relating it the various other operations we haveintroduced and giving a geometric interpretation of the Lie bracket.

Whereas differential forms can be pulled back their dual objects, vector fields,are pushed forward. However, not every smooth map can be used to do this. Forinstance, a smooth map might collapse a curve to a point, preventing a naturalchoice for the push-forward vector at said point. We therefore consider the followingsituation.

Definition 1.27. Let M and N be smooth manifolds and let X ∈ Γ(TM) be asmooth vector field. Further, let φ :M → N be a diffeomorphism. The push-forward

vector field Y on N is given by Y (φ(y)) := (Dxφ)(X(x)).

Lemma 1.28. Retain the notation of Definition 1.27. Then Y is a smooth vectorfield. Furthermore, if LX and LY denote the derivatives associated to X and Yrespectively then LY f = LX(f φ) φ−1 for all f ∈ C∞(N).

Given a smooth map φ : M → N of smooth manifolds, recall that the theassociated algebra homomorphism φ∗ : C∞(N) → C∞(M) is given by φ∗(f) = fφ.In terms of φ∗, Lemma 1.28 states that the following diagram is commutative.

C∞(N)φ∗

//

LY

C∞(M)

LX

C∞(N)φ∗

// C∞(M)

With this in mind, the proof of Lemma 1.28 amounts to a computation.

Proof. (Lemma 1.28). Compute for all f ∈ C∞(N):

(LY f)(y) = Y (y)f = (Dyf)(Y (y)) = (Dyf)(Dxφ(X(x))) = (Dφ(x)f)(Dxφ(X(x)))

= Dx(f φ)(X(x)) = LX(f φ)(x) = LX(f φ)(φ−1(y))

The relation between push-forward vector fields and the bracket operation is thefollowing.

Lemma 1.29. Let M and N be smooth manifolds and let φ : M → N be a diffeo-morphism. Further, let X and Z be smooth vector fields on M . Then

[φ∗(X), φ∗(Z)] = φ∗([X,Z]).

Proof. Here, it is beneficial to think in terms of the associated derivations: We have

Lφ∗X = (φ∗)−1LXφ∗ and Lφ∗(Z) = (φ∗)−1LZφ

∗.

Computing brackets yields

[Lφ∗X , Lφ∗Z ] = (φ∗)−1LXLZφ∗ − (φ∗)−1LZLXφ

∗ = (φ∗)−1[LX , LZ ]φ∗.

Hence the assertion.

Next up is the relation between push-forward vector fields and flows which relieson a uniqueness rather than computation argument.


Lemma 1.30. Let M and N be smooth manifolds and let φ : M → N be a diffeo-morphism. Further, let X be a smooth vector field on M . Set Y := φ∗X and let ψtdenote the flow of Y . Then ψt = φθtφ

−1 for all t ∈ R where θt is the flow of X .

Proof. Set ψt(y) := φθtφ−1(y). Then

d

dt

∣∣∣∣t=0

ψt(y) = Dφ−1(y)φ(X(φ−1(y))) = Dxφ(X(x)) = Y (y)

which implies the assertion by uniqueness.

We end this chapter with the following geometric interpretation of the bracket.

Proposition 1.31. Let M be a smooth manifold and let X and Y be smooth vectorfields on M . Assume that the local group θt of Y is defined. Then

d

dt

∣∣∣∣t=0

((θt)∗X) = [X,Y ].

Note that the expression ((θt)∗X) (x) implicit in Proposition 1.31 is a tangentvector at x ∈M , namely the derivative of X with respect to Y at x ∈M .

Proof. As an exercise, recall that if f : (−ε, ε)×M → R is smooth with f(0, p) = 0for all p ∈M then there is a smooth map g : (−ε, ε)×M → R with f(t, p) = tg(t, p).

Given f ∈ C∞(M), we apply this fact to f(t, p) := f(θt(p)) − f(p) so thatg(0, p) = Y (p)(f). We have

((θt)∗X)(p)(f) = LX(f θt) θ−1t (p)

= X(θ−1t (p))(f θt)

= X(θ−1t (p))(f + tg(t,−)).

We therefore get

(θt)∗X(p)(f)−X(p)(f)

t=X(θ−1

t (p))(f)−X(p)(f)

t+X(θ−1

t (p))(g(t,−))

=X(θ−t(p))(f)−X(p)(f)

t+X(θ−t(p))(g(t,−))

If t tends to zero, we obtain

X(θ−t(p))(f)−X(p)(f)

t→ −LY LX(f)(p)

andX(θ−t(p))(g(t,−)) = LX(g(t,−))(θ−t(p)) → LXLY (f)(p)

which implies the assertion.

1.4. Lie Groups: A very short introduction. Lie groups are particularly in-teresting manifolds and are central to both building large classes of examples ofmanifolds and actions on manifolds. They are also crucial to seemingly unrelatedmathematics such as Fermat’s last theorem.

Definition 1.32. A Lie group is a smooth manifold G endowed with a group struc-ture such that the product map G × G → G, (x, y) 7→ xy and the inverse mapG→ G, x 7→ x−1 are smooth.

We have already seen many example of Lie groups.

Example 1.33.

(i) The manifold Rn is a Lie group with respect to addition.(ii) The manifold GL(n,R) is a Lie group with respect to matrix multiplication.


(iii) Given a quadratic form q on Rn, the manifold O(q) is a Lie group withrespect to matrix multiplication.

(iv) The symplectic groups of example 1.3 are Lie groups as well.(v) It is a highly non-trivial result of E. Cartan that every closed subgroup

G ≤ GL(n,R) is a regular submanifold and hence a Lie group. Note thesharp contrast to the fact that both closed subsets and non-closed subgroupsof GL(n,R) can behave very badly: Consider for instance GL(n,Q) whichcan only be turned into a Lie group by making the topology discrete.

We now illustrate that all the notions we have developed work together nicelyin the case of Lie groups and thereby underline the importance of the latter. Firstof all, note that if G is a Lie group and g ∈ G, the left multiplication Lg : G→ G,x 7→ gx is a preferred diffeomorphism of G sending the identity element e ∈ G tog ∈ G. Indeed, it is smooth by the axioms of a Lie group and admits the smoothinverse Lg−1 . Furthermore, we have the following.

Proposition 1.34. Let G be a Lie group. Then

(i) G is orientable, and(ii) G is parallelizable.

Proof. To show that G is orientable, we construct a nowhere vanishing top formon G: Let ω0 ∈ Λn((TeG)∗) be a non-zero, alternating n-form where n = dimG- recall that if V is an n-dimensional vector space then dimΛk(V ∗) =

(nk

), in par-

ticular dimΛn(V ∗) = 1. We now propagate this form to the whole of G using leftmultiplication: For g ∈ G, set

ωg(v1, . . . , vn) = ωe(DgLg−1v1, . . . , DgLg−1vn).

One checks that ω ∈ Ωn(G) so defined is smooth. By construction it is nowhere van-ishing, i.e. volume form. Hence G is orientable. Let us endow G with the orientationfor which a basis e1, . . . , en is positively oriented if and only if ωg(e1, . . . , en) > 0.

Parallelizabiliy of G is proven as in the case of the torus, see 1.11.

Corollary 1.35. Let G be a Lie group and let ω denote the volume form on Gconstructed in the proof of Proposition 1.34. Then the linear map I : C00(G) → R,defined on the space of continuous, compactly supported functions on G is byI(f) :=

∫Gf ω yields a left-invariant positive Radon measure on G via Riesz rep-

resentation.

Let µ denote the measure obtained on a Lie group via Corollary 1.35. It is calledHaar measure and satisfies µ(gE) = µ(E) for every Borel set E ⊆ G and everyelement g ∈ G. Haar measures exist more generally for locally compact Hausdorffgroups. However, in the general case there is a no smooth structure to work withand hence the proof is based on different ideas.

Proof. (Corollary 1.35). We prove invariance of the functional I from which invari-ance of the associated measure follows. Let f ∈ C00(G) and h ∈ G. Observe that(f Lh)ω = (L∗

h)(f · (L∗

h−1ω) and hence

I(f Lh) =∫

G

(f Lh)ω =

∫

G

L∗hη =

∫

G

η


where η := f · (L∗

h−1ω. Now compute

((Lh−1)∗ω)g(v1, . . . , vn) = ωLh−1(g)(DgLh−1(v1), . . . , DgLh−1(vn))

= ωh−1g(DgLh−1(v1), . . . , DgLh−1(vn))

= ω0(Dh−1gLg−1hDgLh−1(v1), . . . , Dh−1gLg−1hDgLh−1(vn))

= ω0(DgLg−1v1, . . . , DgLg−1vn)

= ωg(v1, . . . , vn).

whence η = fω and therefore I(f Lh) = I(f).

Observe that if G is a compact Lie group then the functional I of Corollary 1.35is defined for all f ∈ C(G).

Corollary 1.36. Every compact Lie subgroup of GL(n,R) is conjugate into O(n).

Proof. LetK ≤ GL(n,R) be a compact Lie subgroup and let µ denote its Haar mea-sure. Choose any scalar product 〈−,−〉 on Rn and define B(v, w) :=

∫G〈gv, gw〉 dµ(g)

for all v, w ∈ Rn. Then it is an easy verification that B is a K-invariant scalarproduct. Now remember from linear algebra that if (−,−) denotes the standardscalar product then there exists A ∈ GL(n,R) such that B(u, v) = (Au,Av) forall u, v ∈ Rn. Since B is K-invariant we obtain (Aku,Akv) = (Au,Av). Settingu′ := Au and v′ := Av we conclude (AkA−1u′, AkA−1v′) = (u′, v′) and henceAKA−1 ⊆ O(n).

We now apply the theory that we have developed for manifolds to the specialcase of Lie groups in which things behave particularly nice.

Definition 1.37. A Lie algebra is a vector space g endowed with a product g× g → g,denoted (A,B) 7→ [A,B] which is

(i) bilinear,(ii) satisfies [A,B] = −[B,A] for all A,B ∈ g, and(iii) satisfies [A, [B,C]] + [B, [C,A]] + [C, [A,B]] = 0 for all A,B,C ∈ g.

Example 1.38. Let M be a smooth manifold. Then g = Γ(TM), endowed with thebracket of vector fields, is a Lie algebra. However, it is infinite-dimensional unlessM is a point.

Definition 1.39. Let G be a Lie group. A smooth vector field X ∈ Γ(TG) is left-

invariant if (Lg)∗X = X for all g ∈ G.

Let Γinv(TG) be the subspace of Γ(TG) consisting of left-invariant vector fields.

Proposition 1.40. Let G be a Lie group. Then Γinv(TG) is a Lie algebra and themap Γinv(TG) → TeG given by X 7→ X(e) is a vector space isomorphism.

In particular, the Lie algebra Γinv(TG) is finite-dimensional. In some way, itencodes the group structure of G. However, it is unclear yet in what way it does so.

Proof. In Lemma 1.28 we have shown that (Lg)∗[X,Y ] = [(Lg)∗X, (Lg)∗Y ] for allX,Y ∈ Γ(TG) and g ∈ G. Hence, if X and Y are invariant smooth vector fields onG then so is [X,Y ].

Injectivity of the evaluation map is due to the fact that left-invariance of Xyields X(g) = (DeLg)(X(e)). Surjectivity follows from defining Y ∈ Γinv(TG) bysetting Y (g) := (DeLg)(v) for a given v ∈ TeG.

The isomorphism of Proposition 1.40 can be used to turn TeG into a Lie algebraas follows: Given u, v ∈ TeG, define [u, v] ∈ TeG to be the evaluation of left-invariant vector field [Xu, Xv] at e where Xu and Xv are the left-invariant vector


fields corresponding to u and v respectively. Note that in order to know [u, v] itdoes not suffice to know u and v, it requires knowledge of the vector fields Xu andXv on a small neighbourhood of the identity in G.

One checks that the bracket on the Lie algebra of Rn is trivial. Generally, itmeasures the degree of non-commutativity and is best used for highly non-abeliangroups, such as semisimple ones.

Definition 1.41. Let G be a Lie group. The Lie algebra of G is the vector spcaeg := TeG endowed with the product [x, y] := [X,Y ](e) where X and Y are theleft-invariant vector fields associated to x and y in TeG.

In the context of general manifolds we have studied flows of vector fields. Thereis a lot to say about these in the context of Lie groups.

Proposition 1.42. Let G be a Lie group and let X ∈ Γinv(TG). Then the local groupθt is defined for all t ∈ R. In addition,

(i) θt(g) = gθt(e) for all t ∈ R and g ∈ G, and(ii) the map R → G, t 7→ θt(e) is a smooth homomorphism whose tangent

vector at e ∈ G is X(e).

Proof. The main part of Proposition 1.42 is to show the existence of integral curvesfor all times. The remaining statements are then easy consequences. For every g ∈ G,let Ig ⊆ R be the largest open interval of definition of cg which contains 0 ∈ R.We are going to show that Ig is independent of g which suffices as we shall see. Foronce, we claim that cg(t) = Lg(ce(t)) which can be verified using uniqueness: Letγ(t) := Lg(ce(t)) for t ∈ Ie. Then

γ′(t) = (Dce(t)Lg)(c′e(t)) = (Dce(t)Lg)(X(ce(t))) = X(gce(t)) = X(γ(t)).

Furthermore, γ(0) = gce(0) = ge = g. hence γ(t) = cg(t) by uniqueness. Thisproves the claim. In particular, we conclude that Ig = Ie =: I for all g ∈ G. Letε > 0 be such that (−ε, ε) ⊆ I and choose t ∈ I. For small enough s we havecg(t+ s) = ccg(t)(s) by uniqueness. Now, the right hand side is defined at least forall s ∈ (−ε, ε) which implies t+ (−ε, ε) ⊆ Ig = Ie = I. Hence I = R.

The claim can be rewritten as θt(g) = gθt(e). Setting g = θs(e) we obtain

θt θs(e) = θs(e)θt(e).

On the other hand, we know that θt θs = θt+s and hence θt+s(e) = θs(e) · θt(e).In other words, the map R → G, t 7→ θt(e) is a homomorphism.

The proof of Proposition 1.42 leads us to the following, fundamental object.

Definition 1.43. Let G be a Lie group and let g be its Lie algebra. The exponential

map expG : g → G is given by x 7→ θ1(e) where θt denotes the one-parameter groupof diffeomorphism associated to the left-invariant vector field X with X(e) = x.

We observe that as a consequence of Definition 1.43, we have θt(e) = exp(tx)which amounts to saying the vector fields can be scaled: Indeed, the map R →Diff(G), s 7→ θst is the one-parameter group associated to tX ; its value at s = 1being exp(tx) coincides with θt(e).

We shall see shortly that the exponential map of a Lie group G with Lie algebrag is a local diffeomorphism at 0 ∈ g. That is, Lie groups come equipped withcanonical charts that are very well adapted to the group structure. For (Rn,+)the exponential map merely associates the position vector to a tangent vector. Itsname, however, stems from the fact that in the case of GL(n,R), the exponentialmap is the usual matrix exponential.


Example 1.44. As an example, we determine the exponential map in the case ofGL(n,R). The manifolds structure of GL(n,R) stems from the fact that it is anopen subset of Mn,n(R). We may hence identify TId GL(n,R) with Mn,n(R). Letx ∈ Mn,n(R). For appropriate ε > 0, the curve t 7→ Id+tx through Id ∈ Mn,n(R)is contained in GL(n,R) for all t ∈ (−ε, ε). Let X be the left-invariant vector fieldon GL(n,R) associated to x ∈Mn,n(R). Then

X(g) = (DeLg)(x) =d

dt

∣∣∣∣t=0

g(Id+tx) = gx.

From this we deduce that (LXf)(g) = X(g)(f) = (Dgf)(gx) for all f ∈ C∞(G)and g ∈ GL(n,R). A computation now implies that for all x, y ∈ Mn,n(R) =TId GL(n,R) we have [x, y] = xy−yx. That is, in this case, the Lie bracket coincideswith the usual commutator bracket operation.

Now, the differential equation given by an invariant vector field X is X(c(t)) =c′(t), that is c(t)x = c′(t). Its solution with initial condition c(0) = Id is given byc(t) = Exp(tx) =

∑∞

k=0(tkxk)/k!. Thus expGL(n,R)(x) = Exp(x) is the usual matrix

exponential, hence the name of the general Lie group exponential.

Proposition 1.45. Let G be a Lie group with Lie algebra g. The exponential mapexpG : g → G is smooth and D0 expG : g → g is the identity map.

Proof. Identifying T0 g with g we have D0 expG(x) =ddt

∣∣t=0

expG(tx) = x.

The importance of Proposition 1.45 is due to the fact that it implies that theexponential map is a diffeomorphism on a neighbourhood of 0 ∈ g. Hence its inversecan be employed as a chart.

Lie subgroups. It is also essential that the exponential map of a Lie group G de-termines the exponential maps of all its Lie subgroups H . The difficulty of thisstatement lies in the definition of Lie subgroup. Whereas this chapter does not aimat being a comprehensive treatment of the foundations of Lie theory, we elaborate abit on this important point: Let G be a Lie group and let H ≤ G be a subgroup. IfH is also a regular submanifold and the map H×H → H , (x, y) 7→ xy−1 is smooththen H is a Lie group. In this case, one can verify that h := TeH ≤ TeG = g is asubalgebra. In general, however, we do not get every subalgebra h of g in this fash-ion which is unacceptable from a categorical point of view. For instance, considerthe torus T2 = Z2 \R2 whose Lie algebra is Lie(T2) = R2 has trivial bracket. Con-sequently, all subspaces of Lie(T2) are subalgebras. But the subalgebra generatedby (1,

√2)T ∈ R2 is not the Lie algebra of a regularly embedded subgroup of T2.

In fact, one has to relax the requirement of the subgroup to be regularly embed-ded: In general, a Lie subgroup of a Lie group G is a pair (H, i) consisting of a Liegroup H and an injective immersion i : H → G, allowing for the topologies on Hand i(H) ⊆ G to differ as in the above case.

Proposition 1.46. Let G be a Lie group with Lie algebra g and let H ≤ G be aregular submanifold. Then the Lie algebra of H is given by

h = TeH = v ∈ TeG = g | expG(tv) ∈ H ∀t ∈ R.

Example 1.47. Proposition 1.46 allows us to compute the Lie algebras of most ourexamples, which are subgroups of GL(n,R). In this case, the general recipe is towrite down the defining equation of the group, substitute exp(tx) for the variable,and take the derivative at t = 0 to obtain the defining equation of the Lie group.


(i) In the case of O(n) we compute

Lie(O(n)) = x ∈Mn,n(R) | Exp(tx) ∈ O(n) ∀t ∈ R= x ∈Mn,n(R) | Exp(tx)T Exp(tx) = Id ∀t ∈ R= x ∈Mn,n(R) | xT + x = 0

as (d/dt)|t=0(Exp(tx)T Exp(tx)) = xT + x.

(ii) Similar to the above, we determine in the case of Sp(2n,R):

Lie(Sp(2n,R)) = x ∈M2n,2n(R) | xT J + Jx = 0.

It is an exercise to think about what happens in cases such as U(n) and SU(n) assubgroups of GL(n,C).

In a sense linear groups such as the examples above already cover almost all Liegroups. This is made precise using representations of Lie groups: Every Lie group Gcomes equipped with a natural action on a finite-dimensional vector space: Recallthat for h ∈ G, we have the right-multiplication diffeomorphism Rh : G→ G givenby g 7→ gh. Any Rh commutes with any Lg (g, h ∈ G) which is due to the easilyoverlooked associativity of the group multiplication. Therefore, givenX ∈ Γinv(TG)and h ∈ G, we have (Rh)∗X ∈ Γinv(TG). Accordingly, we may define for h ∈ Gand x ∈ g = TeG: Ad(h)x := ((Rh−1)∗X)(e). In this way, we get a homomorphismAd : G → GL(g), called the adjoint representation. The kernel of the adjointrepresentation is related to the center of G and is abelian, which is not considereda drama because abelian Lie groups are well-understood.

We end this chapter on Lie groups with the two following results on the adjointrepresentation of a Lie group.

Proposition 1.48. Let G be a Lie group with Lie algebra g. Then for all h ∈ G andx ∈ g we have

Ad(h)(x) =d

dt

∣∣∣∣t=0

h exp(tx)h−1.

The proof of Proposition 1.48 is left as an exercise. Now, the adjoint represen-tation of a Lie group G is a map from G to GL(g). As such it can be differentiatedat t = 0 to produce a homomorphism of the associated Lie algebras g and End(g).This produces the following interpretation of the Lie bracket.

Proposition 1.49. Let G be a Lie group with Lie algebra g. For all x, y ∈ g we have

d

dt

∣∣∣∣t=0

Ad(exp ty)(x) = [y, x].

The proof of Proposition 1.49 is a consequence of Proposition 1.31.

1.5. Coverings and Fibrations. In this section we discuss the important coveringand fibration mechanisms. In the following definition, we adopt the topologicalnotion of covering maps to our smooth setting.

Definition 1.50. Let M and M be smooth manifolds. A map p : M → M is acovering map if

(i) p is smooth and surjective, and(ii) for every m ∈ M there is an open neighbourhood U of m ∈ M such that

p−1(U) =⋃i∈I Ui is a disjoint union of open subsets Ui ⊆ M such that

p|Ui : Ui → U is a diffeomorphism.


b b b

M

b

M

Given a covering map p : M → M the map M → N0 ∪∞ which to m ∈ Massociates the cardinality of its preimage under p is locally constant.

Example 1.51. Here are three of the most important examples and non-examplesof covering spaces.

(i) Let M :=M := S1 and consider p :M →M , z 7→ zd for some d ∈ Z whereS1 is considered as z ∈ C | |z| = 1. Then p is a d-sheeted covering mapand given z ∈M we have p−1(z) = exp(2πik/d) | 0 ≤ k < d.

(ii) The map f : C → C, z 7→ z2 is not a covering: There is no neighbourhoodof 0 ∈ C on which card(p−1(z)) is constant. However, one could remove 0from both M and M to turn f into a covering.

(iii) Set M := R and M = S1. Then the map t 7→ exp(2πit) is a covering.

One way in which covering maps arise is the following: Let M be a locallycompact Hausdorff space on which a group Γ acts by homeomorphisms. That is,for each γ ∈ Γ the map M →M , m 7→ γm is a homeomorphism.

Definition 1.52. Let M be a locally compact Hausdorff space on which a group Γacts by homeomorphisms. That is, for each γ ∈ Γ the map M → M , m 7→ γm is ahomeomorphism.

(i) The action is properly discontinuous if for every compact K ⊆ M the setγ ∈ Γ | γ(K) ∩K 6= ∅ is finite.

(ii) The action is free if for every x ∈M and γ ∈ G\e we have γ(x) 6= x.

Proposition 1.53. Let M be a locally compact Hausdorff space on which a groupΓ acts by homeomorphisms. If the action is properly discontinuous then Γ\M isHausdorff.

In the case of a group Γ acting on a set Γ\M , we denote by Γ\M the set ofequivalence classes for the relation x ∼ y ⇔ Γx = Γy. We remark that in thissituation the map p : M → Γ\M is open: Indeed, if V ⊆ M is any set, thenp−1(p(V )) =

⋃γ∈Γ γ(V ). Hence, if V is open, then p−1(p(V )) is open as a union of

open sets.

Proof. (Proposition 1.53). Since p is open it suffices to show that R = (x, y) ∈M ×M | x ∼ y = (x, γx) | x ∈ M, γ ∈ Γ is a closed subset of M ×M . To thisend, let (x0, y0) ∈ R. For every compact neighbourhood of (x0, y0) ∈M ×M of theform V ×W , define F (V,W ) := γ ∈ Γ | γV ∩W 6= ∅. Observe that F (V,W ) isalways finite: Indeed,

F (V,W ) ⊆ γ ∈ Γ | γ(V ∪W ) ∩ (V ∪W ) 6= ∅and the latter set is finite by the definition of proper discontinuity. Now, fix aparticular neighbourhood V0 ×W0 and consider

F := F (V,W ) | V ×W ∋ (x0, y0) compact neighbourhood and V ×W ⊆ V0×W0Now make the following observations:


(i) F (V,W ) ⊆ F (V0,W0) for all F (V,W ) ∈ F .(ii) F (V,W ) 6= ∅ for all F (V,W ) ∈ F : Indeed, since (x0, y0) ∈ R we have

(V ×W )∩R 6= ∅ and a point (x, γx) ∈ (V ×W ) gives rise to γ ∈ F (V,W ).(iii) Given F (Vi,Wi) ∈ F (i ∈ 1, . . . , n), we have

n⋂

i=1

F (Vi,Wi) ⊇ F

(n⋂

i=1

Vi,

n⋂

i=1

Wi

)6= ∅.

As a consequence, we conclude⋂F (V,W )∈F

F (V,W ) 6= ∅. An element γ ∈ Γ that iscontained in the latter intersection has the property that for any compact neigh-bourhoods V of x0 ∈ M and W of y0 ∈ M we have γV ∩ W 6= ∅. Since M isHausdorff this implies γ(x0) = y0.

Corollary 1.54. Let Γ be a group acting freely and properly discontinuously on amanifold M by diffeomorphisms. Then there is a unique smooth manifold structureon Γ\M such that p :M → Γ\M is a smooth covering.

Proof. (Sketch). Proper discontinuity and freeness of the action imply that for allx ∈ M there is an open neighbourhood Vx of x such that for all γ ∈ Γ\e wehave γVx ∩ Vx = ∅. This is based on the following argument: Given γ ∈ Γ\e,pick disjoint neighbourhoods V of x and W of γx with V ∩W = ∅ using that M isHausdorff. Then V ∩γ−1W is a neighbourhood of x and (V ∩γ−1W )∩(γV ∩W ) = ∅.Adjust this argument to taking into account the finiteness of our situation.

For a neighbourhood Vx of x as above, we observe that p|Vx : Vx → p(Vx) iscontinuous, open, surjective and injective, hence a homeomorphism.

Finally, choose an atlas A on M consisting of charts (Vx, ϕx) as above. A smoothatlas on Γ\M is then given by

A′ = (pVx, φx) | (Vx, φx) ∈ A, φx = ϕx (p|Vx)−1.The smoothness of the transition maps comes from the assumption that Γ acts onM by diffeomorphisms.

Example 1.55. We collect several examples of this efficient construction.

(i) Let Γ = ± Id and M = Sn ⊆ Rn+1. Then Γ\Sn is diffeomorphic to realprojective n-space Pn(R).

(ii) Consider M := H+ := z ∈ C | Im(z) > 0. One verifies that the map

SL(2,R)×M →M,

((a bc d

), z

)7→ az + b

cz + d

defines an action of SL(2,R) on H+. The reader is encouraged to showthat Γ := SL(2,Z), consisting of all the matrices in SL(2,R) with integerentries, acts properly discontinuously onM via the above action (in contrastto having dense orbits on the real line).

(iii) Let G be a Lie group and let Γ ≤ G be a discrete subgroup. Then the leftaction Γ×G→ G, (γ, g) 7→ γg of Γ on G is free and properly discontinuous.Hence Γ\G admits a smooth manifold structure for which p : G → Γ\G isa covering.

For the upcoming discussion of fibrations, we record the following: Given compactsubsets K1 and K2 of a Lie group G, the set K2K

−11 = k2k−1

1 | k2 ∈ K2, k1 ∈ K1is compact: Indeed, it is the image of the compact set K1 ×K2 ⊆ G×G under thesmooth map G×G→ G given by (x, y) 7→ yx−1.

Fibrations are a common generalization of the concepts of vector bundles andcoverings.


Definition 1.56. Let E,B and F be smooth manifolds and let π : E → B be asmooth map. The triple (π,E,B) is a fiber bundle with base B, total space E and

fiber F if

(i) the map π : E → B is surjective, and(ii) there is an open cover Ui | i ∈ I of B and diffeomorphisms

hi : π−1(Ui) → Ui × F with hi(π

−1(x)) = x × F.

Example 1.57.

(i) Coverings yield fiber bundles with discrete fiber.(ii) Vector bundles are fibrations with fiber a vector space.(iii) Consider the following interpretation of S3 ⊆ R4:

E = (z1, z2) ∈ C2 | z21 + z22 = 1The group S1 = ∈ C | || = 1 acts on E via(z1, z2) = (z1, z2). This action is free and the quo-tient identifies with P1(C). The latter is incidentallydiffeomorphic to S2. This is a fibration of S3 over S2

with fiber S1, the so called Hopf fibration. b

b

b

E

B

π

F

Fibrations are particularly fruitful in the setting of Lie groups acting on man-ifolds: An action of a Lie group G on a manifold M is smooth if the action mapG×M →M is smooth. In particular, for every g ∈ G the map M →M , m 7→ gmis a diffeomorphism: It is smooth as a restriction of the smooth action map and hasan inverse arising in the same fashion. The action of G on M is proper if for everycompact set K ⊆M the set g ∈ G | gK ∩K 6= ∅ has compact closure in G.

Theorem 1.58. Let G be a Lie group, M a manifold and G ×M → M a smoothaction of G on M . If the action is free and proper then there is a unique smoothmanifold structure on G\M such that M → G\M is a smooth fibration.

A good reference for this Theorem is [vdB06], in which Theorem is deduced fromthe following, more general equivalence relation version which mostly depends onthe inverse function theorem.

Theorem 1.59. Let M be a smooth manifold and let R ⊆M ×M be an equivalencerelation on M . If R is a closed submanifold of M × M and pr1 : R → M is asubmersion then the quotient R\M has a unique structure of smooth manifoldsuch that π :M → R\M is a submersion.

In particular, we can apply Theorem 1.58 in the case where M = G is a Liegroup and H ≤ G is a closed subgroup of G acting on G on the right: G×H → G,(g, h) 7→ gh.

Lemma 1.60. Let G be a Lie group and let H ≤ G be closed. Then the right actionof H on G is proper and free.

Proof. Freeness is immediate. As to properness, let K ⊆ G be compact. Then

h ∈ H | Kh ∩K 6= ∅ = H ∩K−1K.

Since H is closed and K−1K is compact, so is the above intersection.

As a corollary, we obtain the following powerful mechanism of producing mani-folds and actions on them.


Corollary 1.61. Let G be a Lie group and let H be a closed subgroup of G. Thenthere is a unique smooth manifold structure on G/H such that the quotient mapπ : G→ G/H is a smooth fibration with base G/H and fiber H . With this manifoldstructure, the action G×G/H → G/H is smooth.

Corollary 1.61 produces one of the most enormous classes of manifolds with manyinteresting subclasses depending on properties of G and H .

Example 1.62. To illustrate the usefulness of Corollary 1.61 we consider the follow-ing: Equip Rn with the usual scalar product 〈−,−〉. For 1 ≤ p ≤ n, let

Sp,n := (x1, . . . , xp) ∈ (Rn)p | 〈xi, xj〉 = δij ∀i, j ∈ 1, . . . , p.Observe that O(n) acts transitively on Sp,n as it acts transitively on orthonormalbases. Let (e1, . . . , en) be the standard orthonormal basis of Rn and fix the basepointp := (e1, . . . , ep) ∈ Sp,n. The stabilizer of p in O(n) is given by

Hp =

(Idp 00 B

)∣∣∣∣B ∈ O(n− p)

which is a closed, regularly embedded submanifold of O(n). Hence the bijection

O(n)/Hp∼= O(n)/O(n− p)

can be used to equip Sp,n with a smooth manifold structure. The manifolds Sp,n arecalled Stiefel manifolds and play a fundamental role when it comes to characteristicclasses and vector bundles. A similar reasoning as above, lets us treat Grassmannianmanifolds as quotients of Lie groups and hence equip them with a manifold structurein a very convenient way.

Combining results from above we record the following.

Corollary 1.63. Let M be a manifold and let b ∈ M . Furthermore, let G be a Liegroup acting smootly and transitively on M . Let H denote the stabilizer in G ofb ∈M . Then the bijection G/H →M , gH 7→ gb is a diffeomorphism.

Corollary 1.63 is pleasant compatibility result and in particular states that onecannot find any exotic smooth structures on a manifold by exhibiting it as a homo-geneous space of a Lie group as above.

Example 1.64. We now have a look at further examples of this mechanism.

(i) The Lie group GL(n,R) acts smoothly and transitively on Rn \0. Thestabilizer of e1 = (1, 0, . . . , 0)T ∈ Rn in GL(n,R) is given by

P0 =

(1 v0 A

)∣∣∣∣ v ∈ Rn−1, A ∈ GL(n− 1,R)

which can be identified as a semi-direct product. Anyway, we conclude thatGL(n,R)/P0 is diffeomorphic to Rn \0.

(ii) The Lie group GL(n,R) also acts on the set Pn(R) of all lines passingthrough 0 ∈ Rn+1. Recall that Pn(R) can also be defined as the quotientof Rn+1 \0 by the multiplication action of R∗, and the quotient of Sn bythe antipodal map. Whereas the action of GL(n,R) is not visisble in thesecond definition, the first one clearly shows its transitivity. The stabilizerof [e1] ∈ Pn(R) is given by

P :=

(a v0 A

)∣∣∣∣ a ∈ R \0, v ∈ Rn, B ∈ GL(n,R)

.

As a consequence, GL(n+ 1,R)/P is diffeomorphic to Pn(R).


In the constext of (ii), the following exercises are worthwhile: For g ∈ GL(n,R),study the qualitative behaviour of the action of gn | n ∈ Z on Pn(R). Giventhat linear algebra classifies elements of GL(n,R) up to conjugacy, three particularexamples to look at are

(λ1

λ2

)(λ1 6= λ2),

(cos t sin t− sin t cos t

)and

(1 x

1

).

What do the behaviours say towards a classification of homeomorphisms of S1 upto conjugacy? Finally, one can consider the same question for the case of GL(3,R)where one discovers new phenomena, semi-hyperbolicity.

Also, does there exist an SL(2,R)-invariant probability measure on P1(R)? Theanswer to this question is “no”. However, is this a general phenomenon, in the sensethat no compact homogeneous space SL(2,R)/H of SL(2,R) supports an invariantprobability measure? Again, the answer is “no”.

(iii) Returning to the examples, consider the following construction which is offundamental importance for the remainder of the course. Let q : Rn+1 → R

be the quadratic form given by q(x) = x21+ · · ·+x2n−x2n+1. This quadraticform has three kinds of level sets:(a) For a < 0, the equation x21 + · · · + x2n = x2n+1 + a has no solutions

with −√−a < xn+1 <√−a. For |xn+1| >

√a, there are (rotationally

invariant) solution. In fact, we obtain a two-sheeted hyperboloid.(b) The level set q−1(0) is a cone with singular point 0 ∈ Rn+1.(c) For a > 0, the level set q−1(a) is a one-sheeted hyperboloid.

b

b

Recall that the symmetry group of q is

O(n, 1) = g ∈ GL(n+ 1,R) | q(gx) = q(x) ∀x ∈ Rn+1.A general theorem of Witt states that O(n, 1) acts transitively on all threekinds of level sets, except for the obvious exclusion of 0 ∈ Rn+1 in the casea = 0. We shall now restrict our attention to the upper sheet of q−1(−1)and denote it by Hn := x ∈ q−1(−1) | xn+1 > 0. To this end we introduce

SO0(n, 1) = g ∈ O(n, 1) | g(Hn) = Hn and det g = 1.


The homomorphism O(n, 1) → Sym(π0(q−1(−1)))×−1, 1, given the per-

mutation an element of O(n, 1) induces on the two sheets of q−1(−1) and itsdeterminant, is surjective and continuous for the discrete topology on theright hand side. Therefore, SO0(n, 1) is an open subgroup of index four inO(n, 1). We now show that SO0(n, 1) acts transitively on Hn. The stabilizerin SO0(n, 1) of en+1 is given by

K :=

(A 00 1

)∣∣∣∣A ∈ SO(n)

.

It acts as SO(n) in planes parallel to the plane given by xn+1 = 0: Given

k =

(A 00 1

)∈ K

and x = (x1, . . . , xn+1)T ∈ Hn we have

(x) =

A

x1...xn

xn+1

.

In particular, there is A ∈ SO(n) such that

A(x1

... xn

)=

0...0√

x2n+1 − 1

and hence k(x) =

0...0√

x2n+1 − 1

1

where k ∈ SO0(n, 1) is defined by A. In order to translate vertically, weintroduce

A :=

at :=

Idn−1 0

0

(cosh(t) sinh(t)sinh(t) cosh(t)

)∣∣∣∣∣∣t ∈ R

which is a one-parameter subgroup of SO0(n, 1). Indeed, it preserves the up-per sheet of q−1(1), all its elements have determinant one and one computesq(atx) = q(x) thanks to the identity cosh2(t)− sinh2(t) ≡ 1. Now,

at(en+1) =

0...0

sinh tcosh t

which combined with the K-action shows transitivity of SO0(n, 1) on Hn.

Corollary 1.65. The map SO0(n, 1)/K ∼= K, gK 7→ g(en+1) is a diffeomorphismand SO0(n, 1) is connected.

Proof. The connectedness of SO0(n, 1) is due to the exercise that a total space ofa fiber bundle with connected base and connected fiber is itself connected.

Let us now compute the tangent space of Hn at x ∈ Hn. To this end, let b :Rn+1 ×Rn+1 → R, (x, y) 7→∑n

i=1 xiyi − xn+1yn+1 be the symmetric bilinear formassociated to q. Then Dxq(v) = 2b(x, v) and hence TxH

n = (Rx)⊥ where the


orthogonal complement is taken with respect to b. Observe that since q(x) = −1for all x ∈ Hn the space Rx with form q is non-degenerate, hence the splitting

Rn+1 = Rx⊕ TxHn = Rx⊕ (Rx)⊥.

Silvester’s Inertia Theorem now implies that b restricted to TxHn is positive def-

inite. This constitutes an instance of a Riemannian metric as we shall see shortly.The space Hn is called real hyperbolic space of dimension n, and SO0(n, 1) will acton it by isometries once we have properly defined the metric.

2. Riemannian Metrics, Covariant Derivative and Geodesics

2.1. Definitions and Examples. First of all, we define Riemannian metrics asencountered in the last section and prove their existence.

Definition 2.1. Let M be a smooth manifold. A Riemannian metric on M is a mapg which to every point p ∈ M associates a scalar product gp : TpM × TpM →R, satisfying the following smoothness condition: For any chart (U,ϕ) of M , thefunctions U → R, p 7→ gp(∂ip, ∂jp) are smooth for all 1 ≤ i, j ≤ m.

In the context of Definition 2.1 recall that

∂i(p)(f) =∂(f ϕ−1)

∂xi(ϕ(p))

for every f ∈ C∞(M) and that (∂1(p), . . . , ∂n(p)) is a basis of TpM for all p ∈M .

Remark 2.2. It will be useful to have the bundle definition of Riemannian metrics aswell: For a real vector space V , let S2(V ∗) denote the vector space of all symmetricbilinear maps V ×V → R. Then one can define S2(T∗M) as a smooth vector bundleof symmetric bilinear forms. That is, one introduces a manifold structure on

S2(T∗M) =⋃

p∈M

p × S2((TpM)∗)

as in the case of the tangent and cotangent bundle. A Riemannian metric g on Mis then a smooth section g :M → S2(T∗M) such that gp := g(p) ≫ 0 for all p ∈M .Notationally, given a Riemannian metric g on M , a chart (U,ϕ) and u, v ∈ TpMwith coordinates u =

∑mi=1 ui∂i(p) and v =

∑mj=1 vj∂j(p) respectively, we have

gp(u, v) =∑

i,j

uivjgp(∂i(p), ∂j(p))

where we abbreviate gij(p) := gp(∂i(p), ∂j(p)). Recall that ui = (dxi)p(u) andsimilarly for v ∈ TpM . Then in tensor notation, the bilinear form which maps(u, v) to uivj is (dxi)p⊗(dxj)p. Thus, from the S2(T∗M) viewpoint on Riemannianmetrics, we have

gp =∑

i,j

gij(p)(dxi ⊗ dxj)p.

The traditional (sloppy) way to refer to a Riemannian metric in local coordinatesis g =

∑i,j gij dxidxj .

We now turn to the existence of Riemannian metrics which is unclear, given thatwe are asking for a highly non-zero section of a certain bundle and in view of whatwe have learned about e.g. vector fields on the two-dimensional sphere. Yet we havethe following remarkable result whose proof is not even overly difficult.

Proposition 2.3. Every smooth manifold admits a Riemannian metric.


Proof. First, consider a single chart (U,ϕ) of M . Then we have an associated basis(∂1(p), . . . , ∂m(p)) of TpM for all p ∈ U whence for u, v ∈ TpM we may define

gUp (u, v) :=

m∑

i=1

uivi,

employing the coordinates of u and v with respect to said basis. In other words, weset gUp (∂i(p), ∂j(p)) := δij . This defines a Riemannian metric on U .

As to the whole manifold, let (Ui, ϕi)i∈I be a locally finite covering of M bycharts and let (fi)i∈I be a subordinate smooth partition of unity. For p ∈ M nowdefine

gp :=∑

i∈I

fk(p)gUip .

Since for every p ∈ M admits a neighbourhood Vp such that i ∈ I | Ui ∩ Vp 6= ∅is finite, gp is a scalar product as a convex combination of scalar products.

An alternative proof of Proposition 2.3 is to simply refer to the general versionof Whitney’s Embedding Theorem and restrict the Euclidean scalar product to thetangent spaces of the embedding.

In a sense, Whitney’s theorem states that in order to study smooth manifoldsone has to look no further than submanifolds of Euclidean space. An interestingquestion is whether this also holds for Riemannian manifold: Can every Riemannianmanifold be embedded isometrically into some Euclidean space? The answer is yesand due to Nash.

Example 2.4. Arguably, the most immediate example of a Riemannian manifoldis Rn: For all p ∈ Rn and u, v ∈ TpR

n = Rn set gp(u, v) = 〈u, v〉 where 〈−,−〉denotes the standard scalar product on Rn.

Earlier on, we have seen that differential forms can be pulled back via generalsmooth maps and that vector fields can be pushed forward under some assumptionson the smooth maps. To the assumptions on has to put on a smooth map for itto allow pulling back a Riemannian metric are of intermediary nature: Let M andN be smooth manifolds and let f : M → N be a smooth immersion. If h is aRiemannian metric on N then

gp(u, v) := hf(p)(Dpf(u), Dpf(v))

where p ∈M and u, v ∈ TpM defines a Riemannian metric on g, denoted f∗(h).

Definition 2.5. Let (M, g) and (N, h) be Riemannian manifolds and let f :M → Nbe a smooth map. Then f is an isometry if f is a diffeomorphism and g = f∗(h).

Given a Riemannian manifold (M, g), the set Iso(M, g) of isometries of M isa subgroup of the group of diffeomorphisms of M . Whereas the latter is infinite-dimensional from any point of view, Iso(M, g) is a Lie group by a result of Myersand Steenrod. Later on we will look at the particularly intriguing case in whichIso(M, g) acts transitively on M .

One of the points of having a Riemannian metric is to be able to measure thelength of certain curves.

Definition 2.6. Let (M, g) be a Riemannian manifold and let c : [a, b] → M be aC1-curve. The length l(c) of c is

l(c) :=

∫ b

a

√gc(t)(c′(t), c′(t)) dt.


It is useful to extend Definition 2.6 to piecewise C1-curves: If c : [a, b] → M isC1 on [ai, ai+1] for all i ∈ [0, n− 1] where a0 = a < a1 < · · · < an = b then we setl(c) :=

∑n−1i=0 l(c|[ai,ai+1]).

Note that we are defining the length of a curve using its instant velocities whichis quite the other way around as in calculus where there are length and time first,then their quotient. We will see more instances of this reverse engineering. Anyway,one shows that the length of a piecewise C1-curve does not depend on the choice ofparametrization. Also, one can always reparametrize such a curve by arc length inwhich case ‖c′(t)‖ = 1. One of the first problems in Riemannian geometry is to findcurves of shortest length between given points. We will later on characterize theseas having constant speed and “zero acceleration”. The term “acceleration” howeverrequires clarification as the notion of “second derivative of a curve” is just not there.This will be remedied by introducing connections which provide a preferred way oftransporting tangent vectors along a curve. Whereas there are plenty connections ingeneral, Riemannian manifolds come with a natural one, the Levi-Civita connection.

Now, given a connected Riemannian manifold M and x, y ∈ M we define thedistance between a and b by

d(x, y) := infl(c) | c : [a, b] →M piecewise C1, c(a) = x, c(b) = y.Note that in a connected Riemannian manifold as above, there always is a piece-wise C1-curve between any two given points: Find a sequence of coordinate charts(Ui, ϕi)

ni=0 with x ∈ U0, y ∈ Un and Ui ∩ Ui+1 6= ∅.

b bx y

· · ·U0 Un

Proposition 2.7. Let (M, g) be a connected Riemannian manifold and define themap d :M ×M → R by

d(x, y) := infl(c) | c : [a, b] →M piecewise C1, c(a) = x, c(b) = y.Then d is a metric on M inducing the present topology.

Although this does not present much difficulties we skip the proof and move onto examples of Riemannian manifolds and construction methods.

Example 2.8. This example is to illustrate that for a sub-manifold M of Rn, the distance function induced from therestricted scalar product is generally quite different fromthe ambient distance function: For instance, consider thesphere Sn = x ∈ Rn+1 | ‖x‖ = 1. Then the distanceon Sn between two points x and y on the sphere at anangle α is α whereas their distance in Euclidean spaceis 2(1 − cosα). Also, erasing a neighbourhood of 0 ∈ R2

naturally results in an altered metric.

b b

b

α

b b b

In particular, note that given a metric space one can perfectly restrict the metricto a subset and thereby produce a new metric space, this is not what happens whenone restricts a Riemannian metric to a submanifold.

Example 2.9. The hyperbolic spaces Hn introduced in Example 1.64 provide morekey Riemannian manifolds. Retain the notation of said example. Given x ∈ Hn,set hx(u, v) := b(u, v) for all u, v ∈ TxH

n. Then h is a Riemannian metric on Hn.Observe that for every g ∈ SO0(n, 1) and x ∈ Hn, the map Dxg sends hx to hg(x),i.e. SO0(n, 1) acts by isometries. This yields an injective group homomorphism


SO0(n, 1) → Iso(Hn, h). Later on, we shall see that this homomorphism is almostsurjective. Another model of hyperbolic space Hn via the diffeomorphism

f : Dn := y ∈ Rn | ‖y‖ < 1 → Hn, y 7→(

2yi1− ‖y‖2 ,

1 + ‖y‖21− ‖y‖2

)

One verifies that f indeed ranges in Hn by computing q(f(y)) and that it admits asmooth inverse. Furthermore, one obtains

g := f∗(h) = 4

n∑

i=1

(dyi)2

(1− ‖y‖2 , i.e. gy(u, v) =4〈u, v〉

(1 − ‖y‖2)2 .

In this model the boundary Sn−1 ofDn is infinitely far away from the origin: Indeed,for 0 ≤ r < 1 one computes

lg([0, re1]) = 4

∫ r

0

dt

1− t2= ln

(1 + r

1− r

).

Another interesting observation is that whereas a random walk starting at the originin E2 will return to a neighbourhood of the origin infinitely many times. In the diskmodel of H2, however, random walks do not come back, due to the expanding space:Whereas the area of a ball of radius R in E2 behaves like R2, its behaviour withrespect to the hyperbolic metric is exponential in R. Also, whereas the area of aEuclidean annulus of width R behaves like R, a hyperbolic annulus with largerradius equal to one already contains a large fraction of the entire area.

For later use, we now record two formal definitions that allow us to constructnew Riemannian manifolds out of old.

Definition 2.10. Let (M, g) and (N, h) be a Riemannian manifolds. Then (N, h) isa Riemannian submanifold of (M, g) if

(i) N is a submanifold of M , and(ii) i∗(g) = h where i : N →M is the canonical injection.

Definition 2.11. Let (M, g) and (N, h) be Riemannian manifolds. Then M ×N isa Riemannian manifold with metric g × h defined by

(g × h)(x,y)(u, v) = gx(u1, v1) + hy(u2, v2)

where u = (u1, u2) and v = (v1, v2) with respect to T(x,y)(M ×N) = TxM ×TyN .

It is also important to know how Riemannian metric behave with respect tocovering maps.

Definition 2.12. Let (M, g) and (N, h) be Riemannian manifolds and let p : N →Mbe a smooth map. Then p is a Riemannian covering if

(i) p is a smooth covering, and(ii) h = p∗g.

In particular, a Riemannian covering is a local isometry. It is plain that if p :N → M is a smooth covering of smooth manifolds and g is a Riemannian metricon M then p is a Riemannian covering when N is equipped with the Riemannianmetric h = p∗(g).

We now turn to the question under which circumstances a Riemannian metricon N can be pushed down to M via p.

Proposition 2.13. Let (N, h) be a Riemannian manifold and let Γ be a group. Fur-ther, let Γ×N → N be an action of Γ on N which is free, properly discontinuously,and by isometries. Then there is a unique Riemannian metric g on the quotientmanifold M := Γ\N such that p : N → M is a Riemannian covering.


Proof. Let x ∈ M and suppose y and y′ are elements of p−1(x). Then there is anisometry γ ∈ Γ with γy = y′ and the diagram

Nγ

//

p

N

p~~⑥⑥⑥⑥⑥⑥⑥⑥

M

commutes. So does the diagram obtained by taking derivates.

TxNDyγ

//

Dyp""

Ty′N

Dy′p①①①①①①①①

xM

Since p is a covering map, both Dyp and Dy′p are vector space isomorphisms. Weuse Dyp to introduce a scalar product on TxM by gx := ((Dyp)

−1)∗(hy). Observethat since γ is an isometry we have (Dyγ)

∗(hy′) = hy. Therefore

gx = ((Dyp)−1)∗(hy) = ((Dyp)

−1)∗(Dyγ)∗(hy′)

= (Dyγ (Dyp)−1)∗(hy′) = ((Dy′p)

−1)∗(hy′)

which shows that gx is well-defined. The smoothness of the resulting Riemannianmetric is due to the fact that p is a local diffeomorphism. This way, p becomes aRiemannian covering by construction.

Example 2.14. We equip Rn with the Riemannian metric coming from the standardscalar product 〈x, y〉p :=

∑ni=1 xiyi, denoted can. One can then show that the

Riemannian distance of x, y ∈ Rn given by

d(x, y) := infl(c) | c : [a, b] →M piecewise C1, c(a) = x, c(b) = y.

coincides with the Euclidean distance of x and y given by ‖x−y‖ =√〈x− y, x− y〉.

As a hint, consider the smooth, distance non-increasing projection onto the straightline connecting two points or go back to the definition of the Riemann integral andthe mean value theorem.

Proposition 2.15. Every Riemannian isometry g of (Rn, can) is given by g : v 7→Rv + a for some R ∈ O(n) and a ∈ Rn.

Sketch of Proof. By Example 2.14, an isometry g ∈ Iso(Rn) is a bijection whichsatisfies ‖g(x)− g(y)‖ = ‖x− y‖ for all x, y ∈ Rn. Now one can invoke the Mazur-Ulam theorem.

Theorem 2.16 (Mazur-Ulam). Let V be a normed vector space over R and letT : V → V be a bijection satisfying ‖g(x)− g(y)‖ = ‖x− y‖ for all x, y ∈ V . ThenT is of the form T (x) = Ax+ b where a : V → V is linear.

By this theorem, we have g(v) = R(v) + a for all v ∈ Rn where R is linearand ‖R(x) − R(y)‖ = ‖x − y‖. By the parallelogram law, this implies R ∈ O(n):〈x, y〉 = (‖x+ y‖2 − ‖x− y‖2)/4.

We now examine the group law on Iso(Rn, can): Every g ∈ Iso(Rn) is representedby a pair (R, a) in the set-theoretic cartesian product O(n)×Rn. Given isometriesg1 = (R1, a1) and g2 = (R2, a2) we compute

g1g2(v) = g1(R2v + a2) = R1R2v +R1a2 + a1


and thus g1g2 = (R1R2, R1a2 + a1). Thus, as a group, Iso(Rn, can) is isomorphicto the semidirect product O(n) ⋉ Rn. The map r : O(n) ⋉ Rn → O(n) is a homo-morphism whose kernel is the group of pure translations. Given a ∈ Rn we writeTa(v) = v + a.

Riemannian coverings p : Rn → M are given by subgroups ΓIso(Rn) which actfreely and properly discontinuously on Rn. In this context, we have the following.

Definition 2.17. A subgroup Γ ≤ Iso(Rn, can) is crystallographic if it acts properlydiscontinuously and Γ\Rn is compact. It is Bieberbach if it is crystallographic andacts freely.

In particular, a Bieberbach group Γ ≤ Iso(Rn, can) defines a Riemannian cover-ing pΓ : Rn → Γ\Rn where Γ\Rn is a compact manifold.

Example 2.18. Let P denote the subgroup of O(n) consisting of permutation matri-ces and set Γ := (R, γ) | R ∈ P, γ ∈ Zn is crystallographic but not Bieberbach.

Bieberbach groups are more difficult to construct. The trivial ones are the contentof the following proposition.

Proposition 2.19. Let (a1, . . . , an) be a basis of Rn and Γ = Tγ | γ ∈ Z a1 +· · ·Z an. Then Γ is Bieberbach and Γ\Rn is diffeomorphic to Tn = S1 × · · · × S1.Furthermore, the Riemannian manifolds Γ\Rn and Γ′\Rn are isomorphic if andonly if there is R ∈ O(n) with R(Z a1 + · · · + Z an) = Z a′1 + · · · + Z a′n whereΓ = Tγ | γ ∈ Z a1 + · · ·Z an and Γ′ = Tγ′ | γ′ ∈ Z a′1 + · · ·Z a′n.Proof. Consider the map e : Rn → Tn given by x 7→

(e2πix1 , . . . , e2πixn

)where

x = (x1, . . . , xn)T ∈ Rn. The map e is a smooth covering and we have e(x) = e(y)

if and only if x−y ∈ Z a1+· · ·+Z an, that is if and only if Tγ(x) = y for some Tγ ∈ Γ.This implies that e : Γ\Rn → Tn, being a local diffeomorphism and bijective, is adiffeomorphism.

Assume now that f : Γ\Rn → Γ′\Rn is an isometry. Since pΓ and pΓ′ are cover-ing maps and Rn is simply connected, there exists a diffeomorphism f : Rn → Rn

such that the following diagram commutes:

Γ\Rn f// Γ′\Rn

Rn

pΓ

OO

f

// Rn .

pΓ′

OO

Now observe that f is an isometry: Let gΓ and gΓ′ denote the Riemannian metricson Γ\Rn and Γ′\Rn respectively for which pΓ and pΓ′ are Riemannian coverings.Furthermore, let gcan denote the canonical Riemannian metric on Rn. Then

f∗(gcan) = f∗p∗Γ′(gΓ′) = (pΓ′ f)∗(gΓ′)

= (f pΓ)∗(gΓ′) = p∗Γf∗(gΓ′) = p∗Γ(gΓ) = gcan.

By the above we conclude that f = (R, a) for some R ∈ O(n) and a ∈ Rn. Fromthis one deduces that Γ and Γ′ are conjugate as subgroups of Iso(Rn): fΓf−1 = Γ′.This implies R(Z a1 + · · ·+ Z an) = Z a′1 + · · ·+ Z a′n.

Remark 2.20. The Riemannian manifolds described in Proposition 2.19 are calledflat tori because they are diffeomorphic to a torus and flat from a curvature pointof view as we shall see later. We now ask whether there is a "reasonable space"parametrizing the set of flat tori up to isometry: Let R be the set of all subgroups ofRn of the form Z a1+· · ·+Zan where (a1, . . . , an) is a basis of Rn. Clearly, GL(n,R)


acts on R: Given Λ = Z a1 + · · · + Z an and g ∈ GL(n,R we have g(Λ) = Z ga1 +· · ·+Z gan where (ga1, . . . , gan) is again a basis of Rn. This action is transitive: FixΛ0 = Z e1+ · · ·+Z en. Given Λ = Z a1+ · · ·+Z an set g = (a1, . . . , an) ∈ GL(n,R).Then g(Λ0) = Λ. The stabilizer in GL(n,R) of Λ0 is given by

g ∈GL(n,R) | g(Λ0) = Λ0= g ∈ GL(n,R) | ∀i, j ∈ 1, . . . , n : gij ∈ Z, g−1

ij ∈ Z= g ∈ GL(n,R) | ∀i, j ∈ 1, . . . , n : gij ∈ Z and det g ∈ ±1.

Hence R ∼= GL(n,R)/GL(n,Z) as sets. Observe that GL(n,Z) is a discrete sub-group of the Lie group GL(n,R) whence GL(n,R)/GL(n,Z) has a canonical mani-fold structure. Taking into account isometry classes one sees that at the end of theday, flat tori are parametrized by the double quotient O(n)\GL(n,R)/GL(n,Z)where GL(n,Z) acts on O(n)\GL(n,R) on the right because O(n) is compact.Whereas the above double quotient has a nice structure, equally natural questionslead to double quotients where the left hand subgroup is non-compact, resulting inthe fact that the double quotient is not even a standard Borel space.

A less obvious Bieberbach group is Γ := 〈γ1, γ2〉 ≤ Iso(R2) where

γ1

(xy

)=

(x−y

)+

(10

)and γ2 =

(xy

)=

(xy

)+

(01

).

We claim that Γ is a Bieberbach group, i.e. it acts properly discontinuously, freelyand with compact quotient on R2. Geometrically, we have

(xy

)

γ1

(xy

) (xy

)

γ2

(xy

)

To get a geometric idea of the quotient, consider the domain

12

1D D =

(xy

)∈ R2

∣∣∣∣ 0 ≤ x ≤ 1, |y| ≤ 1

2

Any v ∈ R2 is Γ-quivalent to a point in D: Given v = (x, y)T , applying anappropriate power k of γ2 yields γk2 (x, y)

T = (x′, y′) with |y′| ≤ 12 . Then, consider

γl1γk2 (x, y)

T = γl1(x′, y′)T = (x′ + l, (−1)ly′)

so that 0 ≤ x′ + l ≤ 1 for appropriate l while |(−1)ly′| = |y′| ≤ 1/2. One easilydetermines the identifications that Γ induces on the boundary of D, namely

12

1D

The quotient Γ\R2 is termed Klein bottle. Note that Γ contains the purely trans-lational subgroup Γ′ := Tγ | γ ∈ 2Z e1 + Z e2 and that the torus Γ′\R2 projectswith degree two onto Γ\R2. It is a good exercise to visualize this projection. Lateron, we will study the Riemannian geometries on these examples.


An example of a Bieberbach group in three dimensions is the Hantsche-Wendt

group Γ := 〈γ1, γ2〉 ≤ Iso(R3) where

γ1

xyz

=

−x−yz

+

1/21/21/2

and γ2

xyz

=

x−y−z

+

1/200

.

Interestingly, Γ\R3 and S3 have the same Betti numbers but are not diffeomor-phic: They can be distinguished with integral homology for instance. The followingtheorem of Bieberbach shows that it is always a good idea to look for a purely trans-lational subgroup in order to prove cocompactness. Recall that r : Iso(Rn) → O(n)denotes the map which to an isometry associates its linear part.

Theorem 2.21 (Bieberbach). Let Γ\Iso(Rn) be a crystallographic grup. Then

(i) r(Γ) is a finite subgroup of O(n),(ii) Γ ∩ Ta | a ∈ Rn is Bieberbach, and(iii) Γ acts freely on Rn if and only if it is torsion-free.

The proof of Theorem 2.21 requires several good ideas. We refer the readerto [Cha12] and the classic [Wol11].

Riemannian Submersions. We have seen how Riemannian metrics behave with re-spect to coverings. Now, we look at the more general case of submersions: Letp :M → N be a submersion, i.e. Dxp : TxM → Tp(x)N is surjective for all x ∈M ,and let g be a Riemannian metric on M . Then we can define the horizontal subspace

Hx := (kerDxp)⊥ where the orthogonal complement is taken with respect to the

Riemannian metric. Clearly, Dxp|Hx : Hx → Tp(x)N is an isomorphism.

Definition 2.22. Retain the above notation. The submersion p is Riemannian if themap Dxp|Hx : (Hx, g|Hx) → (Tp(x)N, hp(x)) is an isometry.

Proposition 2.23. Let (M, g) be a Riemannian manifold and let G× M → M bea free, proper Lie group action by isometries of G on M . Then there is a uniqueRiemannian metric g on the quotient M := G\M such that p : M →M is a Rie-mannian submersion.

Proof. The proof is quite analogous to the proof of Proposition 2.13: Let x ∈ Mand y, y′ ∈ p−1(x). Also, let g ∈ G be such that g(y) = y′. Then the followingdiagram commutes:

TyM

Dyp##

Dyg// Ty′M

Dy′p

TxM

In particular, Dyg(kerDyp) = kerDy′p. Since (Dyg)∗(gy′) we get Dyg(Hy) = Hy′

where Hy = (kerDyp)⊥ and Hy′ = (kerDyp)

⊥. Now define

gx = ((Dyp|Hy )−1)∗(gy).

Then the above shows that gx is independent of the choice of y ∈ p−1(x).

Example 2.24. Recall that Pn(C) = C∗ \Cn+1 −0 has a manifold structure withcharts arising from affine hyperplanes in Cn+1. In fact, it is even a complex analyticmanifold. Whereas in the real case, the projection Sn → (Z /2Z)\Sn = Pn(R) is acovering, we obtain a non-discrete fibration in the complex case: Consider S2n+1 =z ∈ Cn+1 |∑n+1

i=1 |zi|2 = 1 and the inclusion S2n+1 → Cn+1 \0. The Lie groupS1 = z ∈ C | |z| = 1 acts on S2n+1 by ζ(z1, . . . , zn+1) = (ζz1, . . . , ζzn+1). Thisaction is free and proper; in fact any smooth action of a compact Lie group is


proper. Now, the inclusion S2n+1 → Cn+1 \0 induces a surjective submersionp : S2n+1 → Pn(C) and one can show that it induces a diffeomorphism

S1\S2n+1 → Pn(C).

Let (z, w) =∑n+1i=1 ziwi be the unitary scalar product on Cn+1. Then the Euclidean

scalar product on the real vector space Cn+1 is given by 〈u,w〉 = Re(u,w). Wenow equip Cn+1 with the standard Riemannian metric coming from 〈−,−〉 andS2n+1 ⊆ Cn+1 with the induced Riemannian metric. Then for z ∈ S2n+1 we have

TzS2n+1 = u ∈ Cn+1 | 〈z, u〉 = 0 = u ∈ Cn+1 | Re(z, u) = 0.

and the orthogonal decomposition Cn+1 = R z ⊕TzS2n+1 where both constituentsof the right hand side are real vector spaces. The S1-orbit through z ∈ S2n+1 canbe parametrized by c : R → S2n+1, t 7→ eit · z. Hence kerDzp = R ·i · z wherep : S2n+1 → S1\S2n+1. Using Hz ⊆ TzS2n+1 we have the following orthogonaldecomposition by definition:

TzS2n+1 = R iz ⊕Hz.

On PnC we put the Riemannian metric coming from S2n+1 or rather the restrictionto Hz . Thus, putting everything together, we have an orthogonal decomposition

Cn+1 = R z + R iz︸︷︷︸=C z

+Hz

where Hz is of real codimension one in TzS2n+1. Hence Hz is in fact orthogonal toCz for the unitary scalar product and hence a complex vector space. This is a goodexample to keep in mind for a submersion with interesting additional structure.

In the context of the above example it is worth mentioning that one can dodifferential geometry over the complex numbers but that things become more rigid.For instance, there is no analogue of Whitney’s embedding theorem or embeddingsinto projective space. In fact, complex submanifolds of complex projective spaceare very special.

Lie Groups. As usual, things work particularly well in the context of Lie groups.

Example 2.25. Let G be a Lie group and let 〈−,−〉e be a scalar product on TeG.Then we can define 〈u, v〉g = 〈DgLg−1(u), DgLg−1(v)〉e for all u, v ∈ TgG, produc-ing a left-invariant Riemannian metric g on G. In particular, G can be viewed as asubgroup of Iso(G, g).

One might as well construct a right-invariant Riemannian metric on a Lie group.In general, however, there need not be one which is bi-invariant.

Proposition 2.26. Let G be a Lie group. Then the following are equivalent.

(i) The Lie group G admits a bi-invariant Riemannian metric.(ii) The image Ad(G) ⊆ GL(g) is contained in a compact subgroup.

Corollary 2.27. Let G be a compact Lie group. Then G admits a bi-invariant Rie-mannian metric.

2.2. Covariant Derivative. This section truly kicks off the theory of Riemanniangeometry by introducing the crucial notion of connection, which was only under-stood in the 1920’s by very few people, including É. Cartan. Nevertheless, it leadsto the Riemannian curvature tensor which was introduced much earlier, namely in1853, by Riemann.

The problem we study is the following: Let X and Y be smooth vector fields onRn; later on we shall be interested in general manifolds, of course. Is there a naturalnotion of taking the derivative of Y with respect to X , that is the “variation of Y


from the point of view of X”? One possible way to deal with this is to consider theintegral curve cp : (−ε, ε) → Rn of X passing through p ∈M at time t = 0,

c′p(t) = X(cp(t))

cp(0) = p,

and to look at the vectors Y (cp(t)) ∈ Tcp(t) Rn. We would like to define the first

order variation of the map t 7→ Y (cp(t)) near t = 0. If we had started out witha general manifold we would be stuck at this point because the different tangentspaces don’t know about each other, that is, we have no means to compare tan-gent vectors at different points. In Rn, however, we can utilize the translations toestablish a privileged identification Tcp(t) R

n → TpRn. Using this identification we

consider Y (cp(t)) to be an element of TpRn and define

(∇XY )(p) := limt→0

Y (cp(t))− Y (p)

t

Writing Y = (Y1, . . . , Yn) and X = (X1, . . . , Xn) one verifies that

(∇XY )(p) =

(. . . ,

n∑

i=1

∂Yj(p)

∂xiXi(p), . . .

),

its merely a directional derivative. The map ∇ : Γ(TRn) × Γ(TRn) → Γ(TRn)satisfies the following formal properties.

(i) ∇XY −∇YX = [X,Y ] for all X,Y ∈ Γ(TRn).(ii) ∇fXY = f∇XY for all X ∈ Γ(TRn) and all f ∈ C∞(Rn).(iii) ∇X(fY ) = (Xf)Y + f∇XY for all X,Y ∈ Γ(TRn) and f ∈ C∞(Rn).

Using these properties, we now define connections in general. This will lead toparallel transport which allows to identify tangent spaces at distinct points of amanifold depending on a path connecting them, taking curvature into account. Thefact that our identification in Rn does not depend on a path connecting given pointsresembles the vanishing of curvature.

Definition 2.28. Let M be a smooth manifold. A connection on M is an R-linearmap ∇ : Γ(TM)×Γ(TM) → Γ(TM) which satisfies the following properties for allX,Y ∈ Γ(TM) and f ∈ C∞(M):

(i) ∇fXY = f∇XY ,(ii) ∇X(fY ) = (Xf)Y + f∇XY , and(iii) ∇XY −∇YX = [X,Y ].

Note that defining ∇ to be the zero map does not yield a connection due to thesecond property. Hence existence of connections is not completely obvious. However,if ∇1 and ∇2 are connections on a manifold M then so is g∇1+(1− g)∇2 for everyg ∈ C∞(M). Therefore, using a partition of unity to patch together connectionsdefined on charts, one shows that every smooth manifolds admits many differentconnections and thus many different notions of acceleration which mathematicianshad a hard time to accept.

Remark 2.29. Let ∇ be a connection on a smooth manifold M . Then the firstcondition implies that for given Y ∈ Γ(TM) the vector (∇XY )(p) (p ∈ M) onlydepends on X(p) rather than the values of X in a neighbourhood of p ∈ M : In-deed, let (U,ϕ) be a chart of M at p and recall that for every q ∈ U we ob-tain a basis (∂1(q), . . . , ∂m(q)) of TqM . Consider ∂i ∈ Γ(TU) (i ∈ 1, . . . ,m)as a smooth vector field. Then the local expression of X on U takes the form


X(q) =∑mi=1Xi(q)∂i(q). In vector field notation: X =

∑mi=1Xi∂i. Therefore

∇XY = ∇∑mi=1Xi∂i

Y =

m∑

i=1

Xi∇∂iY

and hence (∇XY )(p) =∑mi=1Xi(p)(∇∂iY )(p). It follows that for X,X ′ ∈ Γ(TM)

with X(p) = X ′(p) we have Xi(p) = X ′i(p) and hence (∇XY )(p) = (∇X′Y )(p).

The next theorem probably deserves to be called the fundamental theorem ofRiemannian geometry.

Theorem 2.30. Let (M, g) be a Riemannian manifold. Then there exists a uniqueconnection ∇ on M which for all X,Y, Z ∈ Γ(TM) satisfies

Xg(Y, Z) = g(∇XY, Z) + g(Y,∇XZ)

where g(Y, Z) denotes the function M → R, p 7→ gp(Y (p), Z(p)).

Proof. First, we show uniqueness which suggests a formula to prove existence. Con-sider the three equations that arise from the assumption through cyclic permutationof the variables:

Xg(Y, Z) = g(∇XY, Z) + g(Y,∇XZ)

Y g(Z,X) = g(∇Y Z,X) + g(Z,∇YX)

Zg(X,Y ) = g(∇ZX,Y ) + g(X,∇ZY ).

Computing the sum of the first two right hand sides and subtracting the third yields

g(∇XY, Z)+g(∇YX,Z) + g(∇XZ, Y )− g(∇ZX,Y ) + g(∇Y Z,X)− g(∇ZY,X)

= 2g(∇XY, Z)− g([X,Y ], Z) + g([X,Z], Y ) + g([Y, Z], X)

Therefore, we may record that 2g(∇XY, Z) equals

Xg(Y, Z) + Y g(Z,X)− Zg(X,Y ) + g([X,Y ], Z)− g([X,Z], Y )− g([Y, Z], X)

which shows uniqueness since g is non-degenerate. As for existence, let T (X,Y, Z)denote the right hand side of the above equation which we write as

(g([X,Y ], Z)− Zg(X,Y ))︸︷︷︸A(X,Y,Z)

+(Y g(Z,X)− g([Y, Z], X))︸︷︷︸B(X,Y,Z)

+(Xg(Z, Y )− g([X,Z], Y ))︸︷︷︸B(Y,X,Z)

Observe that 2g(∇XY, Z) = 2g(∇XY, Z). Now, it remains to verify that T (X,Y, Z)(p)only depends on Z(p) in which case the expression T (X,Y, Z)(p) is a linear formin Z(p) on TpM for fixed X and Y , to that there exists a unique vector ∇XY (p) ∈TpM such that 2gp(∇XY (p), Z(p)) = T (X,Y, Z)(p).

In order to verify this, we check what happens if we replace Z by fZ for somef ∈ C∞(M). Compute

A(X,Y, fZ) = g([X,Y ], fZ) + fZg(X,Y )

= f(g([X,Y ], Z) + Zg(X,Y ))

= fA(X,Y, Z)

as well as

B(X,Y, fZ) = Y g(fZ,X)− g([Y, fZ], X)

= Y (fg(Z,X))− g(Y (f)Z + f [Y, Z], X)

= Y (f)g(Z,X) + fY g(Z,X)− Y (f)g(Z,X)− fg([Y, Z], X)

= f(Yg(Z,X)− g([Y, Z], X))

= fB(X,Y, Z).


This shows T (X,Y, fZ) = fT (X,Y, Z) and hence as in Remark 2.29 implies thatT (X,Y, Z)(p) only depends on Z(p) for given X and Y . As explained above wemay hence define ∇XY (p) using the equation that showed uniqueness. It is left asan exercise to verify that the map ∇ so defined is indeed a connection.

Definition 2.31. Let (M, g) be a Riemannian manifold. The connection of Theorem2.30 is the Levi-Civita connection of (M, g).

We now aim to get a hold on the Levi-Civita connection by establishing a formulain local coordinates. This dates back to Einstein and his work with Grossmann atETH Zurich around 1910. Recall that given a chart (U,ϕ) of M we have for everyp ∈M a basis (∂1(p), . . . , ∂m(p)) of TpM given by

∂i(p)(f) =∂(f ϕ−1)

∂xi(ϕ(p)) for all f ∈ C∞(M).

Now consider the vector fields ∂i ∈ Γ(TU) and let ∇ denote the Levi-Civita con-nection of M . Then

∇∂j∂k(p) =m∑

l=1

Γljk∂l(p)

where the smooth functions Γljk : U → R are called Christoffel symbols and whichwe now determine: Let g =

∑mi,j=1 gijdxi ⊗ dxj be the expression of the Riemannian

metric in the chart (U,ϕ). Recall in particular that gij(p) = gp(∂i(p), ∂j(p)). Weare going to use the formula established in the proof of Theorem 2.30. To this end,observe that [∂i, ∂j ] = 0 on U because of the definition of the ∂i and the fact that

∂2g

∂xi∂xj=

∂2g

∂xj∂xifor all g ∈ C∞(ϕ(U)).

Employing said formula we obtain

2g(∇∂j∂k, ∂i) = ∂jg(∂k, ∂i) + ∂kg(∂i, ∂j)− ∂ig(∂j , ∂k)

= ∂jgki + ∂kgij − ∂igjk

Substituting the definition of the Christoffel symbols yields

2

m∑

l=1

Γljkgli = ∂jgki + ∂kgij − ∂igjk.

Let g−1 denote the matrix inverse of (gij). Multiplying the equation by (g−1)ir andsumming over i ∈ 1, . . . ,m we obtain using

∑mi=1 gli(g

−1)ir = δlr:

2Γrjk =m∑

i=1

(g−1)ir (∂jgki + ∂kgij − ∂igjk) .

Proposition 2.32. Retain the above notation. Then

∇XY =

m∑

i=1

∑

j

Xj∂jYi +∑

j,k

ΓijkXjYk

∂i

where

2Γljk =m∑

i=1

(g−1)ir (∂jgki + ∂kgij − ∂igjk) .


Proof. We compute

∇XY = ∇∑iXi∂i

(∑

k

Yk∂k

)

=∑

j,k

Xj∇∂j (Yk∂k)

=∑

j,k

Xj∂jYk · ∂k +∑

j,k

XjYk∇∂j∂k

=∑

i

∑

j

Xj∂jYi

∂i +

∑

j,k

XjYk∑

i

Γijk∂i

=∑

i

∑

j

Xj∂jYi +∑

j,k

ΓijkXjYk

∂i.

Example 2.33. Consider the following two examples.

(i) Consider (Rn, can). Then g(p)ij = δij for all p ∈ Rn whence Γijk = 0 for all

i, j, k ∈ 1, . . . , n. We therefore have

∇XY =∑

i

∑

j

Xj∂jYi

∂i.

In fact, ∇∂j∂k = 0 for all j, k ∈ 1, . . . , n.(ii) Consider the upper-half-plane H2 = (x, y) ∈ R2 | y > 0 with the Rie-

mannian metric ((dx)2 + (dy)2)/y2. For the computations, it is useful tothink of x and y in terms of x1 and x2, and accordingly replace ∂x and ∂ywith ∂1 and ∂2 respectively. We then get

2g(∇∂j∂k, ∂i) = ∂j(δkix−22 ) + ∂k(δijx

−22 )− ∂i(δjkx

−22 ).

For instance, let us compute ∇∂1∂1: We have 2g(∇∂1∂1, ∂1) = 0 and

2g(∇∂1∂1, ∂2) = 2x−32 = 2x−1

2 g(∂2, ∂2),

and therefore obtain

∇∂1∂1 = x−12 ∂2

and similarly

∇∂1∂2 = ∇∂2∂1 = −x−12 ∂1 and ∇∂2∂2 = −x−1

2 ∂2.

Canonical Connection on a Riemannian Submanifold. A nice consequenceof the formula for the Levi-Civita connection is that it is simple to compute theLevi-Civita connection of a Riemannian submanifold: Let (M, g) be a Riemanniansubmanifold of a Riemannian manifold (N, h), that is M ⊆ N is embedded and forall p ∈M we have gp = hp|TpM×TpM .

Proposition 2.34. Retain the above notation and let ∇M and ∇N denote the Levi-Civita connections of (M, g) and (N, h) respectively. Further, let X,Y ∈ Γ(TM)and X ′, Y ′ ∈ Γ(TN) be smooth vector fields with X ′|M = X and Y ′|M = Y . Then

(∇MX Y )(p) =

(∇NX′Y ′(p)

)⊥p

for all p ∈M where ⊥p denotes the orthogonal projection from TpN to TpM .


Proof. Let (U,ϕ) be a chart of N at p ∈ M ⊆ N such that ϕ(U) = Cnε (0) andϕ(U ∩ M) = Cmε (0) × (0)n−m. Now consider X,Y, Z ∈ Γ(TM) and let X ′, Y ′

and Z ′ be extensions of X , Y and Z to open neighbourhoods of M in N . ThenX ′ =

∑ni=1X

′i∂i with X ′

i|U∩M = 0 for all i ∈ m + 1, . . . , n and similarly for Y ′

and Z ′. Now recall the formular for the bracket of vector fields in local coordinates:

[X ′, Y ′] =∑

i,j

(X ′i∂iY

′j − Y ′

i ∂iX′j

)∂j

which, taking into account the vanishing of X ′i and Y ′

i on U ∩M for i ≥ m + 1,implies: [X ′, Y ′]U∩M = [X,Y ]. The formula in the proof of Theorem 2.30 nowimplies for p ∈ U ∩M :

2hp(∇NX′Y ′(p), Z ′(p)) = 2gp(∇M

X Y (p), Z(p))

which implies the proposition since Z ′(p) = Z(p).

Covariant Derivative Along a Curve. In this section we discuss the covariantderivative along curves which ultimately leads to the important notion of geodesics.

Definition 2.35. Let c : I → M be a smooth curve defined on some open intervalI ⊆ R. A vector field along c is a smooth map X : I → TM with X(t) ∈ Tc(t)M .

Retaining the above notation, our aim is to define "∇c′(t)X" for a vector fieldX along c. Let Γ(c∗TM) denote the vector space of such vector fields. It is also aC∞(I)-module.

Remark 2.36. The notation c∗TM stands for the pullback of the tangent bundleTM by c. It is defined as

c∗TM = (t, v) ∈ I × TM | v ∈ Tc(t)M.

Then c∗TM is a smooth vector bundle with base I and a smooth vector field alongc is merely a smooth section of c∗TM .

Theorem 2.37. Let (M, g) be a Riemannian manifold with Levi-Civita connection∇ and let c : I → M be a smooth curve defined on an open interval I ⊆ R. Thenthere is a linear map

D

dt: Γ(c∗TM) → Γ(c∗TM)

such that for all f ∈ C∞(I) and Y ∈ Γ(c∗TM):

D

dt(fY ) = f ′Y + f

DY

dt

If furthermore X ∈ Γ(TM) satisfies Y (t) = X(c(t)) for all t ∈ I we have

DY

dt(t) = (∇c′(t)X)(c(t)).

Proof. Let Y ∈ Γ(c∗(TM)) be such that Y (t) = X(c(t)) for some X ∈ Γ(TM). Wecompute (∇c′(t)X)(c(t)) in a chart (U,ϕ): Let ϕ(c(t)) = (x1(t), . . . , xm(t)). Then

c′(t) = (Dtc)(1) =

m∑

i=1

x′i(t)∂i(c(t)).


Indeed, we have

c′(t)(f) = (Dc(t)f)(c′(t))

= Dc(t)((f ϕ−1) ϕ)(c′(t))= Dϕ(c(t))(f ϕ−1)(Dc(t)ϕ)(c

′(t))

= Dx(t)(f ϕ−1)Dt(ϕ c)(1)

=

m∑

i=1

∂(f ϕ−1)

∂xi(x(t))x′i(t)

=

m∑

i=1

x′i(t)∂i(c(t))(f).

Using Proposition 2.32 we therefore have

(∇c′(t)X)(c(t)) =

m∑

i=1

∑

j

x′j(t)∂jXi(c(t)) +∑

j,k

Γijk(c(t))x′j(t)Xk(c(t))

∂i.

Now observe that∑j x

′j(t)∂jXi(c(t)) = Y ′

i (t) whence

(∇c′(t)X)(c(t)) =∑

i

Y ′

i (t) +∑

j,k

Γijk(c(t))x′j(t)Yk(t)

∂i

Given the above computation, we now define for any Y ∈ Γ(c∗TM) and localcoordinates (U,ϕ):

DY

dt(t) :=

m∑

i=1

Y ′

i (t) +∑

j,k

Γijk(c(t))x′j(t)Yk(t)

∂i

and leave the verifications to the reader.

Theorem 2.37 allows us to make the following, central definition.

Definition 2.38. Let (M, g) be a Riemannian manifold with Levi-Civita connection∇ and let c : I →M be a smooth curve defined on an open interval I ⊆ R. Further,let D/dt : c∗(TM) → c∗(TM) be the associated linear map and X ∈ Γ(c∗(TM)).Then X is parallel if D/dt(X) = 0.

Example 2.39. Consider the Riemannian manifold (Rn, can). Let c : R → Rn besmooth and write c(t) = (x1(t), . . . , xn(t)). Then the map R → TRn given byt 7→ c′(t) is a smooth vector field along c. We have

Dc′

dt(t) =

n∑

i=1

x′′i (t)∂i(c(t)) = c′′(t).

Thus, Dc′/dt is the acceleration.If now M ⊆ Rn is a regular submanifold equipped with the induced Riemannian

metric g and if c : R → M is a smooth curve then Dc′/dt = (c′′(t))⊥; that is, thecovariant derivative of c′(t) is the orthogonal projection of c′′(t) onto Tc(t)M .

We now turn to the notion of parallel transport, which in Euclidean space looksas follows.


b

b

Proposition 2.40. Let c : I →M be a C1-curve defined on an open interval I ⊆ R,and let t0 ∈ I. Then for every v ∈ Tc(t0)M there exists a unique parallel vectorfield X : I → TM along c such that X(t0) = 0.

Proof. It suffices to prove the Proposition for every compact subinterval [t0, T ] ⊆ R.Let t0 < t1 < · · · < tn = T be such that c([ti, ti+1]) ⊆ Ui where (Ui, ϕi) is a chart.We now look at the condition for X to be parallel in the chart (U,ϕ): The conditionDX/dt = 0 is equivalent to the system

X ′i(t) +

∑

j,k

Γijk(c(t))x′j(t)Xk(t) = 0 (i ∈ 1, . . . ,m)

for all t in an open interval J ⊆ R with c(J) ⊆ U . For t′ ∈ J and (v1, . . . , vm) ∈ Rn

there is a unique solution of the above system on the whole of J with Xk(t′) = vk.

Applying this argument to each subinterval [ti, ti+1] yields the conclusion.

Definition 2.41. Let c : [a, b] →M be a C1-map. The parallel transport Pc,a,b is thelinear map Tc(a)M → Tc(b)M which to every v ∈ Tc(a)M associates X(b) ∈ Tc(b)where X : I → TM is the parallel vector field with X(a) = v and c is the restrictionof a C1-map defined on an open interval I ⊆ R containing [a, b].

Proposition 2.42. Retain the notation of Definition 2.41. Then the parallel transportPc,a,b : Tc(a)M → Tc(b)M is an isometry. More generally, if X,Y ∈ Γ(c∗TM) then

d

dtgc(t)(X(t), Y (t)) = gc(t)

(D

dtX(t), Y (t)

)+ gc(t)

(X(t),

D

dtY (t)

).

Proof. The general formula is left as an exercise. To deduce that parallel transportis an isometry, note that if X and Y are parallel we obtain (d/dt)g(X(t), Y (t)) = 0and hence g(X(a), Y (a)) = g(X(b), Y (b)).

One can recover the Levi-Civita connection from the parallel transport.

Proposition 2.43. Let (M, g) be a Riemannian manifold, X,Y ∈ Γ(TM), p ∈ Mand c : (−ε, ε) →M an integral curve of X with c(0) = p. Then

(∇XY )(p) =d

dt

∣∣∣∣t=0

P−1c,0,t(Y (c(t)))

Proof. (Sketch). Define an operator D : Γ(TM)× Γ(TM) → Γ(TM) by

DXY (p) =d

dt

∣∣∣∣t=0

P−1c,0,t(Y (c(t)))

and argue that D = ∇ using the uniqueness of the Levi-Civita connection: Consideranother Z ∈ Γ(TM). Since Pc,0,t is an isometry we have

g(P−1c,0,t(Y (c(t))), P−1

c,0,t(Z(c(t)))) = g(Y (c(t)), Z(c(t))).

Taking the derivative at t = 0 yields:

Xg(Y, Z) = g(DXY, Z) + g(Y,DXZ).

at p. Also, one verifies that DfXY = fDXY and DXfY = X(f)Y + fDXY .


Example 2.44. Consider the sphere S2 = x ∈ R3 | x21 + x22 + x23 = 1 with itsinduced Riemannian metric. In this example, we compute the parallel transportalong the circle

c : [0, 2π] → S2, t 7→ (√1− r2 cos t,

√1− r2 sin t, r)

for r ∈ [0, 1). Specifically, we determine the map Tc,0,2π : Tc(0)S2 → Tc(0)S2.

rc

To this end, let x(t) = (x1(t), x2(t), x3(t)) ∈ Tc(t)S2. That is, for all t ∈ [0, 2π]:

(1) x1(t)√

1− r2 cos t+ x2(t)√

1− r2 sin t+ x3(t)r = 0.

According to Example 2.39, the parallelity condition Dx/dt = 0 translates to x′(t)being orthogonal to Tc(t)S2 in this context. That is

(2)x′1(t)√

1− r2 cos t=

x′2(t)√1− r2 sin t

=x′3(t)

r

Taking the derivative of (1) with respect to t and substituting expressions for thex′1(t) and x′2(t) summands in terms of x′3 resulting from (2) yields

(3) − x1(t)√1− r2 sin t+ x2(t)

√1− r2 cos t+

x′3(t)

r= 0.

Taking the derivative of (3) and taking into account (1) as well as (2) yields

x′′3 + rx3 = 0

Hence x3(t) = A cos(rt) + B sin(rt). From this one obtains using (2) formulas forx′1 and x′2 which can be integrated, leading to

x1(t) =−A√1 + r2

(sin(t) sin(rt) + r cos(t) cos(rt))

+B√

1− r2(sin(t) cos(rt) − r cos(t) sin(rt)) + c1

and

x2(t) =−A√1 + r2

(− cos(t) sin(rt) + r sin(t) cos(rt))

− B√1− r2

(cos(t) cos(rt) + r sin(t) sin(rt)) + c2.

The orthogonality condition 〈x(t), c(t)〉 = 0 leads to c1 = 0 = c2. Finally, we get

x(0) =

( −Ar√1− r2

,−Br√1− r

, A

),

x(2π) =

( −r√1− r2

(A cos(2πr) +B sin(2πr)) ,

r√1− r2

(A sin(2πr)−B cos(2πr)) ,

A cos(2πr) +B sin(2πr)) ,


Thus, if we take v1 := (r, 0,−√1− r2) and v2 := (0, 1, 0) as a basis of Tc(0)S2, the

matrix of Tc,0,2π is given by(

cos(2πr) sin(2πr)− sin(2πr) cos(2πr)

).

2.3. Geodesics. In this section, we finally discuss the important notion of geodesics.Let (M, g) be a Riemannian manifold with Levi-Civita connection ∇. Furthermore,let D/dt denote the covariant derivative along a smooth curve c.

Definition 2.45. Retain the above notation. Let c : I → M , defined on an openinterval I ⊆ R, be a smooth curve. Then c is a geodesic if Dc′/dt = 0 wherec′ : I → TM is the tangent velocity vector field along c.

In other words, a geodesic is a curve whose tangent vector field is invariantwith respect to parallel transport along itself. Intuitively, the position vector doesnot experience any acceleration. We will shed light on the exact relation betweengeodesics and distance-minimising curves later on but for the moment, convinceyourself that just as in driving, accelerating corresponds to making a detour.

Geodesics can also be looked at from a variational point of view: Given pointsx, y ∈ M , consider all curves c connecting x to y and the functional L which toa given curve associates its length. If one sets up things correctly then the Euler-Lagrange equations arise from the condition that a curve c be a critical point ofthe functional L, see e.g. [Spi79].

In local coordinates (U,ϕ) with ϕ(c(t)) = (x1(t), . . . , xm(t)), the functions xi(i ∈ 1, . . . ,m) associated to the geodesic c are the solution of the differentialequation

(GLC) x′′i (t) +∑

j,k

Γijk(x(t))x′k(t)x

′j(t) = 0.

Example 2.46. Before going into the theory of geodesics, we collect some examples.

(i) Let M ⊆ (Rn, can) be a regular submanifold with the induced Riemannianmetric. Then a smooth curve c : I →M ⊆ Rn, defined on an open intervalI ⊆ R is a geodesic in M if and only if the acceleration vector c′′(t) isorthogonal to Tc(t)M for all t ∈ I: Indeed, if ∇ is the Levi-Civita connectionon M and ∇ is the one on Rn we know that c′′(t) = Dc′/dt and Dc′/dt =(Dc′/dx)⊥c(x) .

(ii) Geodesics in (Rn, can) are of the form c(t) = tv + w for some v, w ∈ Rn asin this case (GLC) simply says x′′i (t) = 0.

(iii) Consider Sn ⊆ Rn+1 with the induced Riemannian metric. A great circleis determined by a pair u, v ∈ Sn of orthogonal vectors via

c(θ) = cos(θ)u + sin(θ)v.

Then c′(θ) = − sin(θ)u+ cos(θ)v and c′′(θ) = − cos(θ)u− sin(θ)v = −c(θ).As a result, Dc′/dt is orthgonal to Tc(θ)Sn and hence c is a geodesic of Sn.

The local existence and uniqueness of geodesics follows from the following well-known theorem.

Theorem 2.47. Let Ω ⊆ Rn be open and let F : Ω × Rn → Rn be a smooth map.Then for every (s0, v0) ∈ Ω × Rn there is a neighbourhood E × V ⊆ Ω × Rn of(s0, v0) and δ > 0 such that for every (s, v) ∈ E×V there is a unique smooth curvexs,v : (−δ, δ) → Ω satisfying

(i) x′′(t) = F (x(t), x′(t)),(ii) x(0) = s and x′(0) = v.

Furthermore, the map E × V × (−δ, δ) → Ω, (s, v, t) 7→ xs,v(t) is smooth.


We now apply Theorem 2.47 to our situation.

Corollary 2.48. Let (M, g) be a Riemannian manifold with Levi-Civita connection∇. Then every p0 ∈ M admits a neighbourhood U ⊆ M , δ > 0 and ε > 0 suchthat for every p ∈ U and v ∈ TpM with |v| < ε there is a unique geodesic c(p,v) :(−δ, δ) → M such that c(p,v)(0) = p and c′(p,v)(0) = v. Moreover, the map

C : (p, v) ∈ TU | |v| < ε × (−δ, δ) →M, ((p, v), t) → c(p,v)(t)

is smooth.

The following homogeneity lemma often occurs underrated in the literature. Itcritically depends on the form of (GLC).

Lemma 2.49. Retain the notation of Corollary 2.48. Assume that the geodesicc(p,v) is defined on (−δ, δ) and let a > 0. Then c(p,av) is defined on (−δ/a, δ/a) andc(p,av)(t) = c(p,v)(at).

Proof. To simplify notation, we put η(t) = c(p,v)(at) and γ(t) = c(p,v)(t). Observethat η(0) = p and η′(0) = av. We show that η is a geodesic, i.e. Dη′/dt = 0 wherethe covariant derivative is taken along η. Taking a local chart (U,ϕ) and setting

ϕ(γ(t)) + (x1(t), . . . , xm(t)) and ϕ(η(t)) = (y1(t), . . . , ym(t))

We know that yi(t) = xi(at). Therefore:

y′′i (t) =∑

j,k

Γijk(η(t))y′k(t)y

′j(t) = a2x′′i (at) +

∑

j,k

Γijk(γ(at))ax′k(at)ax

′j(at)

= a2

x′′i (at) +

∑

j,k

Γijk(γ(at))x′k(at)x

′j(at)

= 0

which proves the assertion.

Corollary 2.50. Retain the above notation. Then every p0 ∈ M admits a neigh-bourhood U ⊆M and ε > 0 such that for all p ∈ U and v ∈ TpU with |v| < ε thereis a unique geodesic c(p,v) : (−2, 2) →M with c(p,v)(0) = p and c′(p,v)(0) = v.

Proof. This follows from the homogeneity Lemma 2.49: If c(p,v) is defined on (−δ, δ)then ε(p,δv/2) is defined on (−2, 2).

In the context of Corollary 2.50, let

Ω := (p, v) ∈ TM | c(p,v) is defined on (−2, 2).We have seen that Ω is a neighbourhood of the trivial section of TM .

Definition 2.51. Let (M, g) be a Riemannian manifold with Levi-Civita connection∇ and retain the above notation. The exponential map is

exp : Ω →M, exp((p, v)) = c(p,v)(1).

We denote its restriction to Ω ∩ TpM by expp.

As a consequence of Corollary 2.50, the exponential map is smooth.

Proposition 2.52. Retain the above and set φ : Ω → M ×M , (p, v) 7→ (p, expp v).Then φ is smooth and for every (p0, 0) ∈ Ω the map φ is a diffeomorphism from aneighbourhood of (p0, 0) to its image in M ×M .


Proof. The map φ is smooth because its components are. For the second state-ment we employ the inverse function theorem for which end we have to computeD(p0,0)φ: Let (U,ϕ) be a chart of M at p0 ∈ M and recall that it yields a basis(∂1(p), . . . , ∂m(p)) of TpU for all p ∈ U . We will also use the diffeomorphism

U × Rm → TU, (p, y) 7→(p,

m∑

i=1

yi∂i(p)

)

Slightly pedantic, we put ψ(p, y) := φ(p, v) where v =∑m

i=1 yi∂i(p). Finally, weidentify T(p,0)(U × Rm) with TpU ⊕ Rm where we take

(∂1(p), 0), . . . , (∂m(p), 0), (0, ∂1y), . . . , (0, ∂my)

with ∂iy = ∂/∂yi|y=0 as basis of the right hand space. Now, let’s consider thevariation of ψ in the direction of U , keeping the other fixed: We have

ψ(c(p,v)(t), 0) = (c(p,v)(t), c(p,v)(t))

and thereforeD(p,0)ψ(v, 0) = (v, v) ∈ TpM⊕TpM . For the other variation, considerthe smooth path t 7→ (p, ty) in U × Rm with tangent vector (0, y) ∈ TpM ⊕ Rm.Then

ψ(p, ty) = (p, c(p,tv)(1)) = (p, c(p,v)(t))

where v =∑m

i=1 yi∂i(p). Thus D(p,0)ψ(0, y) = (0, v). Identifying TpM ×TpM withRm⊕Rm using the basis

((∂1(p), 0), . . . , (∂m(p), 0), (0, ∂1(p)), . . . , (0, ∂m(p)))

the matrix of D(p,0)ψ is

D(p,0)ψ =

(Id 0Id Id

).

Corollary 2.53. Retain the above notation. For every p0 ∈ M there is an openneighbourhood U of p0 in M and ε > 0 such that the following hold.

(i) For every (x, y) ∈ U × U there is a unique v ∈ TxM with |v| < ε andexpx(v) = y.

In this case, let c(x, y, t) := expx(tv).

(ii) The map U ×U × (−2, 2) →M, (x, y, t) 7→ c(x, y, t) is defined and smooth.(iii) For every x ∈ U the map expx : B(0, ε) → M is a diffeomorphism onto its

image.

Proof. Retaining the above notation, let W ⊆ Ω be an open neighbourhood of(p0, 0) ∈ Ω such that φ : W → φ(W ) is a diffeomorphism. It is easy to see fromlocal triviality of TM that there exists an open neighbourhood V of p0 ∈ M andε > 0 such that T<εV := (p, v) ∈ TM | p ∈ V, |v| < ε ⊆ W . In particular,φ(T<εV ) is an open neighbourhood of (p0, p0) = φ(p0, 0) ⊆M ×M . Hence there isan open neighbourhood U of p0 ∈M with U ×U ⊆ φ(T<εV ). In particular U ⊆ V .This pair (U, ε) satisfies the conclusions.

The infinitesimal definition of geodesics has the disadvantage that its relation todistance-minimising curves between points remains unclear at the moment. On theplus side, it enables one to prove the following which certainly would not hold ifgeodesics had been defined as curves of shortest lengths between points: Think ofthe projection of a straight line onto the torus.

Corollary 2.54. Let (N, h) and (M, g) be Riemannian manifolds. Furthermore, letp : (N, h) → (M, g) be a Riemannian covering. Given a smooth curve c : I → M ,let c : I → N be any lift of c, i.e. p c = c. Then c is a geodesic if and only if c is.


For instance, Corollary 2.54 can be used to understand the geodesics of Bieber-bach manifolds.

2.4. Bi-invariant Metrics on Lie Groups. In this section, we illustrate manyof the notions introduced so far for Lie groups G that admit a bi-invariant metric.Recall that this means that all left- and right-translations of G are isometries. Inturn, this implies that 〈−,−〉e is invariant under Ad(G). Conversely, recall that anAd(G)-invariant scalar product on TeG gives rise to a bi-invariant metric via left-and right-translation.

Example 2.55. We recall that any compact Lie group admits a bi-invariant metric.Thus all orthogonal groups do. For instance, consider G = O(n) ⊆ GL(n,R) with

Lie(O(n)) = TIdG = X ∈Mn,n(R) | X +XT = 0.

As for all matrix Lie groups, we have Ad(g)X = gXg−1. Given X,Y ∈ TeG =Lie(G), set 〈X,Y 〉 = − tr(XY ). This is a bilinear, symmetric form. In addition, wehave

〈X, 〉X = − tr(XX) = tr((−X)X) = tr(XTX) =∑

i,j

x2ij ≥ 0

which proves positive-definiteness. Furthermore,

〈Ad(g)X,Ad(g)Y 〉 = − tr(gXg−1gY g−1) = − tr(gXY g−1) = − tr(XY ) = 〈X,Y 〉.

Hence 〈−,−〉e determines a bi-invariant metric on O(n).

We now show that for the class of Lie groups under consideration, the Rie-mannian and the Lie group exponential maps coincide which yields a very explicitunderstanding of all geodesics, particularly for matrix Lie groups.

Theorem 2.56. Let G be a Lie group admitting a bi-invariant Riemannian metric.Then the Riemannian exponential is defined on the whole of TG and its restrictionexpe : TeG→ G coincides with the Lie group exponential.

Our proof of Theorem 2.56 relies on the following two lemmas.

Lemma 2.57. Let G be a Lie group admitting a bi-invariant Riemannian metric.Then i : G→ G, g 7→ g−1 is an isometry.

Proof. We write the relation gi(g) = e in the following way: The map m ∆ isconstant and equal to e ∈ G where ∆ : G → G × G, g 7→ (g, i(g)). In particular,we have for all h ∈ G and v ∈ ThG: Dh(m ∆)(v) = 0. Note that Dh(m ∆) =D∆(h)m Dh∆ and

D(h1,h2)m(v1, v2) = Dh1Rh2(v1) +Dh2Lh1(v2).

Indeed, consider for instance m(cv1(t), h2) = cv1(t)h2. Also, Dh∆(v) = (v,∆hi(v)).Hence 0 = D(h,h−1)m(v,Dhi(v)) = DhRh−1(v) +Dh−1Lh(Dhi(v)) whence

Dh−1Lh(Dhi(v)) = −DhRh−1(v)

which proves the assertion. More neatly, we record Dhi = −DeLh−1 DhRh−1 . Inparticular, Dei = −Id.

Lemma 2.58. Retain the above notation. For every v ∈ TeG the maximal in-terval of definition of the geodesic c(e,v) is the whole of R, and c(e,v)(t1 + t2) =c(e,v)(t1)c(e,v)(t2) for all t1, t2 ∈ R.


Proof. Fix v ∈ TeG and let (a, b) containing 0 ∈ R be the maximal interval ofdefinition of c(e,v). Since i is an isometry with Dei = −Id we have that i(c(e,v)) isa geodesic with velocity −v at t = 0. Hence we have i(c(e,v)) = c(e,−v) by unique-ness which implies that, whenever defined, c(e,v)(t)−1 = i(c(e,v)(t)) = c(e,−v)(t) =c(e,v)(t). Thus a = −b. Now let ε > 0 be such that expe : B(0, ε) → G is a diffeo-morphism onto its image and consider for 0 < t0 < ε: Γ(s) = c(t0)c(s) = Lc(t0)c(s).By left-invariance of the metric, Γ(s) is a geodesic which is defined on (−b, b). Notethat Γ(−t0) = c(t0)c(−t0) = e and Γ(0) = c(t0) and observe that s 7→ c(t0 + s) is ageodesic which at s = −t0 takes the value e ∈ G and at s = 0 the value c(t0). ByCorollary 2.53 we deduce that Γ(s) = c(s + t0). Since Γ is defined on (−b, b), thiswould extend the domain of definition of c beyond b if b was finite since t0 + b > b.This implies b = ∞, i.e. c(e,v) is defined on the whole of R and

c(e,v)(t0)c(e,v)(s) = c(e,v)(t0 + s) for all s ∈ R, |t0| < ε.

Now let t be arbitrary and take n ∈ N \0 such that |t|/n < ε. Then

c(e,v)(t+ s) = c(e,v)

(t

n+

(n− 1)t

n+ s

)= c(e,v)

(t

n

)c(e,v)

((n− 1)t

n+ s

)

= c(e,v)

(t

n

)nc(s).

In particular, for s = 0 we obtain c(e,v)(t) = c(e,v)(t/n)n which plugged into the

above implies c(e,v)(t+ s) = c(t)c(s).

The proof of Theorem 2.56 is now no longer difficult.

Proof. (Theorem 2.56). Let v ∈ TeG and let Xv be the left-invariant vector fielddetermined by v. Furthermore, let c(e,v) be the geodesic determined by v. We needto show that c(e,v) is an integral curve for Xv: Write c(e,v)(s)c(e,v)(t) = c(s+ t), i.e.Lc(e,v)(s)(c(e,v)(t)) = c(e,v)(s+ t). Taking the derivative with respect to t yields

Dc(e,v)Lc(e,v)(s)(c′

(e,v)(t)) = c′(e,v)(s+ t).

Evaluating at t = 0 proves indeed Xv(c(s)) = DeLc(e,v)(s)(c′

(e,v)(0)) = c′(s).

Example 2.59. Returning to our example O(n), the Riemannian exponential mapis given by Lie(O(n)) → O(n), X 7→∑∞

n=0Xn/n!. In particular,

c(Id,X)(t) =

∞∑

n=0

tnXn

n!.

Therefore, length(c(Id,X)[0, T ]) = T ‖X‖. The matrix exponential can be computedvia diagonalization and conjugation. In the present example, one can for instanceuse this to show that the exponential map is surjective and to determine periodicgeodesics and their length.

Corollary 2.60. Let G be a Lie group admitting a bi-invariant Riemannian metric.Let ∇ denote its Levi-Civita connection and let X,Y be left-invariant vector fieldson G. Then ∇XY = [X,Y ]/2.

Proof. Let Z be any left-invariant vector field and c : R → G any integral curve ofZ. If c(0) = e then c is a geodesic. If c(0) 6= e it is nevertheless a geodesic by left-invariance of Z, thus Dc′/dt = 0. Hence ∇c′(t)Z = 0 since c′(t) = Z(c(t)). For thesame reason,∇ZZ = 0. In particular ∇X+Y (X+Y ) = 0 which by expanding implies0 = ∇XY +∇YX = 0. Combining this with the fact that ∇XY −∇YX = [X,Y ]yields the assertion.

Later on, we will also examine the Riemannian curvature tensor in this settingand use it to derive global properties, e.g. information on the fundamental group.


2.5. Geodesics and Distance. In this section, we characterize geodesics as beinglocally length minimizing. The example of the sphere and the torus show that ingeneral geodesics are not globally length-minimizing.

Now, let (M, g) be a Riemannian manifold and fix p0 ∈M . An open neighbour-hood U of p0 ∈ M for which there is ε > 0 satisfying Corollary 2.53 is totally

normal. Recall that in this case we have for all p ∈ U : The exponential map

expp : B(0, ε) ⊆ TpM → expp(B(0, ε))

is a diffeomorphism onto its image, which contains U . In this context, we now provethe following.

Theorem 2.61. Retain the above notation. In particular, let U be a totally normalneighbourhood of p0 ∈M . Then

(i) for all p, q ∈ U there is a unique geodesic c of length l(c) < ε connecting pto q, and

(ii) for any piecewise C1-curve γ : [0, 1] → M with γ(0) = p and γ(1) = q wehave l(γ) ≥ l(c), with equality if and only if γ equals c up to reparametri-sation.

The main ingredient in the proof of Theorem 2.61 is a lemma of Gauss whichexpresses the Riemannian metric in polar coordinates. As preparation, considerp ∈ M and ε > 0 such that expp : B := B(0, ε) → M is a diffeomorphism ontoits image B′. Further, let Sn−1 := v ∈ TpM | ‖v‖ = 1. We introduce polarcoordinates on B\0 via

B\0expp

// B′\p

(0, ε)× Sn−1

OO

f

88qqqqqqqqqq

where f(r, v) := expp(rv).

Lemma 2.62 (Gauss). Retain the above notation. The decomposition

T(r,v)((0, ε)× Sn−1) = Tr(0,∞)⊕ TvSn−1

is orthogonal with respect to f∗(g)(r,v). In these coordinates,

f∗(g)(r,v) = dr2 + h(r,v)

where h(r,v) is a scalar product on TvSn−1 depending on r.

Proof. First of all, observe that the restriction to Tr(0,∞) of f∗(g)(r,v) equals dr2.Indeed, the map

cv : (0, ε) → B′, r 7→ expp(rv) = f(r, v)

is a geodesic parametrized by arc length. To show the asserted orthogonality, weconsider certain vector fields: Let X be a smooth vector field on Sn−1, considered as


a map Sn−1 → TpM , v 7→ X(v) satisfying 〈v,X(v)〉 = 0 for all v ∈ Sn−1. It inducesthe vector field X(r, v) := rX(r, v) on B\0 via the polar coordinatization.

b

v

X(v)

rX(v)

Furthermore, we let Y := expp,∗(X) be the direct image of X via the diffeomorphismf , that is

Y (f(r, v)) = D(r,v)f(X(r, v)) = Drv expp(rX(v)).

Now, let ∂/∂r denote the radial vector field on (0, ε× Sn−1.Then the lemma follows if we show that f∗(∂/∂r) and Y are orthogonal at every

point f(r, v). To this end, observe that

f∗

(∂

∂r

)(f(r, v)) = f∗

(∂

∂r

)(cv(r)) = c′v(r) ∈ Tf(r,v)M.

We therefore have using Proposition 2.42 and the properties of the Levi-Civitaconnection

d

drg

(Y (f(r, v)), f∗

(∂

∂r

)(f(r, v))

)=

d

drg(Y (cv(r)), c

′v(r))

= g(∇c′vY, c′v) + g(Y,∇c′vc

′v︸︷︷︸

=0

)

= g(∇Y c′v, c

′v) + g([c′v, Y ], c′v)

=1

2Y g(c′v, c

′v) + g([c′v, Y ], c′v).

Since cv is a geodesic, g(c′v, c′v)cv(t) is constant in t with initial value g(v, v) = 1 at

t = 0. Thus the map (r, v) 7→ g(c′v, c′v)cv(r) is constant and hence Y g(c′v, c

′v) = 0. It

therefore remains to evaluate [c′v, Y ]: We have

[c′v, Y ] = f∗

([∂

∂r,X

])

A simple computation yields [∂/∂r,X ] = 0. Overall, we conclude

d

drg

(Y, f∗

(∂

∂r

))= 0.

Since Y (0) = 0 this concludes the lemma.

We now turn to Theorem 2.61

Proof. (Theorem 2.61). Regarding the first assertion, we already know that thereis a unique v ∈ TpM with ‖v‖ < ε and cv(1) = q. Clearly, l(cv([0, 1])) = ‖v‖ <ε. Conversely, assume there is w ∈ TpM and t > 0 such that cw(t) = q andl(cw([0, t])) < ε. Since l(cw([0, t])) = t‖w‖, setting u := tw yields

cu(1) = ctw(1) = cw(t) = q

by homogeneity of geodesics. And since ‖u‖ = t‖w‖ < ε we conclude u = v byuniqueness of geodesics.

For the second assertion, assume that γ : [0, 1] →M is a piecewise C1-curve withγ(0) = p and γ(1) = q. Then either γ([0, 1]) is contained in B′ or there is s ∈ (0, 1)


such that γ[0, s) ⊆ B′ and γ(s) 6∈ B′. For 0 ≤ t ≤ s, write γ(t) = f(r(t), v(t)). ThenLemma 2.62 implies

l(γ) >

∫ s

0

√gγ(t)(γ′(t), γ′(t)) dt

≥∫ s

0

|r′(t)| dt ≥∣∣∣∣∫ s

0

r′(t) dt

∣∣∣∣ = ε.

Therefore, l(γ)) > ε > l(c). Suppose now that γ([0, 1]) is contained in B′. Again,using polar coordinates, we have

l(γ) =

∫ 1

0

√[r′(t)2 + h(r(t),v(t))(v′(t), v′(t))] dt ≥ r(1)− r(0) = l(c).

Moreover, equality occurs if and only if h(r(t),v(t))(v′(t), v′(t)) = 0 for all 0 ≤ t ≤ 1,i.e. v′(t) = 0 whence v(t) = v for all t ∈ [0, 1], and r is monotone. Assuming thatγ is parametrized proportional to arc length with velocity l, we get γ(t) = f(lt, v)and γ(t/l) = cv(t).

Corollary 2.63. Retain the above notation. A piecewise C1-curve γ : I → Mparametrized proportional to arc length is a geodesic if and only if for all t ∈ Ithere is ε > 0 such that d(c(t − ε), c(t+ ε)) = l(c([t− ε, t+ ε])).

Proof. This is due to Theorem 2.61 if γ is C1 and is left as an exercise otherwise.

Remark 2.64. As mentioned in the introduction, the examples of the sphere andthe torus show that in general geodesics are not globally length minimizing.

Definition 2.65. Retain the above notation. A geodesic c : [a, b] →M is minimal ifl(c([a, b])) = d(c(a), c(b)).

Proposition 2.66. Retain the above notation. Let c : [a, b] → M be piecewise C1.If l(c) = d(c(a), c(b)) and c is parametrized proportional to arc length then c is ageodesic.

Proof. Observe that if a ≤ a′ ≤ b′ ≤ b then l(c([a′, b′])) = d(c(a′), c(b′)). Indeed,otherwise l(c([a′, b′])) > d(c(a′), c(b′)) and there is by definition a piecewise C1-curve γ : [a′, b′] → M with l(γ) < l(c([a′, b′])), γ(a′) = c(a′) and γ(b′) = c(b′). Butthen we have for the concatenation η of c|[a,a′], γ and c|[b′,b]:

l(η) = l(c([a, a′])) + l(γ) + l(c([b′, b]))

= l(c)− l(c([a′, b′])) + l(γ)

< l(c) = d(c(a), c(b)).

Combining this observation with Corollary 2.63 implies the proposition.

Normal Coordinates. As before, let (M, g) be a Riemannian manifold with Levi-Civita connection ∇. Recall that in a local coordinate system (U,ϕ) of M we havedefined the Christoffel symbols Γijk by

∇∂j∂k =∑

i

Γij,k∂i.

Since ∇∂j∂k − ∇∂k∂j = [∂j , ∂k] = 0 we get ∇∂k∂j =∑i Γ

ikj∂i =

∑i Γ

ijk∂i and

hence Γikj = Γijk.Now let p ∈M and ε > 0 such that expp : B(0, ε) → U ⊆M is a diffeomorphism

onto an open neighbourhood U of p. Choosing an orthonormal basis e1, . . . , em of(TpM, gp) and setting ϕ := exp−1

p : U → TpM , we obtain local coordinates atp ∈M . In these coordinates we have

(i) gp(∂i(p), ∂j(p)) = δij , and


(ii) ∇∂i∂j(p) = 0

Indeed, for (i) note that by definition D0 expp(ei) = ∂i(p) and since D0 expp =Id we get gp(∂i(p), ∂j(p)) = gp(ei, ej) = δij . For the second assertion, let v ∈TpM and cv : I → M , t 7→ cv(t) the corresponding geodesics. Then by definitionexp−1

p (cv(t)) = tv, i.e. ϕ(cv(t)) = (x1(t), . . . , xm(t)) with xi(t) = tvi. Since cv is ageodesic we have for all i ∈ 1, . . . ,m:

x′′i (t) +∑

j,k

Γikj(cv(t))x′k(t)x

′j(t) = 0.

Evaluating at t = 0 yields ∑

k,j

Γikj(p)vkvj = 0.

Since Γikj is symmetric in (k, j) and the above holds for any vector v ∈ B(0, ε) weconclude Γikj(p) = 0.

2.6. The Hopf-Rinow Theorem. Let (M, g) be a connected Riemannian mani-fold. We have seen that one can define a distance on M by setting

d(p, q) := infl(c) | c : [0, 1] →M is piecewise C1, c(0) = p, c(1) = q

for all p, q ∈M . In addition, d induces the given topology on M .

Definition 2.67. A Riemannian manifold (M, g) is geodesically complete if for everyp ∈M , the expoential map expp is defined on the whole of TpM .

That is, in the case of geodesically complete Riemannian manifold M , given(p, v) ∈ TM the geodesic c(p,v)(t) is defined for all t ∈ R. In this section, we provethe following fundamental theorem, linking the properties of (M,d) as a metricspace with geodesic completeness.

Theorem 2.68 (Hopf-Rinow). Let (M, g) be a Riemannian manifold, p ∈ M and dthe metric induced by g. Then the following statements are equivalent.

(i) The map expp is defined on the whole of TpM .(ii) Closed and bounded sets in M are compact.(iii) The metric space (M,d) is complete.(iv) The Riemannian manifold (M, g) is geodesically complete.

In addition, any of the above statements implies the following.

(v) For any q ∈M there is a minimal geodesic joining p to q.

Proof. The main point of the proof lies in the implication (i)⇒(v). For this, wefirst need to find a candidate for the tangent vector v ∈ TpM giving the minimalgeodesic. To this end, let δ > 0 be such that B(p, δ) is a normal ball, that is, theexponential map expp : B(0, δ) ⊆ TpM → M is a diffeomorphism onto its image.Let S := exp(S(0, δ)) be the image of the boundary of B(0, δ) and let x0 ∈ S bea point such that d(x0, q) = mind(x, q) | x ∈ S which exists by compactness ofS. Now, let v ∈ TpM be the unique vector with ‖v‖ = 1 and expp(δv) = x0, andlet γ : R → M , s 7→ expp(sv) be the corresponding geodesic which by assumptionis defined on the whole of R. We proceed to show that γ(r) = q for r := d(p, q):Consider the set A := s ∈ [0, r] | d(γ(s), q) = r − s. Then A is non-empty since0 ∈ A and closed by continuity. Now, let s0 ∈ A with s0 < r and δ′ > 0 such that

(i) s0 + δ′ ≤ r, and(ii) B(γ(s0), δ

′) is a normal ball.


In this setting, we show that s0 + δ′ ∈ A. Since A is closed, this will imply thatr ∈ A which concludes this part of the proof. Consider the following figure.

bp

b qb

γ(s0)

δ′ bx′

0

γ

Let x′0 ∈ S′ := expγ(s0)(B(0, δ′)) be such that d(x′0, q) = mind(x, q) | x ∈ S′.We claim that x′0 = γ(s0 + δ′): Let v′ ∈ Tγ(s0)M with ‖v′‖ = 1 be such thatexpγ(s0) δ

′v′ = x′0. The concatenation η of the geodesic γ([0, s0]) with γ′([0, δ′])

where γ′(t) = expγ(s0)(tv′) for 0 ≤ t ≤ δ′ is a piecewise C1-curve joining p to x′0

and has length l(η) = s0 + δ′. On the other hand, we have

(1) d(p, x′0) ≥ d(p, q)− d(q, x′0)

and

(2) d(γ(s0), q) = δ′ +mind(x, q) | x ∈ S′ = δ′ + d(x′0, q).

Equation (2) implies

(3) r − s0 = δ′ + d(x′0, q).

Substituting (3) back into (1) yields

d(p, x′0) ≥ r − (r − s0 − δ′) = s0 + δ′.

Since l(η) = s0 + δ′ we conclude that d(p, x′0) = s0 + δ′ and that η is a minimalgeodesic joining p to x′0. Given that η′(0) = γ′(0) we conclude that η(t) = γ(t) forall t ∈ [0, s0 + δ′]. In particular, γ(s0 + δ′) = x′0 which proves the claim.

We proceeed by showing how the claim implies s0 + δ′ ∈ A: Indeed, (3) nowimplies

r − s0 = δ′ + d(x′0, q) = δ′ + d(γ(s0 + δ′), q).

Equivalently, d(γ(s0 + δ′), q) = r − (s0 + δ′), i.e. s0 + δ′ ∈ A.

For the implication (i)⇒(ii), let F be a closed and bounded set. Then

supd(p, x) | x ∈ F <∞.

We may hence choose T > 0 with d(p, x) ≤ T for all x ∈ F . By (v), we haveexpp(B(0, T )) ⊇ F . That is, F is a closed subset of a compact set and hencecompact itself.

The implication (ii)⇒(iii) is a general topology statement.

To prove that (iii) implies (iv), assume that (M, g) is not geodesically complete.Then there is q ∈ M and v ∈ TqM with ‖v‖ = 1 such that γ(s) := expq(sv) isdefined on [0, s0) but not for s = s0. Pick a sequence (sn)n∈N with sn < s0 forall n ∈ N such that limn→∞ sn = s0. Then d(γ(sn), γ(sm)) ≤ |sn − sm| and hence(γ(sn))n∈N is a Cauchy sequence in M , which by assumption converges to a point,say p0 = limn→∞ γ(sn). Let (W, δ) be a totally normal neighbourhood of p0. Thatis, for all p ∈W , the map

expp : B(0, δ) → expp(B(0, δ))

is a diffeomorphism and expp(B(0, δ)) ⊇ W . For large enough n,m, the imagespoints γ(sn) and γ(sm) lie in W and |sn− sm| < δ. Now there is a unique geodesic


g joining γ(sn) to γ(sm) of length strictly less than δ. Clearly, γ coincides with gwhenever defined. Now expγ(sn) : B(0, δ) → M is a diffeomorphism onto its imagewhich contains W . Hence g extends γ beyond s0.

Corollary 2.69. Every compact connected Riemannian manifold ist geodesicallycomplete.

Corollary 2.70. Let G be a compact connected Lie group. Then the Lie groupexponential expG : Lie(G) → G is surjective.

Corollary 2.71. Every closed submanifold of a complete connected Riemannianmanifold is complete.

3. Curvature

In this section we finally discuss various locally defined notions of curvatureand their global implications. The first was introduced by Gauss for surfaces in R3

and later on generalized by Riemann to sectional curvature: Given a Riemannianmanifold (M, g) and a point p ∈M , look at two-dimensional subspaces E of TpMand apply Gauss’ notion of curvature to the surface pieces that arise from lookingat small geodesics with initial velocity in E. This yields a local notion of curvatureon M depending on the choice of a tangent plane.

There is a notion of curvature which subsumes all the above and many oth-ers, called Riemannian curvature. Although not invented by Riemann himself it isdetermined by all the sectional curvatures above.

3.1. Definition and Formal Properties. Given a Riemannian manifold (M, g)with Levi-Civita connection ∇, consider the map

R(X,Y ) : Γ(TM) → Γ(TM), Z 7→ ∇Y∇XZ −∇X∇Y Z +∇[X,Y ]Z

One way to at it is to study the extent to which the map Γ(TM) → End(Γ(TM))which maps X to ∇X fails to be a Lie algebra homomorphism.

Proposition 3.1. Retain the above notation. The map

Γ(TM)× Γ(TM)× Γ(TM) → Γ(TM), (X,Y, Z) 7→ R(X,Y )Z

is a tri-linear map of C∞(M)-modules.

Proof. As prepration for the proof, we compute for f, ϕ ∈ C∞(M):

[fX, Y ](ϕ) = (fX)(Y (ϕ))− Y (fX)(ϕ)

= fX(Y (ϕ)) − Y (f)Y (ϕ) − fY X(ϕ) = (f [X,Y ]− Y (f)X)(ϕ).

Therefore, [fX, Y ] = f [X,Y ] − Y (f)X and [X, gY ] = g[X,Y ] + X(g)Y . Now, wetreat the C∞(M)-linearity of the given map in its three slots individually. For thefirst one, we have

R(fX, Y )Z = ∇Y∇fXZ −∇fX∇Y Z +∇[fX,Y ]Z

= ∇Y (f∇XZ)− f∇X∇Y Z + f∇[X,Y ]Z − Y (f)∇XZ

= Y (f)∇XZ + f(∇Y∇XZ −∇X∇Y Z +∇[X,Y ]Z)− Y (f)∇XZ

= fR(X,Y )Z.

A similar computation works in the case of the second variable. Concerning theZ-variable, first compute

∇Y∇X(fZ) = ∇Y (X(f)Z + f∇XZ)

= Y (X(f))Z +X(f)∇Y Z + Y (f)∇XZ + f∇Y∇XZ


and similarly ∇X∇Y (fZ). The difference of the two equals

Y (X(f))Z + f∇Y∇XZ −X(Y (f))Z − f∇X∇Y Z

= (Y (X(f))−X(Y (f)))Z + f(∇Y∇XZ −∇X∇Y Z).

On the other hand, we have

∇[X,Y ]fZ = [X,Y ](f)Z + f∇[X,Y ]Z = (X(Y (f))− Y (X(f)))Z + f∇[X,Y ]Z.

Overall, this completes the proof.

Now, let (U,ϕ) be a chart of M at p ∈ M and let (∂i)mi=1 be the associated

coordinate vector fields on U . Then

R(∂i, ∂j)∂k =∑

l

Rlijk∂l

where Rlijk : U → R are smooth functions on U . Decomposing X =∑

iXi∂i,Y =

∑j Yj∂j and Z =

∑k Zk∂k on U we get

R(X,Y )Z = R

∑

i

Xi∂i,∑

j

Yj∂j

(∑

k

Zk∂k

)

=∑

i,j,k

XiYjZkR(∂i, ∂j)∂k

=∑

i,j,k,l

XiYjZkRlijk∂l.

As a corollary we record that R(X,Y )Z ∈ Γ(TM) is a tensor on M in thefollowing sense.

Corollary 3.2. Retain the above notation. The value of R(X,Y )Z(p) only dependson X(p), Y (p) and Z(p).

Definition 3.3. Let (M, g) be a Riemannian manifold with Levi-Civita connection∇. The Riemannian curvature tensor of (M, g) is the collection of trilinear maps

Rp : TpM × TpM × TpM → TpM, (v, w, u) 7→ R(X,Y )Z(p) (p ∈M)

where X,Y, Z are vector fields defined on open neighbourhoods of p ∈ M withX(p) = v), Y (p) = w and Z(p) = u.

Equivalently, one can view the Riemannian curvature as a collection Rp(v, w) ∈End(TpM) of endomorphisms, v, w ∈ TpM , p ∈M .

All curvature notions that we discuss are descendents of the Riemannian cur-vature tensor but may have a more explicit geometric interpretation. In order towork with these, we establish certain symmetry properties of R, coming from theLie algbera structure on Γ(TM) as well as the relation between the Levi-Civitaconnection and the metric.

Proposition 3.4 (Bianchi). Retain the above notation, let X,Y, Z ∈ Γ(TM). Then

R(X,Y )Z +R(Y, Z)X +R(Z,X)Y = 0.

Proof. Expanding the definition, the proposition amounts to proving

0 = ∇Y∇XZ −∇X∇Y Z +∇[X,Y ]Z

+∇Z∇YX −∇Y∇ZX +∇[Y,Z]X

+∇X∇ZY −∇Z∇XY +∇[Z,X]Y.


Pairing appropriate expressions, this retranslates to

0 = ∇Y (∇XZ −∇ZX) +∇X(∇ZY −∇Y Z) +∇Z(∇YX −∇XY )

+∇[X,Y ]Z +∇[Y,Z]X +∇[Z,Y ]Y

Using relations like ∇XZ−∇ZX = [X,Z] the above reduces to the Jacobi identity.

For the following, consider X,Y, Z, T ∈ Γ(TM) on M and define for p ∈M :

(X,Y, Z, T )p := 〈R(X,Y )Z(p), T (p)〉TpM .This way, we hope to deduce further properties of R taking into account the fact∇ is not just a connection but behaves nicely with respect to the metric.

Proposition 3.5. Retain the above notation. Then the following hold.

(i) (X,Y, Z, T ) + (Y, Z,X, T ) + (Z,X, Y, T ) = 0.(ii) (X,Y, Z, T ) = −(Y,X,Z, T ).(iii) (X,Y, Z, T ) = −(X,Y, T, Z).(iv) (X,Y, Z, T ) = (Z, T,X, Y ).

A way to remember Proposition 3.5 is to note the simple behaviour of the form(−,−,−,−) under the action of the subgroup of S4 which preserves the blockdecomposition 1, 2, 3, 4 of 1, 2, 3, 4 depicted by (ii), (iii) and (iv). If one isconcerned with oher permutations one has to use Bianchi’s equality (i).

Proof. (Proposition 3.5). As remarked above, (i) is merely a reformulation of Bianchi’sidentity. The second assertion is an immediate consequnce of the definition. For (iii),note that by doubling variables, it suffices to show that (X,Y, Z, Z) = 0. We aretherefore looking at 〈∇Y∇XZ − ∇X∇Y Z + ∇[X,Y ]Z − Z〉. Using that ∇ is theLevi-Civita connection, we have

〈∇Y∇XZ,Z〉 = Y 〈∇XZ,Z〉 − 〈∇XZ,∇Y Z〉and

〈∇X∇Y Z,Z〉 = X〈∇Y Z,Z〉 − 〈∇Y Z,∇XZ〉.Taking the difference of the above equalities we obtain Y 〈∇XZ,Z〉 −X〈∇Y Z,Z〉.Observing that

〈Z,Z〉 = 〈∇XZ,Z〉+ 〈Z,∇XZ〉 = 2〈∇XZ,Z〉the above difference is

1

2Y X〈Z,Z〉 − 1

2XY 〈Z,Z〉 = −1

2[X,Y ]〈Z,Z〉 = 〈∇[X,Y ]Z,Z〉.

Hence the assertion. Finally, for (iv) compute all instances of Bianchi’s identity (i)arising from cyclic permutation of the variables and sum the result. Due to (ii) and(iii), one obtains

2(X,Z, T, Y )− 2(T, Y,X,Z) = 0

which is (iv).

If so inclined, one can make the functions Rlijk∂l occuring in the equality

R(∂i, ∂j)∂k =∑

l

Rlijk∂l

explicit in terms of the metric as in the case of the Christoffel symbols: Given that[∂i, ∂j ] = 0 for all i, j ∈ 1, . . . ,m we have R(∂i, ∂j)∂k = ∇∂j∇∂i∂k − ∇∂i∇∂j∂k.


Recall that ∇∂i∂k =∑l Γ

lik∂l. Plugging this into the above and comparing coeffi-

cients one gets the following, seldomly used, formula

Rlijk = (∂jΓlik − ∂iΓ

ljk) +

∑

s

(ΓsikΓ

ljs − ΓsjkΓ

lis

)

Going back to the formula of the Christoffel symbols in terms of the Riemannianmetric, one obtains an expression of Rlijk in terms of the same.

3.1.1. Sectional Curvature. We now turn to sectional curvature, introduced by Rie-mann, which is simpler than the Riemannian curvature tensor above but still deter-mines it. Given vectors x and y in a Euclidean space, let |x∧y| =

√|x|2|y|2 − 〈x, y〉2

denote the area of the parallelogram spanned by x and y.

Proposition 3.6. Let (M, g) be a Riemannian manifold, p ∈ M and E ≤ TpMtwo-dimensional. Given a basis (x, y) of E, the quantity

K(x, y) :=(x, y, x, y)

|x ∧ y|2does not depend on the choice of basis of E.

Definition 3.7. Retain the above notation. The sectional curvature of M at p with

respect to E is given by K(E) := K(x, y) where (x, y) is any basis of E.

Proof. We show that K is invariant under the following transformations:

(i) (x, y) 7→ (x, y),(ii) (x, y) 7→ (λx, y), λ ∈ R \0,(iii) (x, y) 7→ (x, y + µx), µ ∈ R.

This suffices since the associated matrices(1

1

),

(λ

1

)and

(1 µ

1

)

generate GL(2,R) for the given ranges of λ and µ. In particular, they realize anypossible change of basis. For the first transformation, note that

(y, x, y, x) = −(x, y, y, x) = (x, y, x, y)

and |x ∧ y|2 = |y ∧ x|2. For (ii), observe

(λx, y, λx, y) = λ2(x, y, x, y)

and |λx ∧ y|2 = λ2|x ∧ y|2. Finally, for the third assertion, compute

(x, y + µx, x, y + µx) = (x, y, x, y) + (x, y, x, µx) + (x, µx, x, y) + (x, µx, x, µx)

= (x, y, x, y) + µ(x, y, x, x) + µ(x, x, x, y) + µ2(x, x, x, x)

= (x, y, x, y)

and |x ∧ (y + µx)|2 = |x ∧ y|2.

Remark 3.8. The Riemannian curvature tensor can be recovered from sectionalcurvature in the following way: For x, y, z, t ∈ TpM we have

∂2

∂α∂β(x+ αz, y + βt, x+ αz, y + βt) = 6(x, y, z, t)

In order to give some examples, we make the following preliminary remarks:Let (M, g) be a Riemannian manifold and f ∈ Iso(M, g). Using the uniqueness ofthe Levi-Civita connection or the formula for the same developed in the proof oneverifies that for any X,Y, Z ∈ TM we have

∇f∗Xf∗Y = f∗(∇XY )


from which we deduce R(f∗X, f∗Y )f∗Z = f∗(R(X,Y )Z) by definition of R. Hencefor x ∈ M and E ≤ TxM two-dimensional we have Kx(E) = Kf(x)(Dxf(E)). Inparticular, the map

Kx : Gr2(TxM) → R

is invariant under stabIso(M,g)(x).

Example 3.9. The above remarks facilitate the computation of sectional curvatureenormously in the following examples.

(i) The Riemannian curvature of (Rn, can) vanishes identically: Indeed, forX,Y ∈ TRn we have ∇XY = LXY if Y is considered as a map from Rn

to Rn. Consequently,

R(X,Y )Z = ∇Y∇XZ −∇X∇Y Z +∇[X,Y ]Z

= LY LXZ − LXLY Z + L[X,Y ]Z = 0

(ii) Now consider Sn ⊆ Rn+1 with the induced Riemannian metric. We showthat K : Gr2(Sn) → R is constant: Recall that O(n+1) acts transtively byisometries on Sn since it does so on Rn+1. Next,

stabO(n)(en+1) =

(A 00 1

)∣∣∣∣A ∈ O(n)

acts on Ten+1Sn = (v, 0)T ∈ Rn+1 | v ∈ Rn. As remarked above, the map

Ken+1 : Gr2(Tn+1 Rn) → R is invariant under stabO(n+1)(en+1). Observing

that the action of O(n) on Gr2(Rn) is transitive we deduce that Ken+1 is

constant. Hence so is K by transitivity of O(n+ 1).(iii) Finally, consider hyperbolic space

Hn = x ∈ Rn+1 | x21 + · · ·+ x2n − x2n+1 = −1, xn+1 > 0.We know that SO0(n, 1) = g ∈ O(n, 1) | det g = 1, g(Hn) = Hn actstransitively on Hn by Riemannian isometries. As before, we look at thebase point en+1 ∈ Hn: We have Ten+1 H

n = (v, 0)T ∈ Rn+1 | v ∈ Rn andthe stabilizer

stabSO0(n,1)(en+1) =

(A 00 1

)∣∣∣∣A ∈ SO(n)

.

The action of this stabilizer on Ten+1 Hn is equivalent to the SO(n)-action

on Rn which is again transitive on two-dimensional subspaces of Rn+1.Hence, as before, the sectional curvature of Hn is constant.

As a matter of fact, the three examples above are the only complete, simply con-nected Riemannian manifolds with constant sectional curvature.

Example 3.10. As one might expect, things turn out particularly nice in the settingof Lie groups: Let G be a Lie group which admits a bi-invariant Riemannian metric.We have seen that for X,Y ∈ Γinv(TG) we have ∇XY = 1

2 [X,Y ]. Hence we obtainfor X,Y, Z ∈ Γinv(TG):

R(X,Y )Z = ∇Y∇XZ −∇X∇Y Z +∇[X,Y ]Z

=1

4[Y, [X,Z]]− 1

4[X, [Y, Z]] +

1

2[[X,Y ], Z]

=1

4[[X,Y ], Z]

by Jacobi’s identity. Notice that since R is a tensor we do not have to worry aboutonly knowing its values on invariant vector fields: For x, y, z ∈ g = TeG we haveRe(x, y)z = 1

4 [[x, y], z].


Now suppose that x, y ∈ g are orthonormal. Then

(x, y, x, y) =

⟨1

4[[x, y], x], y

⟩

g

= −1

4〈[x, [x, y]], y]〉g.

Applying the invariance of 〈−,−〉 under the adjoint representation, i.e. for allv, w, u ∈ g we have

0 =d

dt

∣∣∣∣t=0

〈Ad(exp(tv))w,Ad(exp(tv))u〉

=

⟨0 =

d

dt

∣∣∣∣t=0

Ad(exp(tv))w, v

⟩+

⟨w,

d

dt

∣∣∣∣t=0

Ad(exp(tv))u

⟩

= 〈[v, w], u〉+ 〈w, [v, u]〉,we obtain

(x, y, x, y) =1

4〈[x, y], [x, y]〉 = 1

4‖[x, y]‖2.

In particular, the sectional curvature of any two-dimensional subspace E ⊆ g isgreater or equal to zero. Equality holds if and only if E is an abelian subalgebra.These considerations lead to root systems of Lie algebras and flat subspaces ofmanifolds.

3.1.2. Ricci Curvature and Scalar Curvature. Although sectional curvature is easierto deal with than the Riemannian curvature tensor, it is still fairly complicated giventhat it determines the latter. Two weaker and simpler curvature notions are Ricci

curvature and scalar curvature which we define in this section.Let (M, g) be a Riemannian manifold, p ∈ M and x ∈ TpM a unit tangent

vector. Pick an orthonormal basis (z1, . . . , zm) of TpM such that zm = x.

Definition 3.11. Retain the above notation. Then the Ricci curvature of M at pwith respect to x is

Ricp(x) :=1

m− 1

m−1∑

i=1

K(x, zi)

and the scalar curvature of M at p is

K(p) :=1

m

m∑

i=1

Ricp(zi)

Notice that whereas sectional curvature depends on a point and a two-dimensionalsubspace of that point’s tangent space, Ricci curvature only requires a point and atangent vector, and scalar curvature only depends on a point. We need to show thatthe above definitions do not depend on the choice of orthonormal basis involvingthe given vector x ∈ TpM . To this end, consider the bilinear map

Q : TpM × TpM → R, (x, y) 7→ tr(z 7→ Rp(x, z)y).

Then Q is symmetric since

Q(x, y) =

m∑

i=1

〈Rp(x, zi)y, zi〉 =m∑

i=1

(x, zi, y, zi) =

m∑

i=1

(y, zi, x, zi) = Q(y, x).

Thus Q is a symmetric bilinear form and

Q(x, x) =

m∑

i=1

(x, zi, x, zi) =

m−1∑

i=1

(x, zi, x, zi) =

m−1∑

i=1

K(x, zi) = (m− 1)Ricp(x).


For scalar curvature, let K ∈ End(TpM) be the symmetric endomorphism withQ(x, y) = 〈K(x), y〉. Then

tr(K) =

m∑

i=1

〈K(zi), zi〉 =m∑

i=1

Q(zi, zi) = (m− 1)

m∑

i=1

Ricp(zi) = m(m− 1)K(p).

3.2. The first and second variation formula. We now aim to deduce globalproperties of a manifold from local curvature properties. To this end, we first needto generalize the notions "vector field along a curve" and "covariant derivative" tothe setting of "parametrized submanifolds". Let N and M be a manifolds and leth : N →M be a smooth map. In practice, N is a domain in R2. Recall that

h∗(TM) := (x, v) ∈ N × TM | v ∈ Th(x)Mis the pullback bundle of TM under h. Its sections are vector fields along h. Thefollowing two Lie algebra homomorphisms come with the above definition:

Γ(TN) → Γ(h∗TM), X 7→ X, X(x) := Dxh(X(x)), and

Γ(TM) → Γ(h∗TM), Y 7→ h∗(Y ), h∗(Y )(x) = Y (h(x)).

The following generalizes Theorem 2.37.

Proposition 3.12. Retain the above notation. There is a bilinear map

∇h : Γ(TN)× Γ(h∗(TM)) → Γ(h∗(TM))

such that for all X ∈ Γ(TN), Y ∈ Γ(h∗(TM)) and f ∈ C∞(N):

(i) ∇hfXY = f∇h

XY , and(ii) ∇h

X(fY ) = X(f)Y + f∇hXY .

Furthermore, we have for X ∈ Γ(TN) and Y ∈ Γ(TM):

(iii) ∇hXh

∗(Y ) = h∗(∇XY ).

Proof. We proceed as in the case of the covariant derivative along curves. If (U,ϕ)is a chart of M and V := h−1(U) then for Y ∈ Γ(h∗(TM)) and y ∈ V we have

Y (y) =m∑

i=1

Yi(y)h∗(∂i)(y).

Assuming that ∇h with the asserted properties exists we thus have

∇hXY (y) =

m∑

i=1

X(Yi)(y)∂i(h(y)) +m∑

i=1

Yi(y)(∇hX∂i

)(h(y))

Using(∇h

X∂i)(h(y)) = h∗ (∇X∂i) (y) =(∇Dyh(X(y))∂i

)(h(y))

we thus have

∇hXY (y) =

m∑

i=1

X(Yi)(y)∂i(h(y)) +

m∑

i=1

Yi(y)(∇Dyh(X(y))∂i

)(h(y))

which shows uniqueness. As before, the above can be used to define ∇hXY locally

from which one deduces the asserted properties.

One can now pretend that ∇h produces a curavture notion and obtain the fol-lowing.

Proposition 3.13. Let M and N be Riemannian manifolds and let h : N → M bea smooth map. Further, let X,Y ∈ Γ(TN) and U, V ∈ Γ(h∗TM). Then

(i) ∇hXY −∇h

YX = [X,Y ].(ii) X〈U, V 〉 = 〈∇h

XU, V 〉+ 〈U,∇hXV 〉.


(iii) ∇hY∇h

XU −∇hX∇h

Y U +∇h[X,Y ]U = R(X,Y )U

where R(X,Y )U(y) := Rh(y)(X(y), Y (y)

)(U(y)).

To prove e.g. part (i) of Proposition 3.13 one shows that ∇hXY −∇h

YX− [X,Y ] isC∞(N)-linear in X and Y and conludes by evaluating on coordinate vector fields.The same works for part (ii) and (iii).

3.2.1. First Variation Formula. Let (M, g) be a Riemannian manifold. Recall that

for a piecewise C1-curce c : [a, b] → M , we have defined l(c) :=∫ ba ‖c′(t)‖ dt. For

many reasons, the energy of c, defined by

E(c) :=1

2

∫ b

a

‖c′(t)‖2 dt,

is a better object to work with due the smoothness and strict convexity of thesquare of the absolute value, although it depends on the parametrization of c. Notethat by Cauchy-Schwarz we have

l(c) =

∫ b

a

1 · ‖c′(t)‖ dt ≤(∫ b

a

12 dt

)1/2(∫ b

a

‖c′(t)‖2 dt)1/2

=√b − a

√E(c).

In other words, l(c)2 ≤ (b − a)E(c) with equality if and only if ‖c′(t)‖ is constant,i.e. c being parametrized proportional to arc length.

We are interested in the minimal and critical points of thi energy functional.

Lemma 3.14. Let γ : [a, b] →M be a minimizing geodesic connecting p to q. Thenfor any piecewise C1-curve c : [a, b] →M joining p to q we have E(γ) ≤ E(c) withequality if and only if c is a minimizing geodesic.

Proof. By the above, (b − a)E(γ) = l(γ)2 ≤ l(c)2 ≤ (b − a)E(c).

Formalizing the above question, consider the map

E : Ωp,q := c : [a, b] →M | x is smooth, c(a) = p, c(b) = q → R .

Then by the above the absolute minima of E arise through the minimal geodesics.We now examime critical points which is made precise by the following.

Definition 3.15. Let M be a manifold and let c : [a, b] →M be a smooth curve. Avariation of c is a smooth map

h : [a, b]× (−ε, ε) →M

such that h(s, 0) = c(s). For t ∈ (−ε, ε), set ct : [a, b] → M , s 7→ h(s, t). Thevariation h has fixed endpoints if h(a, t) = c(a) = p and h(b, t) = c(b) = q for allt ∈ (−ε, ε).

s

t

a b

ε

-ε

h

c(a)

c(b)


In the context of Definition 3.15, smoothness of h means that it is the restrictionof a smooth map defined on an open neighbourhood of [a, b]× (−ε, ε). We let ∂/∂tand ∂/∂s denote the coordinate vector fields on R2.

Lemma 3.16. Let M be a manifold and c : [a, b] → M be a smooth curve. Further,let h : [a, b]× (−ε, ε) →M a variation of c. Then Y (s) := D(s,0)h(∂/∂t) is a vectorfield along c. Conversely, given Y ∈ Γ(c∗TM) there is a variation h of c satisfyingD(s,0)h(∂/∂t).

Proof. The first assertion is immediate. Conversely, given Y ∈ Γ(c∗TM) set

h(s, t) := expc(s) tY (s)

which by a compactness argument is defined for t ∈ (−ε, ε) for some ε > 0.

Theorem 3.17. Let (M, g) be a connected Riemannian manifold.

(i) Let c : [a, b] → M a smooth curve and let h be a variation of c. DefineY ∈ Γ(c∗TM) by Y (s) := D(s,0)h(∂/∂t). Then

d

dt

∣∣∣∣t=0

E(ct) = g(Y (s), c′(s))|ba −∫ b

a

g(Y (s),∇c′(s)c′(s)) ds.

(ii) The critical points of E : Ωp,q → R, i.e. curves c for which

d

dt

∣∣∣∣t=0

E(ct) = 0

for all variations of c with fixed endpoints, are exactly the geodesics con-necting p to q.

Proof. Given the definition of the energy functional, we compute g(c′t(s), c′t(s)) and

integrate the result over s ∈ [a, b]. LetN be an open neighbourhood of [a, b]×(−ε, ε)in R2 so that h : N →M is smooth. Then

(∂

∂s

)

(s0,t0)

= D(s0,t0)h

(∂

∂s

)= c′t0(s0).

We therefore obtain using Proposition 3.12 and 3.13:

d

dtg(c′t(s), c

′t(s)) =

d

dtg

(∂

∂s,∂

∂s

)= 2g

(∇h

∂∂t

∂

∂s,∂

∂s

).

Also, by Proposition 3.13 we have

∇h∂∂t

∂

∂s−∇h

∂∂s

∂

∂t=

[∂

∂t,∂

∂s

]= 0.

Hence the above computation continues as

= 2g

(∇h

∂∂s

∂

∂t,∂

∂s

)= 2

(d

dsg

(∂

∂t,∂

∂s

)− g

(∂

∂t,∇h

∂∂s

∂

∂s

))

Finally, note that

∂

∂t

∣∣∣∣t=0

= D(s,0)h

(∂

∂t

)= Y (s),

∂

∂s

∣∣∣∣t=0

= c′(s), and ∇h∂∂s

∂

∂s= ∇c′(s)c

′(s).

Overall, we therefore obtain

d

dt

∣∣∣∣t=0

g(c′t(s), c′t(s)) = 2

(d

dsg(Y (s), c′(s))− g(Y (s),∇c′(s)c

′(s))

)

which is the assertion after integrating with respect to s.


For the second assertion, note that by (i) we have for every variation of c withfixed endpoints:

d

dt

∣∣∣∣t=0

E(ct) = −∫ b

a

g(Y (s),∇c′c′(s)) ds

If c is a geodesic, then ∇c′c′ = 0 and hence d

dt

∣∣t=0

E(ct) = 0. Conversely, assumethat the above integral vanishes for all variations of c with fixed endpoints. Letf : [a, b] → R be a smooth function with f(a) = 0 = f(b) and f(s) > 0 for alls ∈ (a, b). Set Y (s) := f(s)∇c′c

′(s). Then

d

dt

∣∣∣∣t=0

E(ct) = −∫ b

a

f(s)‖∇c′c′(s)‖2 ds = 0

Then ∇c′c′ vanishes on (a, b) and hence also at a and b by continuity which shows

that c is a geodesic.

3.2.2. Second Variation Formula. The second variation formula concerns the sec-ond derivative of the energy functional. Retain the above notation and define inaddition: Y (s, t) := D(s,t)h(∂/∂t). Then Y is a smooth vector field along h.

Theorem 3.18. Retain the above notation. Let c : [a, b] →M be a geodesic. Then

1

2

d2

dt2

∣∣∣∣t=0

E(ct) = g(∇h

∂∂t

Y (s, 0), c′(s))∣∣∣b

a+

∫ b

a

∥∥∥∇h∂∂s

Y (s)∥∥∥2

− (Y, c′, Y, c′)(s, 0) ds

In the context of Theorem 3.18, recall that

(Y, c′, Y, c′)(s, 0) = g(Rc(s)(Y (s, 0), c′(s))Y (s, 0), c′(s)).

Proof. We determine an expression for

d2

dt2

∣∣∣∣t=0

g(c′t(s), c′t(s))

which we integrate with respect to s in order to obtain the second variation of theenergy functional. In view of Proposition 3.13 we have

∂

∂s(s, t) = D(s,t)h

(∂

∂s

)= c′t(s) and

∂

∂t(s, t) = D(s,t)h

(∂

∂t

)= Y (s, t).

To begin with, we compute

d

dtg(c′t(s), c

′t(s)) =

d

dtg

(∂

∂s,∂

∂s

)= 2g

(∇h

∂∂t

∂

∂s,∂

∂s

)= 2g

(∇h

∂∂s

∂

∂t,∂

∂s

).

Therefore we obtain

1

2

d2

dt2=

d

dtg

(∇h

∂∂s

∂

∂t,∂

∂s

)

= g

(∇h

∂∂t

∇h∂∂s

∂

∂t,∂

∂s

)

︸︷︷︸(1)

+ g

(∇h

∂∂s

∂

∂s,∇h

∂∂t

∂

∂s

)

︸︷︷︸=

∥∥∥∥∇h∂∂s

Y

∥∥∥∥2

The second term in the above sum is in its final term. We continue with the firstone using Proposition 3.13

(1) = g

(∇h

∂∂s

∇h∂∂t

∂

∂t,∂

∂s

)

︸︷︷︸(2)

+ g

(R

(∂

∂s,∂

∂t

)∂

∂t,∂

∂s

)

︸︷︷︸=−(Y,c′,Y,c′)


Again, the second term is final. We continue with (2):

(2) =d

dsg

(∇ ∂

∂t

∂

∂t,∂

∂s

)− g

(∇h

∂∂t

∂

∂t,∇h

∂∂s

∂

∂s

).

Evaluating at t = 0 and using that

∇h∂∂s

∂

∂s= ∇h

∂∂s

c′(s) = ∇c′(s)c′(s) = 0

since c is a geodesic we obtain overall:

1

2

d2

dt2

∣∣∣∣t=0

E(ct) =∥∥∥∇h

∂∂s

Y (s, 0)∥∥∥2

− (Y, c′, Y, c′)(s, 0) +d

dsg(∇h

∂∂t

Y (s, 0), c′(s))

which implies the assertion.

3.3. Curvature and Topology. Finally, we are in a position to discuss some in-teresting applications. In fact, a large amount of recent research in RiemannianGeometry has focused on the relation between the curvature of a Riemannian man-ifold and its global topological properties. Ricci curvature in particular is involvedin many interesting statements. Recall that given a Riemannian manifold (M, g),p ∈M and x ∈ TpM we have defined

Ricp(s) =1

(m− 1)

m−1∑

i=1

〈R(x, zi)x, zi〉

where z1, . . . , zm−1, zm = x is an orthonormal basis of TpM . As an example, con-sider Sm(r), the sphere of radius r. In this case, Ricp(x) = 1/r2 for all p ∈ Sm(r)and x ∈ TpSm(r). Furthermore, the diamater of Sm(r) is given by diamSm(r) = πr.This example is the extremal case in the following Theorem.

Theorem 3.19 (Bonnet-Myers). Let M be a complete connected Riemannian man-ifold with Ricp(x) ≥ 1/r2 > 0 for all (p, x) ∈ TM . Then M is compact and in factdiam(M) ≤ πr.

It is a theorem of Schenk that equality in the second inequality fo Theorem 3.19implies that M is isometriec to a sphere of radius r. This relates to eigenvalues ofthe Laplacian among other things.

Proof. (Theorem 3.19). By Theorem 2.68 it suffices to show that all minimizinggeodesics have length at most πr. Let p, q ∈ M and let γ : [0, 1] → M bea minimizing geodesic with γ(0) = p and γ(1) = q. Let l = d(p, q). Then wemay extend γ′(0)/l to an orthonormal basis e1, . . . , em−1, em = γ′(0)/l of TpM .Let e1(s), . . . , em−1(s) be the parallel vector fields along γ with initial conditionse1, . . . , em−1 and define Yj(s) = sin(πs)ej(s). Then Yj(0) = 0 and Yj(1) = 0. Fi-nally, let hj : [0, 1] × (−ε, ε) → M be a variation of c with fixed end points andD(s,0)h

(∂∂t

)= Yj(s). We now compute the second variation of the energy of the

map s 7→ hj(s, t), denoted by E′′j (0). Since

∇h∂∂s

Yj(s, 0) = ∇h∂∂s

(sin(πs)ej(s)) = π cos(πs)ej(s) + sin(πs)∇h∂∂s

ej(s)︸︷︷︸

=0

we have ‖∇h∂/∂sYj(s, 0)‖2 = π2 cos(πs)2. Furthermore,

(Yj , γ′, Yj , γ

′)(s,0) = (sin πs)2l2 (ej(s), em(s), ej(s), em(s)) (s, 0)

and hence

1

2E′′j (0) =

∫ 1

0

π2 cos(πs)2 − l2 sin(πs)2K(ej(s), em(s)) ds


Partial integration of the π2 cos(πs)2 term yields

1

2E′′j (0) =

∫ 1

0

sin(πs)2(π2 − l2K(ej(s), em(s))

)ds

and therefore

1

2(m− 1)

m−1∑

j=0

=

∫ 1

0

sin(πs)2(π2 − l2Ricc(s)(em(s))

)ds ≤

∫ 1

0

sin(πs)2(π2 − l2/r2

)

Now, if l πr then∑m−1

i=1 E′′j (0) 0. Hence E′′

j (0) 0 for some j ∈ 1, . . . ,m−1which contradicts γ being minimizing.

Corollary 3.20. LetM be a complete connected Riemannian manifold with Ricp(x) ≥δ > 0 for all (p, x) ∈ T1M . Then the universal covering M of M is compact andπ1(M) is finite.

Proof. Since the covering map is a local isometry, M satisfies the same curvaturebounds as M . Hence M is compact by Theorem 3.19 and π1(M) is finite.

As usual, Lie groups with bi-invariant metrics constitute a particularly nice classof examples.

Corollary 3.21. Let G be a connected Lie group which admits a bi-invariant metricand let g := Lie(G). Assume that Z(g) = 0. Then both G and G are compact andπ1(G) is finite.

The assumption of Corollary 3.21 that Z(g) be trivial is equivalent to the centerof G being zero-dimensional.

Proof. (Corollary 3.21). Since G acts on itself by isometric left-translations, wehave Ricg(DeLg(v)) = Rice(v) for all v ∈ g with |v| = 1. Hence it suffices to boundRice(v) from below by a positive quantity. Recall that for orthonormal X,Y ∈ g

we have K(X,Y ) = 14‖[X,Y ]‖2. Now, let Y1, . . . , Ym−1, Ym = X be an orthonormal

basis of g. Then

Rice(X) =1

m− 1

m−1∑

i=1

K(X,Yi) =1

m− 1

m−1∑

i=1

‖[X,Y ]‖2 ≥ 0

with equality if and only if [X,Yi] = 0 for all i ∈ 1, . . . ,m − 1 which in turn isequivalent to X ∈ Z(g), i.e. X = 0 which contridicts ‖X‖ = 1. Therefore, the Riccicurvature being a positive function on a compact sphere, it is bounded below bysome δ which implies the assertions by Theorem 3.19.

As a remark in the context of the proof of Corollary 3.21 we state that

Rice(X) =1

m− 1

m∑

i=1

1

4‖[X,Yi]‖2

=1

4(m− 1)

m∑

i=1

〈ad(X)Yi, ad(X)Yi〉

= − 1

4(m− 1)

m∑

i=1

〈ad(X)2Yi, Yi〉

= − 1

4(m− 1)tr(ad(X)2)

= − 1

4(m− 1)Kg(X,X)


where Kg denotes the Killing form of g defined by

Kg : g× g → R, (X,Y ) 7→ tr(ad(X) ad(Y ))

Using this, one obtains more information about the diameter of such Lie groups interms of root systems.

Corollary 3.22. Let G be a connected compact Lie group with zero-dimensionalcenter. Then G is compact and π1(G) is finite.

Proof. A compact Lie group always admits a bi-invariant metric.

Another application of the second variation formula is the following.

Theorem 3.23 (Weinstein). Let M be a compact connected oriented Riemannianmanifold which has everywhere strictly positive sectional curvature. Further letf ∈ Iso(M) be orientation-preserving if dimM is even and orientation-reversing ifdimM is odd. Then f has a fixed point.

The example of the two-dimensional sphere and the antipodal map shows thatthe assumptions on f are necessary.

Before turning to the proof of Theorem 3.23, consider the following linear version.

Lemma 3.24. Let A ∈ O(m− 1) and suppose that detA = (−1)m. Then A fixes anon-trivial vector.

Proof. We show that 1 ∈ R is an eigenvalue of A: Let λ1, . . . , λr, µ1, µ1, . . . , µk, µkbe the eigenvalues of A, listed with multiplicity, with λi ∈ R for all i ∈ 1, . . . , r andµi 6= µi for all i ∈ 1, . . . , n. Since A is orthgonal, λi ∈ ±1 for all i ∈ 1, . . . , rand all µi have unit length. Therefore

∏ri=1 λi = (−1)m. Also we have r+2k = m−1.

Now, assume that m is even. Then m− 1− 2k is odd and hence so is r. Hence thereis i ∈ 1, . . . , r such that λi = 1. Conversely, if m is odd, then −1 = λ1 · · ·λrimplies r ≥ 1. Also, r is even since r = m − 1 − 2k. Consequently there is againi ∈ 1, . . . , r such that λi = 1.

Proof. (Theorem 3.23). We argue by contradiction: Choose p ∈M with

d(p, f(p)) = mind(q, f(q)) | q ∈Mand assume that 0 < d(p, f(p)) =: l. In this case, let γ : [0, l] →M be a minimizinggeodesic connecting p to f(p). First, we claim that Dpf(γ

′(0)) = γ′(l): To this end,consider 0 < t′ < l and p′ = γ(t′). Then

l ≤ d(p′, f(p′)) ≤ l([p′, f(p)] ∪ [f(p), f(p′)]) = l.

Since the concatenation [p′, f(p)]∪ [f(p), f(p′)] is a minimizing geodesic and henceC1 we conclude that γ′(l) = Dpf(γ

′(0)).

b

b

b

p

f(p)

f2(p)

γ

f γ

γ′(0)γ′(l)

Dpf(γ′(0))

Now consider A := Pγ,f(p),p Dpf ∈ GL(TpM). We have A(γ′(0)) = γ′(0) as wellas detA = (−1)m. Applying Lemma 3.24 to A ∈ O(γ′(0)⊥), there is e1 ∈ TpM ,orthogonal to γ′(0) which is fixed by A. Let e1(s) ∈ Tγ(s)M be the vector fieldparallel to γ with e1(0) = e1. Then Dpf(e1(0)) = e1(l).


Now look at the variation h(s, t) := expγ(s)(te1(s)). The above implies thatf(h(0, t)) = h(l, t), i.e. f(γt(0)) = γt(l). Hence

l(γt) ≥ d(γt(0), γt(l)) = d(γt(0), f(γt(0))) ≥ l.

This implies that γ = γ0 is a local minimum of t 7→ l(γt). Hence d2/dt2|t=0l(γt) ≥ 0.On the other hand, the second variation formula yields

1

2

d2

dt2

∣∣∣∣t=0

E(γt) = g(∇ ∂

∂te1(s, 0), γ

′(s))∣∣∣l

0+

∫ l

0

∥∥∥∇ ∂∂te1

∥∥∥2

− (e1, γ′, e1, γ

′)(s, 0) ds

= −∫ l

0

K(e1(s), γ′(s)) ds < 0

which contradicts the above.

We now discuss further applications of the second variation formula in bothpositive and negative curavture.

Corollary 3.25 (Synge). Let M be a compact, connected Riemannian manifoldof dimension m with everywhere strictly positive sectional curvature. Then thefollowing hold.

(i) If m is even and M is orientable then M is simply connected.(ii) If m i sodd then M is orientable.

As a preliminary remark towards the proof of Corollary 3.25 be note that everyconnected Riemannian manifold M admits an orientable cover. Indeed, set

M := (p,Op) | p ∈M, Op orientation on TpM.It is obvious that one can equip M with the structure of a smooth manifold suchthat p : M → M , (p,Op) → p is a two-sheeted covering which is connected ifand only if M is not orientable. It is a Galois covering with non-trivial coveringtransformation σ : (p,Op) 7→ (p,Op) where Op is the orientation opposite Op.

Proof. (Corollary 3.25). For the first assertion, let p : M →M be the universalcover of M and let g be the Riemannian metric on M turning p into a Riemanniancovering. In particular, if δ > 0 is a lower bound on the sectional curvature ofM thenδ is also a lower bound on the sectional curvature of (M, g). Therefore, Theorem3.19 implies that M is compact. Now, let π1(M) act by covering transformations onM ; they are orientation-preserving isometries since M is orientable. If π1(M) 6= ethen there is f ∈ π1(M)\Id which by Theorem 3.23 has at least one fixed point.This, however, contradicts the fact, that non-trivial covering transformations donot have fixed points.

For part (ii), let m be odd and assume that M is not orientable. In this case,consider the orientable Riemannian double cover M of M with Riemannian metricg making the covering map Riemannian. Then the generator σ of π1(M) can beviewed as an isometry of M and by Theorem 3.23 has a fixed point which yieldsthe same contradiction as before.

3.3.1. Jacobi Fields. In this section we introduce another important, namely Jacobivector fields, which will allow us to give a description of the exponential map. Tomotivate this, let’s rewrite the integral part of the second variation formula: Since

⟨∇ ∂

∂sY,∇ ∂

∂sY⟩=

d

ds

⟨Y,∇ ∂

∂sY⟩−⟨Y,∇2

∂∂s

⟩,

the integral term reads∫ b

a

‖∇ ∂∂sY ‖2−(c′, Y, c′, Y )(s, 0) ds =

⟨Y,∇ ∂

∂s

⟩∣∣∣b

a−∫ b

a

⟨Y,∇2

∂∂s

Y +R(c′, Y )c′⟩ds.


Therefore,

d2

dt2

∣∣∣∣t=0

E(ct) =stuff depending on

endpoints only−∫ b

a

⟨Y,∇2

∂∂s

Y +R(c′, Y )c′⟩ds

Here, the symmetric bilinear form −∫ ba

⟨Y,∇2

∂∂s

Y +R(c′, Y )c′⟩ds can be viewed

as the second derivative of the energy functional at c and Jacobi vector fields arevector fields in directions where “nothing happens”, reflecting degeneracy.

Definition 3.26. Let M be a Riemannian manifold and let c be a geodesic into M .A Jacobi vector field is a vector field Y along c satisfying

∇ ∂∂sY +R(c′, Y )c′ = 0.

The following are the main statement we shall prove about Jacobi vector fields.

Theorem 3.27. Let M be a Riemannian manifold and let c be a geodesic into M .Further, let u, v ∈ Tc(0)M . Then there is exactly one Jacobi vector field Y along cwith Y (0) = u and Y ′(0) := ∇∂/∂sY (0) = v.

Proposition 3.28. Let M be a Riemannian manifold, c : [a, b] → M a geodesic andh : [a, b] × (−ε, ε) → M a variation of c such that for every t ∈ (−ε, ε), the curvect : [a, b] → M is a geodesic. Then Y (s) := D(s,0)h(∂/∂t) is a Jacobi vector fieldalong c. Coversely, every Jacobi vector field can be obtained in this way.

Proof. (Theorem 3.27). Fix an orthonormal frame X1, . . . , Xm of parallel vectorfields along c using that parallel transport is isometric. Then any vector field Yalong c can be expressed as

Y (s) =

m∑

i=1

yi(s)Xi(s)

for some smooth functions yi. Since the Xi are parallel along c we conclude

∇2∂∂s

Y (s) =

m∑

i=1

y′′i (s)Xi(s).

Therefore,

∇2∂∂s

Y +R(c′, Y )c′ = 0 ⇔⟨∇2

∂∂s

Y +R(c′, Y )c′, Xi

⟩= 0 ∀1 ≤ i ≤ m.

Replacing Y by its expression in the Xi leads to the following system of ordinarydifferential equations:

y′′i (s) +

m∑

j=1

yj(s)〈c′, Xj, c′, Xi〉(s) = 0.

Standard results now imply the theorem.

We now turn to Proposition 3.28.

Proof. (Proposition 3.28). For the first part, assume that h : [a, b] × (−ε, ε) is avariation of c such that all ct are geodesics and compute

∇2∂∂s

Y = ∇ ∂∂s∇ ∂

∂s

∂

∂t= ∇ ∂

∂s∇ ∂

∂t

∂

∂s= ∇ ∂

∂t∇ ∂

∂s

∂

∂s+R

(∂

∂t,∂

∂s

)∂

∂s.

Now note that∂

∂s(s, t) = D(s,t)h

∂

∂s= c′t(s).

Because of this and the fact that all ct are geodesics we obtain evaluating at t = 0:

∇2∂∂s

Y = R(Y, c′)c′ = −R(c′, Y )c′.


For the converse, let Y be a Jacobi vector field along c and let γ : (−ε, ε) →M bea geodesic with γ(0) = c(0). Also, let X be a vector field along γ. Both are going tobe determined later. In any case, h(s, t) := expγ(t)(sX(t)) has the property that foreach t the map s 7→ h(s, t) is a geodesic. We have h(s, 0) = expc(0)(sX(0)). We aregoing to choose X(t) = X1(t) + tX2(t) for some parallel vector fields X1, X2 alongγ with X1(0) = c′(0). For the moment, compute the Jacobi vector field associatedto this geodesic variation.

∂

∂t= D(s,0)h

∂

∂t.

Unable to do this directly, we look at the initial conditions

∂

∂t(0, 0) and ∇ ∂

∂s

∂

∂t(0, 0).

Regarding the first, we have

∂

∂t(0, 0) = D(0,0)h

∂

∂t=

d

dt

∣∣∣∣t=0

(t 7→ expγ(t)(0) = γ(t)) = γ′(0)

Therefore, we set, γ′(0) = Y (0). Regarding the second, we have

∇ ∂∂s

∂

∂t(s = 0) = ∇ ∂

∂t

∂

∂s(s = 0).

Using that

∂

∂s

∣∣∣∣s=0

= D(0,t)h∂

∂s= X(t) = X1(t) + tX2(t)

we conclude

∇ ∂∂t

∂

∂s(0, t) = X2(t)

and hence choose X2(0) = Y ′(0).

The following result finally links Jacobi vector fields to the derivative of theexpoential map.

Proposition 3.29. Retain the above notation. Let u, v ∈ TpM and c(s) := exp(su).Further, let Y be the Jacobi vector field along c with Y (0) = 0 and Y ′(0) = 0.Then Dsv(expp)(su) = Y (s).

Proof. Consider the variation of c given by h(s, t) := expp(s(v+ tu)). Then we haveh(s, 0) = exp(sv) = c(s) and h(0, t) = p = c(0) = ct(0). We compute the Jacobivector field Z associated to h. In fact, in view of Theorem 3.27, we determine Z(0)and Z ′(0). Obviously, Z(0) = 0. Next,

Z ′(0) = ∇ ∂∂s

∂

∂t(0, 0) = ∇ ∂

∂t

∂

∂s(0, 0)

where∂

∂s= D(s,t)h

∂

∂s= Ds(v+tu) expp(v + tu)

At s = 0 we therefore have ∂/∂s(0, t) = v + tu. Hence Z ′(0) = u. Therefore,Z(s) = Y (s) but by definition

Z(s) =∂

∂t

∣∣∣∣t=0

expp(s(v + tu)) = Dsu expp(su).


As an application of the theory of Jacobi fields, we present the following argu-ments to determine the sectional curvature of spheres and hyperbolic space, by-passing several pages of computations.

Consider the sphere Sn. Let x ∈ Sn and pick v ∈ Sn with 〈x, v〉 = 0. Then

c(s) := (cos s)x+ (sin s)v

is a geodesic. Now pick u ∈ Sn with 〈u, x〉 = 0 = 〈u, v〉 and consider the variation

h(s, t) := cos s+ sin s((cos t)v + (sin t)u).

Then h(s, 0) = c(s) and h(0, t) = x. Also,

Y (s) :=∂

∂t= (sins)U(s)

for a vector field U along c with U(s) = u. Then

∇ ∂∂sY = (cos s)U(s) and ∇2

∂∂s

Y = −(sin s)U(s)

because U is parallel along c given that parallel transport is an orientation-preservingisometry. We thus have

0 = 〈∇2∂∂s

Y +R(c′, Y )c′, Y 〉 ⇔ 0 = − sin2 s+ sin2 s(c′u, c′, u)

for all s with sin(s) 6= 0, implying (c′, u, c′, u) = 1 for such s and by continuity(u, v, u, v) = 1.

The case of hyperbolic space is left as an exercise.

3.3.2. The Cartan-Hadamard Theorem. In this section, we see an example of hownegative curvature impacts on the topology of manifolds.

Theorem 3.30 (Cartan-Hadamard). Let (M, g) be a complete Riemannian manifoldwith everywhere non-positive sectional curvature. Then expp : TpM → M is acovering map.

In the context of Theorem 3.30 note that if M is simply connected then expp isa diffeomorphism. Its proof relies on the following lemma.

Lemma 3.31. Let (M, g) be a complete Riemannian manifold with everywhere non-positive sectional curvature. Then expp : TpM →M is a local diffeomorphism.

Proof. Here, we utilize our knowledge of the derivative of the exponential map aswell as the inverse function theorem. Using the notation of Proposition 3.29, letY (s) = Dsv expp(su) and assume u 6= 0. We know that Y (s) = 0 and Y ′(0) = u.Now consider the function f(s) := 〈Y (s), Y (s)〉. Then

f ′(s) = 2⟨∇ ∂

∂sY (s), Y (s)

⟩

and′′(s) = 2

⟨∇ ∂

∂sY (s),∇ ∂

∂sY (s)

⟩+ 2

⟨∇2

∂∂s

Y (s), Y (s)⟩.

where ⟨∇2

∂∂s

Y, Y⟩= −(c′, Y, c′, Y ) ≥ 0

which implies f ′′(s) ≥ 0. Also, we have f(0) = 0 and f ′(0) = 0. From the first,we conclude that f ′ is increasing. In particular, since f ′(0) = 0 we get f ′(s) ≥ 0for all s ∈ [0,∞). Hence f is increasing and positive. Assume there is s0 > 0with f(s0) = 0 which implies f(s) = 0 for all s ∈ [0, s0]. Then f ′′(0) = 0 for alls ∈ (0, s0), i.e.

‖∇ ∂∂sY (s)‖ = 0 ∀s ∈ (0, s0).

and by continuity‖∇ ∂

∂sY (0)‖ = 0.


That is, u = 0, a contradiction.

In order to pass from a local diffeomorphism to a covering map, we shall use thefollowing lemma whose proof is left as an exercise.

Lemma 3.32. Let (N1, g1) and (N2, g2) be connected Riemannian manifolds andlet p : N1 → N2 be a local isometry. Assume that (N1, g1) is complete. Then p iscovering map.

Proof. (Theorem 3.30). Let g be the pullback of g via expp, so that expp is a localisometry. The metric g is complete because straight lines in TpM issuing from0 ∈ TpM are geodesics by the Lemma 2.62.

4. What’s Beyond

First, we record the following important theorem, see [Wol11].

Theorem 4.1. Let M be a simply connected, complete Riemannian manifold ofconstant sectional curvature k. Then M is, up to rescaling of the metric, isometricto either Sm, Em or Hm.

What about classifying Riemannian manifold of constant sectional curvature ingeneral? To avoid pathologies like taking the complement of a closed set in one ofthe above, one should require completeness at least. The positive curvature casethen amounts to determine, up to conjugacy, finite subgroups G of O(m+ 1) suchthat for every g ∈ G\Id, the number one is not an eigenvalue. This is fairlycomplicated and was done accomplished by Vincent, a student of de Rham in the1940’s. In the flat case, this is still an open question. For instance, Bieberbachgroups have only been classified for m ≤ 6, the case m = 3 being particularlyclassical and used by chemist’s ever since. There is a mechanism due to Calabi bywhich one can understand n-dimensional flat manifolds with β1(M) > 0 in termsof lower dimensional ones N with β1(N) = 0.

In negative curvature, the story has a completely different flavour. For instance,recall the classification of compact orientable manifolds.

Sphere Torus Surface with two holes

There is the sphere, the torus, and surfaces Sg (g ≥ 2) with g holes. The Gauss-Bonnet theorem states that for a compact orientad surface S with sectional curva-ture k :M → R we have

1

2π

∫

S

kg dω = χ(S) = 2− 2g

where χ(S) is the Euler characteristic of S. In particular, only the sphere carriesa positively curved metric, only the torus carries a flat metric and only the highergenus surfaces potentially carry hyperbolic metric. In this case, the Gauss-BonnetTheorem implies that area(S) = 4π(g − 1).

In fact, there are many hyperbolic metrics on higher genus surfaces. To describethese one can either turn to the study of discrete subgroups of the isomorphismgroup of two-dimensional hyperbolic space or use the following explicit approachdue to Buser in 1978 involving right-angled hexagons in two-dimensional hyperbolic


space. To describe these, we use the Poincaré disk model of the hyperbolic planewhich has the advantages that the Euclidean angles one sees are the same as theactual (hyperbolic) angles since at every point in the disk, the hyperbolic metric isa multiple of the Euclidean metric.

b

b

b

b

b

b

a

b

c

In a right-angled hexagons, bounded by geodesic rays, the side lengths a, b and care free parameters. Let H(a, b, c) denote the associated hexagon and H(a, b, c) theone with opposite orientation. Glueing H(a, b, c) to H(a, b, c) yields a pair of pantswhich closes up nicely due to the fact that the hexagons are right-angled.

2a

2b

2c

A surface of genus two now arises through gluing two pairs of pants along thegeodesic boundaries. Here, three more rotational parameters are introduced.

In total, there are six parameters. In the case of arbitrary higher genus g, there are6g − 6 parameters.

In higher dimensions, the story yet takes a completely different flavour.

Theorem 4.2 (Mostow). Let M1 and M2 be compact Riemannian manifolds ofconstant sectional curvature k = −1 and of dimension at least three. Then anyisomorphism of π1(M1) and π1(M2) is induced by an isometry of M1 and M2.

In particular, such a manifold carries only one hyperbolic metric. Understandingthe isomorphism classes of such π1(M) inside SO(n, 1) remains a challenge though.

We also did not touch at all pinching theorems in the spirit of Berger-Klingenbergmentioned in the introduction.

A revolutionary type of result was introduced by Gromov. The basic idea dates backto Ulam who introduced the study of small perturbations of notions in algebra. Forinstance, instead of studying the homomorphism equation ϕ(ab)−ϕ(a)−ϕ(b) = 0,one may, in the presence of a metric, ask for the left-hand-side to be bounded and see


what remains. In a Riemannian geometric setting, one may want to study manifoldsfor which the absolute value of the sectional curvature is bounded. Or rather, sincethe metric can always be scaled, ask for the product of the absolute value of thesectional curvature and the diameter of the manifold to be bounded. If the manifoldhas zero curvature and is compact then its fundamental group contains Zm and isvirtually abelian. If one only asks for a small bound, nilpotent fundamental groupsarise and it is an influential theorem of Gromov stating that this is always the casein a certain sense.

Theorem 4.3. For every n ≥ 1 there is a constant εn > 0 such that if (M, g) is acompact Riemannian n-manifold with

|k|diam(M)2 < εn

then there is a finite covering of M of the form Γ\N where N is a simply connectednilpotent group and Γ is a lattice. In particular, π1(M) is virtually nilpotent.

For instance, the manifold

N :=

1 x y

1 z1

∣∣∣∣∣∣x, y, z ∈ R

admits such an almost flat metric and the quotient N(Z)\N has nilpotent funda-mental group N(Z).


References

[BT15] M. Burger and S. Tornier, Differential geometry, 2015.[Cha12] L. S. Charlap, Bieberbach groups and flat manifolds, Springer Science & Business Media,

2012.[Kön13] K. Königsberger, Analysis 2, Springer, 2013.[Lee10] J. Lee, Introduction to topological manifolds, vol. 940, Springer, 2010.

[Rie54] B. Riemann, Über die Hypothesen, welche der Geometrie zugrunde liegen, Werke 2

(1854), 272–287.[Spi79] M. Spivak, Differential Geometry, vol. I-II, Publish or Perish, 1979.[vdB06] E. van den Ban, Notes on quotients and group actions, 2006.[Wol11] J. A. Wolf, Spaces of constant curvature, vol. 372, American Mathematical Society, 2011.

Documents

Contents Introduction - ETH · ETH Zurich in 2016. Disclaimer. This is a preliminary version. Please report any typos, mistakes, comments etc. to [email protected]. Acknowledgements