Differential operators on Manifolds

and positive scalar curvature

Quinn PattersonSupervised by Adam Rennie and Alan Carey

University of Wollongong

Summer 2017-2018

Vacation Research Scholarships are funded jointly by the Department of Education and Training

and the Australian Mathematical Sciences Institute.

Contents

1 Foundations 4

1.1 The cotangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.2 A Monstrous Curvature Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4 The Dirac Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Topology 35

2.1 de Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2 Euler Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Bochner’s method and Betti numbers 38

3.1 Positive curvature and the cotangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Positive curvature and the exterior bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Positive curvature and the spinor bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Abstract

This work was completed in the summer of 2017/18 as part of an AMSI vacation research scholarship

supervised by Adam Rennie and Alan Carey. We construct the Clifford bundle on a manifold, which

we then use to define Dirac operators. We prove some basic properties about Dirac operators, and then

introduce some topology we will need for our major results. We then prove Bochner’s identity for a Dirac

operator, and apply this to different Dirac bundles to deduce interesting results about curvature and the

topology of a compact, oriented manifold. We finish with a brief outline of spin structures and spinor

bundles, and look at the Lichnerowicz formula and the positive scalar curvature conjecture in dimensions

4k.

Introduction

This project aims to explore the relationship between topology and geometry on manifolds arising from

imposing conditions on curvature. In particular we wish to prove rigorously the notion that if a manifold

has a hole, then it must curve negatively around it. For this we will require some foundation knowledge of

the geometry of manifolds and their topology.

Our aim will be to construct and analyse the Clifford bundle over a manifold, and then define Dirac

operators using Clifford bundles. Dirac operators contain important geometric information, as they are

natural ‘Square roots’ of the Laplacian, and in particular they have the same kernel as the laplacian. We then

are able to use the group structure of the Clifford bundle in the definition Dirac operators to produce a formula

for the Laplacian in terms of a positive operator and a curvature operator. Using Hodge’s decomposition

theorem for the de Rham cohomology groups of a compact manifold which relates these groups to the kernel

of the Laplacian, we are then able to deduce results about the de Rham cohomology groups arising from

restrictions on curvature.

Throughout this document we will denote by (M, g) a compact and oriented Riemannian manifold. By

Riemannian we mean M is equipped with a smoothly varying family of positive definite inner products

gx : TxM×TxM → R on each tangent space TxM . We also will assume M has no boundary unless explicitly

stated.

3

1 Foundations

1.1 The cotangent bundle

Recall that given a vector space V , its dual is the vector space V ∗ of linear functionals φ : V → K where K

is the underlying field of V . On a manifold (M, g) we may form the cotangent bundle denoted T ∗M which is

the bundle whose fibres are the dual to those of the tangent bundle. Given a Riemannian metric on M , there

is a canonical isomorphism using the metric called the ‘musical isomorphism’ between TM and T ∗M given

by the map X ∈ TM 7→ g(X, ·) ∈ T ∗M . The Riesz lemma tells us that this map has a well defined inverse

which one can check is in fact a bijection. The image of an element ω ∈ T ∗M under the inverse of this map

is denoted ω] and satisfies g(ω], X) = ω(X) for X ∈ TM . We are able to use the musical isomorphism to

give T ∗M its own smoothly varying family of inner products. More can be found about this in [1, Chapter

7 section 7.8]

Lemma 1.1. Given the Riemannian metric g on TM we may define a family of inner products g on the

cotangent bundle T ∗M via the formula

g(ω1, ω2) = g(ω]1, ω]2)

and in local coordinates we have (g)kl = (g−1)kl where (g)kl := g(dxk, dxl) and gkl := g( ∂∂xk ,

∂∂xl ).

Proof. Clearly this definition gives a family of inner products on T ∗M since it inherits the desired properties

from the existing metric g. It remains to show that g = g−1. Let

(dxi)] =

n∑j=1

ηij∂

∂xj.

Since (dxi) is the dual basis to ∂xi we may form the expression

δij = dxi(∂

∂xj) = g((dxi)],

∂

∂xj)

=

n∑k=1

g(ηik∂

∂xk,∂

∂xj)

=

n∑k=1

ηikgkj

4

which tells us that ηji = gij where gij refers to the ij-th component of g−1. Thus

g(dxi, dxj) = g((dxi)], (dxj)]) = dxi((dxj)])

=

n∑k=1

ηjkdxi(∂

∂xk) = ηji = gij .

as required.

1.2 Clifford algebras

Now we wish to construct a vector bundle over M called the clifford bundle. We will do this by defining and

analysing the clifford algebra for a single (finite dimensional) vector space and the way that it acts on the

exterior alggebra of that same vector space, then extend these notions to a vector bundle whose fibres are

the vector spaces we have analysed.

Definition 1.2. Let V be a finite dimensional vector space with an inner product 〈 , 〉. Let T (V ) ⊂

⊕∞n=0V⊗n be the span of all finite sums of elements v1 ⊗ · · · ⊗ vn with multiplication given by a · b = a⊗ b

and declaring that this multiplication be distributive. We may define an inner product on tensor powers

by 〈a ⊗ b, c ⊗ d〉 := 〈a, c〉〈b, d〉 which then descends to an inner product on T (V ) by declaring 〈a, b〉 = 0

when a, b are in differing tensor powers. Let J(V ) ⊂ T (V ) be the ideal generated by elements in of the form

v · v + 〈v, v〉1. Then the Clifford algebra Cl(V ) is the quotient space T (V )/J(V ).

This definition leaves us with a vector space having a natural multiplication satisfying the fundamental

relation v · v = −〈v, v〉1. By applying this to 〈v +w, v +w〉 we learn that v ·w +w · v = −2〈v, w〉. We have

a natural inclusion V ↪→ Cl(V ) by including v ∈ V into T (V ) and composing with the quotient map and so

we may make sense of the expression v ·w in the Clifford algebra for elements v, w ∈ V . A basis of Cl(V ) is

then given by {vi1 · vi2 · · · vik : 1 ≤ i1 < · · · < ik ≤ dim(V )} where {vi} is a basis for V . The Clifford algebra

is canonically isomorphic to the exterior algebra Λ∗(V ) as a vector space (not as an algebra) via the map

φ : Λ∗V → Cl(V ) given by

φ(v1 ∧ · · · ∧ vk) =1

k!

∑σ∈Sk

(−1)|σ|v1 · · · · · vk.

Note that for an orthogonal basis {ei} of V , the relations v · w + x · w = −2〈v, w〉 give eiej = −ejei when

i 6= j. This then tells us that eσ(1) · · · eσ(k) = (−1)|σ|e1 . . . ek which says that for the basis {ei1 · · · eik} of

5

Cl(V ) we have

φ(ei1 l ∧ · · · ∧ eik) = ei1 · · · eik .

Of interest to us will be the operator v· : Λ∗V → Λ∗V , v · x = φ−1(v · φ(x)) for v ∈ V .

Lemma 1.3. Let v ∈ V ⊂ Cl(V ) and w ∈ Λ∗(V ). Then

v · w = v ∧ w − vxw (1)

where x is the interior product. This is the map such that if w = w1 ∧ · · · ∧ wk ∈ ΛkV and v ∈ V then

vxw =

k∑j=1

(−1)j−1〈v, wj〉w1 ∧ · · · ∧ wj ∧ · · · ∧ wk

where wj denotes deletion of wj from the expression w1 ∧ · · · ∧wk. In particular, since v ∧ · and vx· are both

linear maps, left multiplication by elements in V is linear.

Proof. We first check that both sides agree on an orthonormal basis. Let {ei} be an orthonormal basis of

V , giving {ei1 ∧ · · · ∧ eik}i1<···<ik a basis of Λ∗V . Fix i and fix an ordered subset i1 < · · · < ik of 1, . . . , n.

Then since ei ∧ ej = −ej ∧ ei we have

ei · (ei1 ∧ · · · ∧ eik) =

(−1)jei1 ∧ · · · ∧ eij ∧ · · · ∧ eik if i = ij

(−1)jei1 ∧ · · · ∧ eij ∧ ei ∧ eij+1∧ · · · ∧ eik if i 6= il for any l, and j < i < j + 1.

Now on the other hand we compute the right hand side of (1) to check we get the same. Suppose i = ij for

some j. Then

ei ∧ (ei1 ∧ · · · ∧ eik)− eix(ei1 ∧ · · · ∧ eik) = 0−n∑l=1

(−1)l+1〈ei, eil〉ei1 ∧ · · · ∧ eil ∧ · · · ∧ eik

= (−1)j+2ei1 ∧ · · · ∧ eij ∧ · · · ∧ eik since 〈ei, eil〉 = δiil

= (−1)jei1 ∧ · · · ∧ eij ∧ · · · ∧ eik

= ei · (ei1 ∧ · · · ∧ eik) .

Now instead, suppose that i 6= il for any l, and let j be such that j < i < j + 1. Then 〈ei, eil〉 = 0 and so

6

eix(ei1 ∧ · · · ∧ eik) = 0, and so we have

ei ∧ (ei1 ∧ · · · ∧ eiK )− eix(ei1 ∧ · · · ∧ eik) = ei ∧ ei1 ∧ · · · ∧ eik

= (−1)jei1 ∧ · · · ∧ eij ∧ ei ∧ eij+1 ∧ · · · ∧ eik

= ei · (ei1 ∧ · · · ∧ eik)

and thus we conclude ei ·(ei1 ∧ · · · ∧ eik) = ei∧(ei1 ∧ · · · ∧ eiK )−eix(ei1 ∧ · · · ∧ eik) for any i and subsequence

i1 . . . ik of 1, . . . , n.

Now since · ∧ · and ·x· are both bilinear, expressing v and w in the bases {ei} and {ei1∧, · · · ∧ eik}i1<···<ik

gives the desired result.

Remark. This multiplication on Λ∗V is an example of a representation of Cl(V ), and we will define and prove

this below. There are multiple different ways we could represent Cl(V ) as a collection of linear operators on a

vector space, for example we could define a different multiplication v ·w = v∧w+vxw on Λ∗(V ), or we could

define a multiplication on another vector space entirely. The ‘Dirac Operator’ which we will define soon

depends on the choice of representation of Cl(M) on a given vector bundle, however of particular interest

to us is the representation we have analysed above, which we will refer to as the exterior representation of

Cl(M) on Λ∗M .

Definition 1.4. A representation of an algebra A is an algebra homomorphism ρ : A → GL(V ) for some

vector space V , where GL(V ) refers to the collection of invertible linear maps T : V → V . A subspace

W ⊆ V is called invariant if ρ(a)(W ) ⊆ W for all a ∈ A. A representation is called irreducible if the only

invariant subspaces are V and {0}.

Proposition 1.5 (Universal property of Clifford Algebras). Given a linear map f : V → A such that

f(v) · f(v) = −〈v, v〉 · 1

where A is a unital associative algebra, there exists a unique algebra homomorphism f : Cl(V )→ A which

agrees with f on V .

See [3] Proposition 1.1 for a proof of this.

Proposition 1.6. Left multiplication by elements in Cl(V ) defines a representation of Cl(V ) on Λ∗V .

7

Proof. We wish to show for elements v ∈ V ↪→ Cl(V ) that v2 = −〈v, v〉1, then use the universal property

of Clifford algebras to lift this to an an algebra homomorphism on all of Cl(V ) (that is, a representation of

Cl(V ) on Λ∗V ). Take {ei} an orthonormal basis of V , and let v ∈ V . First we show ei∧eixφ+eixei∧φ = φ.

Linearity says it suffices to show this on a basis element φ = ei1 ∧ · · · ∧ eip . Suppose eij = ei for some j.

Then

ei ∧ eixφ+ eixei ∧ φ = ei ∧ eixφ

= ei ∧n∑l=1

(−1)l−1〈ei, eij 〉ei1 ∧ · · · ∧ eil ∧ · · · ∧ eip

= ei(−1)j−1ei1 ∧ · · · ∧ ei ∧ · · · ∧ eip

= (−1)2j−2φ = φ.

Suppose then that ei 6= eij for any j. Then

ei ∧ eixφ+ eixei ∧ φ = eixei ∧ φ

= 〈ei, ei〉φ+

n∑l=1

(−1)l〈ei, eil〉ei1 ∧ · · · ∧ eil ∧ · · · ∧ eip

= φ.

Now we quickly note that for i 6= j

eix(ej ∧ φ) = 〈ei, ej〉φ−n∑l=1

(−1)l−1〈ei, eil〉ej ∧ ei1 ∧ · · · ∧ eil ∧ · · · ∧ eip

= −ej ∧n∑l=1

(−1)l−1〈ei, eil〉ei1 ∧ · · · ∧ eil ∧ · · · ∧ eip

= −ej ∧ eixφ (2)

8

which allows us to calculate by swapping indices and using (2)

n∑i 6=j

〈v, ei〉〈v, ej〉 [ei ∧ ejxφ+ eixej ∧ φ] =

n∑j 6=i

〈v, ej〉〈v, ei〉 [ej ∧ eixφ+ ejxei ∧ φ]

= −n∑i 6=j

〈v, ei〉〈v, ej〉 [ei ∧ ejxφ+ eixej ∧ φ]

=⇒n∑i 6=j

〈v, ei〉〈v, ej〉 [ei (∧ejxφ) + eix(ej ∧ φ)] = 0.

Finally we calculate

v · v · φ =

n∑i,j=1

〈v, ei〉〈v, ej〉ei · ejφ

=

n∑i,j=1

〈v, ei〉〈v, ej〉ei ∧ (ej ∧ φ− ejxφ)− eix(ej ∧ −ejyφ)

= −∑i,j=1

〈v, ei〉〈v, ej〉 [ei ∧ ejxφ+ eixej ∧ φ]

= −n∑i=1

〈v, ei〉2 [ei ∧ eixφ+ eixei ∧ φ]−n∑i 6=j

〈v, ei〉〈v, ej〉 [ei ∧ ejxφ+ eixej ∧ φ]

= −n∑i=1

〈v, ei〉2φ = −〈v,n∑i=1

〈v, ei〉ei〉φ

= −〈v, v〉φ

as required, and so that the map which takes v ∈ V to the linear operator v· extends to a representation of

Cl(V ) on Λ∗(V ).

Remark. We can similarly define a right multiplication of Cl(V ) on Λ∗V by sending an element of Λ∗V to

Cl(V ), multiplying on the right instead of left, and sending this product back to Λ∗V . This techincally does

not define a representation since if vR denotes multiplication on the right and φ the canonical ismomorphism

9

between Λ∗V and Cl(V ) described at the start of section 1.2, then

(vw)Rϕ = φ−1(φ(ϕ)vw)

Whilst

(v(wRϕ)R) = vRφ−1(φ(ϕ)w)

= φ−1(φ(φ−1φ(ϕ)w)v)

= φ−1(φ(ϕ)wv)

6= (vw)Rϕ.

One might think we can fix this by defining vRϕ = φ−1(φ(ϕ)v−1), however not every element of the clifford

algebra is invertible! We will leave the definition of right multiplication as is and there will not be any huge

dramas, we will just keep in mind to be careful when right multiplying multiple elements. One can show

that for e ∈ Cl(V ) and v ∈ ΛpV we have

eR · v = (−1)p(e ∧ v + exv).

We will just write eR · v = v · e from now on. With a right and left multiplication, we may then make sense

of the expresion [e, v] for e ∈ Cl(V ) and v ∈ Λ∗V .

Given an inner product space V [1, section 1.6.5] shows that we may define an inner product on ΛkV by

〈v1 ∧ · · · ∧ vk, w1 ∧ · · · ∧ wk〉 =

∣∣∣∣∣∣∣∣∣∣〈v1, w1〉 . . . 〈v1, wk〉

......

〈vk, w1〉 . . . 〈vk, wk〉

∣∣∣∣∣∣∣∣∣∣. (3)

This inner product is well defined since swapping two rows or columns results in a change of sign of the

determinant which corresponds to swapping two elements in the wedge product. This definition can then be

extended to Λ∗V by declaring 〈v, w〉 = 0 whenever v and w are in different exterior powers. With respect

to this inner product we have the following lemma:

10

Lemma 1.7. For v ∈ V and w, x ∈ Λ∗V , we have

〈v ∧ w, x〉 = 〈w, vxx〉.

In particular, left multiplication by ei is an orthogonal map.

Proof. We show this first on simple elements then bootstrap up to arbitrary vectors. Let {ei} be an or-

thonormal basis of V , let w = w1 ∧ · · · ∧wk ∈ ΛkV and let ei1 ∧ · · · ∧ eij , i1 < · · · < ij be a basis element of

Λ∗V . Since 〈y, z〉 = 0 whenever y and z are in different exterior powers, the expression

〈ei ∧ w, ei1 ∧ · · · ∧, eij 〉 = 〈w, eix(ei1 ∧ · · · ∧ eij

)〉 (4)

is trivially true if j 6= k + 1 since e1 ∧ · raises the exterior power by 1 and e1x· lowers by one whence both

sides of the above equation are zero. So then let us consider the case when j = k + 1. Let A be the matrix

A =

〈ei, ei1〉 . . . 〈ei, eik+1

〉...

. . ....

〈wk, ei1〉 . . . 〈wk, eik+1〉

so that 〈e1∧w, ei1∧· · ·∧eik+1

〉 = detA. If i 6= il for any l then orthogonality says the top row A1,l is zero and

therefore so too are∣∣A∣∣ and the left hand side of (4). Similarly orthogonality says that eixei1 ∧· · ·∧eik+1

= 0

and so the right hand side of (4) is zero as well, so we are left with the case that i = il for some l. Using the

notation Aj for the k × k sub-matrix of A with the first row and jth column deleted, we compute

〈ei ∧ w, ei1 ∧ · · · ∧, eij 〉 = detA

=

n∑j=1

(−1)jA1,j detAj

= (−1)l−1 detAl by orthogonality

= 〈w, (−1)l−1ei1 ∧ · · · ∧ eil ∧ · · · ∧ eik+1〉

= 〈w, eix(ei1 ∧ · · · ∧ eik+1

)〉

and so (4) holds for all ei, w ∈ Λk and ei1 ∧ · · · ∧ eij . Now writing v =∑ni=1 viei, w =

∑j,ij

wijei1 ∧ · · · ∧ eij

11

and x =∑j,ij

xijei1 ∧ · · · ∧ eij we have

〈v ∧ w, x〉 =∑

m,nm,j,ij ,p

vpwijxmn〈ep ∧ ei1 ∧ · · · ∧ eij , en1 ∧ · · · ∧ enm〉

=∑

m,nm,j,ij ,p

vpwijxmn〈ei1 ∧ · · · ∧ eij , epxen1 ∧ · · · ∧ enm〉

= 〈w, vxx〉

as required.

Now to see that left multiplication by ei is orthogonal, we calculate

〈ei · v, w〉 = 〈ei ∧ v, w〉 − 〈eixv, w〉

= 〈v, eixw〉 − 〈v, ei ∧ w〉

= −〈v, ei · w〉

and so e∗i = −ei. However −ei · ei = ei · −ei = 1 so e−1i = −ei = e∗i and so ei is an orthogonal operator.

Remark. Everything we have done so far has been with vector spaces V . We wish to use our results on vector

bundles over a manifold, namely the vector bundles Cl(M) and Λ∗M . The defintion of Cl(M) is the vector

bundle whose fibres are Cl(M)x = Cl(T ∗M)x. The defintion of Λ∗M is the vector bundle whose fibres are

Λ∗Mx = Λ∗T ∗Mx. We denote by Cl0(M) and Cl1(M) the products of odd and even elements of Cl(M). We

have a representation of Cl(M)x on Λ∗Mx at each point, which we then extend to a bundle representation

by defining for ϕ ∈ Γ(Cl(M)), φ ∈ Γ(Λ∗M) that

(ϕ · φ)(x) = ϕ(x) · φ(x).

Similarly since the manifolds we are considering are Riemannian, we have from Lemma 1.1 a smoothly

varying inner product on T ∗M which we can use to create a smoothly varying inner product on Λ∗M via

(3).

12

1.3 Connections

Definition 1.8. A connection ∇ on a vector bundle E over M is a mapping ∇ : Γ(E)→ Γ(T ∗M⊗E) where

Γ(E) denotes smooth sections of E, such that

∇(fe) = df ⊗ e+ f∇e (5)

for all f ∈ C∞(M) and all e ∈ Γ(E).

Since ∇ maps into Γ(T ∗M ⊗E), given a vector field X and section σ we may compose ∇σ with X in its

left T ∗M slot to obtain a map ∇X : Γ(E) → Γ(C⊗ E), which we denote ∇Xσ. If we write c : C⊗ E → E

as c(a⊗ φ) = aφ then we may consider ∇X as a map from Γ(E) into itself via the compositions

Γ(E)∇−→ Γ(T ∗M × E)

Evaluate at X−−−−−−−−−→ Γ(C⊗ E)c−→ Γ(E),

or we may consider ∇ as a map

∇ : Γ(TM ⊗ E)→ Γ(E).

Property (5) can then be restated as

∇Xfe = df(X)e+ f∇Xe = Xf + f∇Xe.

It is a well known theorem [7, Theorem 3.6 chapter 2] that any manifold admits a unique connection over its

tangent bundle such that the torsion tensor T (V,W ) = ∇VW−∇WV −[V,W ] is identically zero and such that

g(V,W ) is constant whenever ∇VW ≡ 0. These two properties are called ‘symmetry’ and ‘compatibility’.

The Christoffel symbols Γkij for this connection, which are defined by ∇∂i∂j =∑nk=1 Γkij∂k, are

Γkij =

n∑r=1

1

2gkr (∂jgri + ∂igrj − ∂rgij) . (6)

This connection is known as the ‘Levi-Civita connection’. Given a connection ∇ on a vector bundle S, we

may extend ∇ to a connection on the tensor powers S⊗p, p 6= 0 by defining

∇(v1 ⊗ · · · ⊗ vp) = (∇v1)⊗ v2 ⊗ · · · ⊗ vp + · · ·+ v1 ⊗ · · · ⊗ vp−1 ⊗ (∇vp).

13

For p = 0 we make the definition ∇f = df . This formula indeed defines a connection since

∇fv1 ⊗ · · · ⊗ vp = (df ⊗∇v1 + f∇v1)⊗ v2 ⊗ · · · ⊗ vp + · · ·+ v1 ⊗ · · · ⊗ vp−1 ⊗ (df ⊗∇vp + f∇vp)

= df ⊗ [(∇v1)⊗ v2 ⊗ · · · ⊗ vp + · · ·+ v1 ⊗ · · · ⊗ vp−1 ⊗ (∇vp)] +

f [(∇v1)⊗ v2 ⊗ · · · ⊗ vp + · · ·+ v1 ⊗ · · · ⊗ vp−1 ⊗ (∇vp)]

= df ⊗∇v1 ⊗ · · · ⊗ vp + f∇v1 ⊗ · · · ⊗ vp,

and in the case p = 0 we have the product rule

∇fg = d(fg) = (df)g + f(dg) = df ⊗ g + f∇g.

By declaring ∇(a + b) = ∇a + ∇b when a and b are in differing tensor powers we may extend this to a

connection on the tensor algebra ⊕∞p=0S⊗p (where sequences in this direct sum are eventually zero). This

now descends to a connection on Λ∗S (which is the quotient of this space with the ideal J generated by

{v ⊗ v}) because it preserves J . That is, for v ⊗ v ∈ J , ∇(v ⊗ v) is also in J , which can be seen by the

calculation

∇X(v ⊗ v) = ∇Xv ⊗ v + v⊗ = (∇Xv + v)⊗ (∇Xv ⊗ v)− v ⊗ v −∇Xv ⊗∇Xv ∈ J.

Since ∇ preserves J , this connection satisfies

∇(a ∧ b) = ∇a ∧ b+ a ∧∇b. (7)

Similarly to above, given the metric on T ∗M , for any connection which is compatible (so g(V,W ) = 0

whenever ∇VW = 0), the induced connection on the tensor algebra of T ∗M descends to a connection on

14

Cl(M) because it too preserves the ideal J spanned by {v ⊗ v + 〈v, v〉1}, as can be seen by the calculation

∇X(v ⊗ v + g(v, v)1)

= ∇Xv ⊗ v + v ⊗∇Xv + dg(v, v)(X)1

= ∇Xv ⊗ v + v ⊗∇Xv + g(∇Xv, v)1 + g(v,∇Xv)1 a consequence of being compatible

= (v +∇Xv)⊗ (v +∇Xv) + g(v +∇Xv, v +∇Xv)1− (v ⊗ v + g(v, v)1)− (∇Xv ⊗∇Xv + g(∇Xv,∇Xv)1) ∈ J.

Usually we will be interested in the induced Levi-Civita connection on Cl(M).

Lemma 1.9. Let v ∈ Cl(M) and φ ∈ Λ∗. Then for the Levi-Civita connections on Cl(M) and Λ∗M

∇(v · φ) = (∇v) · φ+ v · (∇φ). (8)

Proof. Let {ei} be an orthonormal basis of T ∗M . Since {ei} generates Cl(M) it suffices to demonstrate (8)

for ei ∈ Cl(M) and since ∇(a+ b) = ∇a+∇b it suffices to demonstrate for φ = aei1 ∧ · · · ∧ eip . On the left

of (8) we have

∇(ei · φ) = ∇(ei ∧ φ− eixφ)

= (∇ei) ∧ φ+ ei ∧ (∇φ)−∇(eixφ) using (7)

so if we can show that

∇(eixφ) = (∇ei)xφ+ eix(∇φ) (9)

then we will be done. For compact notation we make the convention that if v = v1 ∧ · · · ∧ vp then vk =

v1 ∧ · · · ∧ vk ∧ · · · ∧ vp. Let ϕ = ei1 ∧ · · · ∧ eip so that φ = aϕ. Supposing that i = ij for some j, on the left

15

of (9) we have

∇(eixφ) = ∇(eixaei1 ∧ · · · ∧ eip)

= ∇(a

n∑l=1

(−1)l−1〈ei, eil〉ϕl

= (−1)j−1∇(aϕj).

For the below calculation we use the equation

∇(aϕ) = da⊗ ϕ+ a(∇ei1 ∧ · · · ∧ eip + · · ·+ ei1 ∧ · · · ∧ ∇eip)

to pick out the components (∇aϕ)l. On the right hand side of (9) we have

(∇ei)xφ+ eix(∇φ) = a∑l=1

(−1)l−1〈∇ei, eil〉ϕl +

n∑l=1

(−1)l−1〈e, (∇aφ)j〉∇aϕl

= −a∑l=1

(−1)l−1〈ei,∇eil〉ϕl +

n∑l=1

(−1)l−1〈e, (∇aφ)j〉∇aϕl using compatibility

= −a∑l=1

(−1)l−1〈ei,∇eil〉ϕl + a

n∑l=1

(−1)l−1〈e,∇eil〉ϕl +∇aϕj

= ∇aϕj = ∇(ei · φ)

as required. In the case that ei 6= eij for any j the same argument shows that both sides of (9) are zero and

so we are done.

1.3.1 Curvature

As the name of this project suggests, we will be very interested in results concerning curvature. In this

section we will briefly outline the definition of the different forms of curvature we will use in this report. We

will be interested in three main forms of curvature:

Definition 1.10. Let ∇ be a connection on a Riemannian vector bundle S → M . The map R : Γ(TM) ×

Γ(TM)→ End(Γ(S)) given as

RV,W = ∇V∇W −∇W∇V −∇[V,W ]

16

is called the curvature tensor for S. In particular, when S = Λ∗M and ∇ is the Levi-Civita Connection, we

call R the Riemann curvature tensor for M

Remark. The most important case for us will be when S = Λ∗M and ∇ is the Levi-Civita connection,

because its special properties give us information about the metric and geometry of M . Generically ∇ does

not carry any relation to the metric on S, and so will not tell us any decent information about the geometry

of M .

By taking V and W to be basis vectors ∂i and ∂j , one can see

R∂i,∂jX =(∇∂i∇∂j −∇∂j∇∂i

)X,

and so R can be thought of as how much differentiation fails to commute in different directions. As one

would expect, for Rn we have R ≡ 0, and so loosely we may think of R as measuring how ‘non-Euclidean’

a manifold is, or ‘how much it curves’. A perhaps surprising property of R is that despite the fact that

∇fX 6= f∇X in general, the maps x 7→ Rx,·, y 7→ R·,y and z 7→ R·,·z are all linear over functions. That is,

we have

RfX+gY,ZW = fRX,ZW + gRY,ZW

RX,fY+gZW = fRX,YW + gRX,ZW

RX,Y (fV + gW ) = fRX,Y V + gRX,YW.

It is immediate from the above definition that

RV,W = −RW,V and RV,V = 0 (10)

and for the Levi-Civita connection on Λ∗M , by playing with lots of simultaneous equations ([7] Chapter 4

Proposition 2.4) one can also show that

〈RV,WX,Y 〉 = 〈RX,Y V,W 〉 and 〈RV,WX,Y 〉 = −〈RV,WY,X〉. (11)

17

Definition 1.11. The map Ric : Γ(TM)→ Γ(TM) given as

Ric(X) = −n∑i=1

Rei,Xei

for {ei} a local orthonormal frame field and R the Riemannian curvature tensor is called the Ricci transform

of M . This then gives rise to a bilinear form

Ric(X,Y ) = 〈Ric(X), Y 〉

called the Ricci curvature tensor. From (11) this can be seen to be symmetric.

Note that unlike R which takes inputs from Γ(TM)×Γ(TM)×Γ(Λ∗M), Ric is defined only on Γ(TM)×

Γ(TM).

Definition 1.12. The map κ : M → R given by

κ = −n∑

i,j=1

〈Rei,ejei, ej〉

for {ei} a local orthonormal frame field is called the scalar curvature of M . This is just the trace of the map

X 7→ Ric(X,X).

We also have one last curvature the ‘sectional curvature’. This is the curvature that takes a 2 dimensional

subspace of TxM and measures how a basis of the subspace varies according to the Riemman curvature tensor:

Definition 1.13. Let x ∈M and σ ⊂ TxM be a 2-dimensional subspace spanned by a basis {X,Y }. Then

the sectional curvature of σ at x is given by

RX,Y Y,X

|X ∧ Y |2.

This definition is independent of chosen basis.

1.3.2 A Monstrous Curvature Calculation

It can be shown [2, Chapter 4 equation 4.1.13] that locally every connection on Cl(M) is of the form∇ = d+ω

where d is the exterior derivative and ω is a matrix of one forms. One can deduce that if ∇ is compatible

with the metric on Cl(M) that ω must be skew symmetric, that is, ωT = −ω. We have seen earlier in this

18

section that if we start with a connection on T ∗M then there is a standard way of lifting it to Cl(M). In

[3, section ii chapter 4] we learn that if locally ∇T∗M = d + ω then choosing an orthonormal frame {ei} of

T ∗M and setting ωij = 〈ωei, ej〉, the lifted connection on Cl(M) is given locally by ∇Cl(M) = d+ ω where

ωφ =1

2

∑i<j

ωij [eiej , φ]. (12)

We can see that these two connections agree on T ∗M since

ωek =1

2

∑i<j

ωij [eiej , ek] =1

2

∑i<j

ωij(2δikej − 2δjkei)

=∑k<j

ωkjej −1

2

∑i<k

ωikei

=∑k<j

ωkjej −1

2

∑j<k

ωjkej swapping i with j in the second sum.

Since ω is skew symmetric, we obtain ωij = −ωji, so in particular ωii = 0 for all i. This allows us to

change the index of the right sum from j < k to j ≤ k since we are only adding a zero term. Resuming the

calculation we have

ωek =∑k<j

ωkjej +∑j≤k

ωkjej since ωjk = −ωkj

=

n∑j=1

ωkjej

=

n∑j=1

〈ωek, ej〉ej = ωek.

19

We will now refer to ω just as ω. We can use the form ∇ = d+ω of the connection to compute the curvature:

∇V∇Wϕ−∇W∇V ϕ−∇[V,W ]ϕ = ∇V (Wϕ+ ω(W )ϕ)−∇W (V ϕ+ ω(V )ϕ)− [V,W ]ϕ− ω([V,W ])ϕ

= VWϕ+ (V ω(W ))ϕ+ ω(W )V ϕ+ ω(V )Wϕ+ ω(V )ω(W )ϕ−WV ϕ− (Wω(V ))ϕ

− ω(V )Wϕ− ω(W )V ϕ− ω(W )ω(V )ϕ− [V,W ]ϕ− ω([V,W ])ϕ

= (V ω(W ))ϕ− (Wω(V ))ϕ− ω([V,W ])ϕ+ ω(V )ω(W )ϕ− ω(W )ω(V )ϕ

= d(ω)(V ∧W )ϕ+ ω(V )ω(W )ϕ− ω(W )ω(V )ϕ. (13)

We will now use (13) to prove the following identity for the curvature on the clifford bundle:

Lemma 1.14. Using RCl and RT∗M to denote the curvature operators on Cl(M) and T ∗M , if ∇ is metric

compatible then we may write RCl as

RClV,Wϕ =

1

2

∑i<j

〈RT∗M

V,W ei, ej〉[eiej , ϕ] (14)

where {ei} is an orthnormal local frame.

Proof. Lemma 1.14 features in [3, Chapter 4], our proof is a very long unpacking and calculation to show it.

This calculation will be quite long, so we will break it down into component parts one by one. Fix {ei} an

orthonormal basis of T ∗M . Recalling (12) we may write

ω(V )ω(W ) =1

4

∑i<j,k<l

ωij(V )ωkl(W )[eiej , [ekel, ϕ]]. (15)

By expanding the commutators above and using the product rule for commutators we can see that

[eiej , [ekel, ϕ]] = [eiej , ekelϕ]− [eiej , ϕekel]

= ekel[eiej , ϕ] + [eiej , ekel]ϕ− ϕ[eiej , ekel]− [eiej , ϕ]ekel

= [ekel, [eiej , ϕ]] + [[eiej , ekel], ϕ].

20

Inserting this into (15) we obtain

ω(V )ω(W )ϕ =1

4

∑i<j,k<l

ωij(V )ωkl(W )[[eiej , ekel], ϕ] +1

4

∑i<j,k<l

ωij(V )ωkl(W )[ekel, [eiej , ϕ]]

=1

4

∑i<j,k<l

ωij(V )ωkl(W )[[eiej , ekel], ϕ] + ω(W )ω(V )ϕ

=⇒ ω(V )ω(W )− ω(W )ω(V ) =1

4

∑i<j,k<l

ωij(V )ωkl(W )[[eiej , ekel], ϕ]. (16)

We now wish to work with the right hand side of (16). Using the anticommuting properties of {ei} one can

see that

[eiej , ek] = 2δikej − 2δjkei, (17)

which in turn gives

[eiek, ekel] = ek[eiej , el] + [eiej , ek]el = 2δilekej − 2δjlekei + 2δikejel − 2δjkeiel. (18)

We then insert this into (16) to obtain

ω(V )ω(W )ϕ− ω(W )ω(V )ϕ =1

4

∑i<j,k<l

ωij(V )ω(W )kl[[eiej , ekel], ϕ]

=1

2

( ∑l<j,k<l

ωlj(V )ωkl(W )[ekej , ϕ]−∑

i<l,k<l

ωil(V )ωkl(W )[ekei, ϕ]

+∑

k<j,k<l

ωkj(V )ωkl(W )[ejel, ϕ]−∑

i<k,k<l

ωik(V )ωkl(W )[eiel, ϕ]

)

=1

2

( ∑l<j,k<l

ωlj(V )ωkl(W )[ekej , ϕ]−∑

j<l,k<l

ωjl(V )ωkl(W )[ekej , ϕ]

+∑

k<j,k<l

ωkj(V )ωkl(W )[ejel, ϕ]−∑

j<k,k<l

ωjk(V )ωkl(W )[ejel, ϕ]

).

=1

2

( ∑l<j,k<l

ωlj(V )ωkl(W )[ekej , ϕ] +∑

j≤uul,k<l

ωlj(V )ωkl(W )[ekej , ϕ]

+∑

k<j,k<l

ωkj(V )ωkl(W )[ejel, ϕ] +∑

j≤k,k<l

ωkj(V )ωkl(W )[ejel, ϕ]

)

21

Note that in the second and fourth sums we have changed the inequality in the index to a greater than or

equal to since ωjj = 0 and we are adding only a zero term. Having written this we can see that the first and

second sums are the same quantity being summed over the mutually exlusive indices l < j and j ≤ l. We

may combine them into the one sum over all j and l and do the same for the third and fourth sums, so we

continue

ω(V )ω(W )ϕ− ω(W )ω(V )ϕ =1

2

n∑j,l=1

∑k<l

ωlj(V )ωkl(W )[ekej , ϕ] +1

2

n∑j,k=1

∑k<l

ωkj(V )ωkl(W )[ejel, ϕ]

=1

2

n∑j,l=1

∑k<l

ωlj(V )ωkl(W )[ekej , ϕ] +1

2

n∑j,l=1

∑l<k

ωlj(V )ωlk(W )[ejek, ϕ]

=1

2

n∑j,l=1

∑k<l

ωlj(V )ωkl(W )[ekej , ϕ] +1

2

n∑j,l=1

∑l<k

ωlj(V )ωkl(W )[ekej , ϕ] (19)

=1

2

n∑j,k,l=1

ωlj(V )ωkl(W )[ekej , ϕ]. (20)

In line (19) we have used that ωkl = −ωlk, and the fact that ejek = −ekej when j 6= k. One may note that

j and k can be equal in the sum where we have interchanged ejek with −ekej , however in this case, we have

[ekej , φ] = [e2k, ϕ] = −[1, ϕ] = 0,

so we are justified in carrying the minus sign. Now we will calulate some portion of the right hand side of

(14). Using the identity (17), the antisymmetry of ω and by relabelling indices as in the calculations above

22

we obtain

1

2

∑i<j

〈ω(V )ω(W )ei, ej〉[eiej , ϕ] =1

8

∑i<j,k<l,s<t

ωkl(V )ωst(W )〈[ekel, [eset, ei]], ej〉[eiej , ϕ]

=1

4

n∑t=1

∑i<j,k<l

ωkl(V )ωit(W )〈[ekel, et], ej〉[eiej , ϕ]

=1

2

n∑t,l=1

∑i<j

ωtl(V )ωit(W )〈el, ej〉[eiej , ϕ]

=1

2

n∑t=1

∑i<j

ωtj(V )ωit(W )[eiej , ϕ]

=1

2

n∑l=1

∑k<j

ωlj(V )ωkl(W )[ekej , ϕ]. (21)

Combining (21) and (20) then gives us

1

2

∑i<j

〈ω(V )ω(W )ei − ω(W )ω(V )ei, ej〉[eiej , ϕ] =1

2

n∑l=1

∑k<j

(ωlj(V )ωkl(W )− ωlj(W )ωkl(V )

)[ekej , ϕ]

=1

2

n∑l,k,j=1

ωlj(V )ωkl(W )[ekej , ϕ]

= ω(V )ω(W )ϕ− ω(W )ω(V )ϕ. (22)

Lastly we make the calculations

1

2

∑i<j

〈V ω(W )ei, ej〉[ei, ej , ϕ] =1

4

∑i<j,k<l

V ωkl(W )〈[ekel, ei], ej〉[ei, ej , ϕ]

=1

2

n∑l=1

∑i<j

V ωil(W )〈el, ej〉[ei, ej , ϕ]

=1

2

∑i<j

V ωij(W )[ei, ej , ϕ]

= V ω(W )ϕ (23)

23

and

1

2

∑i<j

〈ω([V,W ])ei, ej〉[eiej , ϕ] =1

4

∑i<j,k<l

ωkl([V,W ])〈[ekel, ei], ej〉[ei, ej , ϕ]

=1

2

∑i<j

ωij([V,W ])[ei, ej , ϕ]

= ω([V,W ])ϕ. (24)

To complete the proof we then combine (23), (24) and (22) to obtain

1

2

∑i<j

〈RV,W ei, ej〉[eiej , ϕ] =1

2

∑i<j

〈V ω(W )ei −Wω(V )ei − ω([V,W ])ei + ω(V )ω(W )ei − ω(W )ω(V )ei, ej〉[eiej , ϕ]

= V ω(W )ϕ−Wω(V )ϕ− ω([V,W ])ϕ+ ω(V )ω(W )ϕ− ω(W )ω(V )ϕ

= RV,Wϕ

as required.

1.4 The Dirac Operator

Definition 1.15. A Dirac bundle is a vector bundle S →M with a bundle metric, a connection on S and a

left representation of Cl(M) such that left multiplication by orthonormal elements of Cl(M) is an orthogonal

map and such that the connection ∇ obeys the product rule

∇e · σ = (∇e) · σ + e · (∇σ)

for e ∈ Cl(M) and σ ∈ S, where ∇e refers to the induced Levi-Civita on Cl(M).

Example 1.16. Proposition 1.6 together with Lemmas 1.7 and 1.9 show that the exterior bundle Λ∗M with

the Levi-Civita connection and the exterior representation of Cl(M) on Λ∗M is a Dirac bundle. This bundle

will be of primary interest for us.

Definition 1.17. Let M be Riemannian manifold with Clifford bundle Cl(M) and let S be a Dirac bundle

24

over M with connection ∇. Then the first order differential operator D : Γ(S)→ Γ(S) given by

Dσ =

n∑i=1

ej · ∇ejσ

where {ei} is an orthonormal basis of TM , is called the Dirac Operator of S. The operator D2 is called

the Dirac Laplacian.

Remark. Note that we have written ej above to mean both a basis of TM and its corresponding dual basis

of T ∗M . The Dirac operator of S depends on the vector bundle S over M , along with choice of connection

∇ on S, and choice of representation of Cl(M) on S (that is, how the multiplication ei· is defined). We will

often refer to ‘The’ Dirac operator of S with the choice of these other items being implicit. We may also

define a right Dirac operator by multiplying on the right instead of left. Of particular interest to us will

be the Dirac operator arising from the Levi-Civita connection on Λ∗(M) and the exterior representation of

Cl(M) on Λ∗(M).

Proposition 1.18. The Dirac operator D is independent of choice of orthonormal basis.

Proof. Let {ei} and {fi} both be orthonormal bases of TM . Then

n∑i=1

ei · ∇eiσ =

n∑i,j,k=1

(〈ei, fj〉fj) ·(∇〈ei,fk〉fkσ

)expressing ei in the basis {fi}

=

n∑i,j,k=1

〈ei, fj〉〈ei, fk〉fj · ∇fkσ using linearity of left multiplication and ∇

=

n∑i,j,k=1

〈〈ei, fj〉ei, fk〉fj · ∇fkσ

=

n∑j,k=1

〈fj , fk〉fj · ∇fkσ =

n∑k=1

fk · ∇fkσ.

We may lift the metric on the Dirac bundle S to an inner product on Γ(S) by defining for a, b ∈ Γ(S)

〈a, b〉 :=

∫M

〈a, b〉.

We wish to use some Hilbert space techniques in the analysis of D, so we take the completion of Γ(S) under

25

this inner product to make Γ(S) into a Hilbert space which we denote L2(S, g). With respect to this metric

we have the following proposition.

Proposition 1.19. Dirac operator is symmetric on L2(S, g).

Proof. Take {ei} to be parallel around x ∈M . We calculate at the point x.

〈Da, b〉L2 =

∫M

〈Da, b〉x =

∫M

n∑i=1

〈ei · ∇eia, b〉

= −∫M

n∑i=1

〈∇e1a, ei · b〉

= −∫M

n∑i=1

ei〈a, ei · b〉 −∫M

〈a,∇e1ei · b〉

= −∫M

n∑i=1

ei〈a, ei · b〉 −∫M

〈a,∇e1ei · b〉

= −∫M

n∑i=1

ei〈a, ei · b〉 −∫M

〈a, ei · ∇e1b〉 −∫M

〈a, (∇eiei) · b〉

=

∫M

div(V ) +

∫M

〈a,Db〉

where V is the section of T ∗M defined implicitly by

〈V,W 〉 = −〈a,W · b〉

26

For all sections W of T ∗M . The last line in the above calculation can be seen by the following

div(V ) =

n∑j=1

〈∇ejV, ej〉

=

n∑j=1

ej〈V, ej〉 − 〈V,∇; aejej〉

=

n∑j=1

ej〈V, ej〉

= −n∑j=1

ej〈a, ej · b〉.

Now the divergence theorem will turn the divergence term into an integral over the boundary of M , which

is zero since M has empty boundary and we are done. If we allow M to have boundary then provided either

a or b are compactly supported, V will be the zero vector field on ∂M and the same result will hold.

Corollary 1.20. The Dirac operator D is essentially self-adjoint on Γ(S).

Proof. This is due to [4, Corollary 10.2.6] which states that any symmetric differential operator on a compact

manifold without boundary is essentially self-adjoint.

Example 1.21. We now write down explicitly the Dirac operator on R2 with the usual metric gij = δij for

the exterior representation of Cl(R2) on Λ2(R2) with ∇ the Levi-Civita connection on R2.

On R2, the cotangent and tangent bundle are both the trivial bundle R2 × R2. Picking the standard

basis {e1, e2} of R2 gives a global orthonormal frame field of TM . That is we obtain two smooth sections

x 7→ e1, x 7→ e2 of TM which form an orthonormal basis in each fibre. This basis coincides exactly with

the basis {∂1, ∂2} of TM obtained by using the global identity chart. We use the same two sections to

obtain a global orthonormal frame field of T ∗M and which correspond precisely to the dual basis {dx1, dx2}

obtained from the identity chart. The Clifford algebra Cl(T ∗R2, g) = Cl(R2, g) is the algebra generated by

{1, dx1, dx2, dx1dx2} subject to the relations

(dx1)2 = (dx2)2 = −1 dx1dx2 = −dx2dx1.

27

These relations give us an algebra isomorphism from Cl(R2) to H via

1 7→ 1 dx1 7→ i dx2 7→ j dx1dx2 7→ k.

We will write 1, i, j, k for the basis of Cl(R2) and 1, dx1, dx2, dx1 ∧ dx2 for the corresponding basis of Λ∗R2.

The canonical isomorphism between Cl(R2) and Λ2R2 can then be written as

1 7→ 1 i 7→ dx1 j 7→ dx2 k 7→ dx1 ∧ dx2.

Now sitting down and writing what i, j and k do to each basis vector 1, dx1, dx2 and dx1 ∧ dx2 under the

representation we obtain by going over to the clifford algebra, left multiplying and coming back, we obtain

the following matrix representations with respect to this basis:

1 =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

i =

0 −1 0 0

1 0 0 0

0 0 0 −1

0 0 1 0

j =

0 0 −1 0

0 0 0 1

1 0 0 0

0 −1 0 0

k =

0 0 0 −1

0 0 −1 0

0 1 0 0

1 0 0 0

.

Using (6) we find that the Christoffel symbols of the Levi Civita connection on R2 all vanish and so ∇∂idxj =

0. If X = Xi∂i and Y = Y idxi then the connection on T ∗M is given by

∇XY = Xi(∂iYj)dxj .

28

This tells us that the induced connection on the second exterior power is

∇Xfdx1 ∧ dx2 = [X1(∂1f) +X2(∂2f)]dx1 ∧ dx2 + fXi[(∇∂idx1) ∧ dx2 + dx1 ∧ (∇∂idx2)]

= [X1(∂1f) +X2(∂2f)]dx1 ∧ dx2.

Writing σ = σ11 + σ2dx1 + σ3dx2 + σ4dx1 ∧ dx2 for σ a section of Λ∗R2, the induced connection on Λ∗R2 is

then

∇Xσ = (X1∂1σ1+X2∂2σ1)1+(X1∂1σ

2+X2∂2σ2)dx1+(X1∂1σ3+X2∂2σ3)dx2+(X1∂1σ

4+X2∂2σ4)dx1∧dx2

Finally we are ready to write for D the Dirac operator on this bundle that

Dσ = i∇∂1σ + j∇∂2σ

= i[(∂1σ

1)1 + (∂1σ2)dx1 + (∂1σ

3)dx2 + (∂1σ4)dx1 ∧ dx2

]+ j

[(∂2σ

1)1 + (∂2σ2)dx1 + (∂2σ

3)dx2 + (∂2σ4)dx1 ∧ dx2

]= (∂1σ

1)dx1 − (∂1σ2)1 + (∂1σ

3)dx1 ∧ dx2 − (∂1σ4)dx2 + (∂2σ

1)dx2 − (∂2σ2)dx1 ∧ dx2 − (∂2σ

3)1 + (∂2σ4)dx1

= −[(∂1σ2) + (∂2σ

3)]1 + [(∂1σ1) + (∂2σ

4)]dx1 + [−(∂1σ4) + (∂2σ

1)]dx2 + [(∂1σ3)− (∂2σ

2)]dx1 ∧ dx2.

If we then use the above to write down the matrix of the Dirac operator with respect to the basis {1, dx1, dx2, dx1∧

dx2} we obtain

D =

0 −∂1 −∂2 0

∂1 0 0 ∂2

∂2 0 0 −∂1

0 −∂2 ∂1 0

.

Recall that Λ∗M has a first order differential operator d : Γ(Λ∗M) → Γ(Λ∗M) called the exterior

derivative, and its L2(M, g) adjoint d∗ is given by

d∗ = (−1)np+n+1 ∗ d∗

on ΛpM where ∗ : ΛpM → Λn−pM is the Hodge * operator. The Hodge * operator takes φ ∈ ΛpM to the

29

element ∗φ ∈ Λn−pM such that the linear operator ψ 7→ ψ∧∗φ on ΛnM is given by 〈ψ, φ〉e1 ∧ · · · ∧ en. Note

that the Hodge * operator depends up to a minus sign on a choice of orientation.

Lemma 1.22. Fix x ∈ M and choose an orthonormal frame field {e1, . . . , en} in an neighbourhood U of x

such that ∇eiej = 0, for ∇ the Levi-Civita connection on M . The operators d and d∗ are given in U by the

formulas

d =

n∑j=1

ej ∧∇ej (25)

d∗ = −n∑j=1

ejx∇ej . (26)

Proof. Proposition 1.18 tells us that the above expressions are independent of choice of orthonormal frame

field. Using [1, Theorem 2.5.1] to prove (25) we need only to show that the right hand side satisfies the

following axioms for d

(i) d2φ = 0

(ii) d(ϕ ∧ φ) = dϕ ∧ φ+ (−1)pϕ ∧ dφ

(iii) df =grad(f)

for all smooth functions f , p-forms ϕ and q-forms φ. Property (iii) is immediate from our definition of the

extension of ∇ to Λ∗M . Using equation (7), property (ii) can be seen by the calculation

n∑i=1

ei ∧∇ei(ϕ ∧ φ) =

n∑i=1

ei ∧ ((∇eiϕ) ∧ φ+ ϕ ∧∇eiφ)

=

(n∑i=1

ei ∧∇ei

)∧ φ+ (−1)pϕ ∧

(n∑i=1

ei ∧∇eiφ

).

Since any ϕ ∈ Λ∗M can be written as ∑ij ,j

ϕijei1 ∧ · · · ∧ eij

where each ϕij is a smooth function, it suffices to show (i) on elements of the form ϕσ where σ = ei1∧· · ·∧eij .

30

Since {ei} is parallel,

∇eiσ = ∇eiei1 ∧ ei2 ∧ · · · ∧ eij + · · ·+ ei1 ∧ · · · ∧ eik−1∧∇eieik = 0,

and so we compute

n∑i,j=1

ej ∧∇ej (ei ∧∇eiϕσ) =

n∑i,j=1

ej ∧∇ej (ei ∧ [eiϕσ + ϕ∇eiσ])

=

n∑i,j=1

ej ∧∇ej ((eiϕ)ei ∧ σ)

=

n∑i,j=1

ej ∧([ej(eiϕ)ei + (eiϕ)∇ejei

]∧ σ + (eiϕ)ei ∧∇ejσ

)=

n∑i,j=1

(ejeiϕ)ej ∧ ei

Since we are using the Levi-Civita connection, ∇VW − ∇WV = [V,W ]. Letting V and W be our parallel

vector fields ei and ej , this tells us that eiej = ejei. Now by swapping indices and using antisymmetry we

have

n∑i,j=1

(ejeiϕ)ej ∧ ei =

n∑j,i=1

(eiejϕ)ei ∧ ej

= −n∑

j,i=1

(ejeiϕ)ej ∧ ei

=⇒ 0 =

n∑i,j=1

(ejeiϕ)ej ∧ ei =

n∑i,j=1

ej ∧∇ej (ei ∧∇eiϕσ) .

Now for the second statement of the proposition, we make the required calculation on a single basis element

ϕe1 ∧ · · · ∧ ep for simplicity. With a bit of counting it can be seen that ∗(e1 ∧ · · · ∧ ep) = ep+1 ∧ · · · ∧ en and

that for j ≤ p,

∗(ej ∧ ep+1 ∧ · · · ∧ en) = (−1)(n−p)(p+1)+j−1e1 ∧ · · · ∧ ej ∧ · · · ∧ ep.

Since n(p + 1) ≡ (n − p)(p + 1) mod 2, we may rewrite this as (−1)n(p+1)+j−1e1 ∧ · · · ∧ ej ∧ · · · ∧ ep. Now

31

finally we calculate

∗(d ∗ (ϕe1 ∧ · · · ∧ ep)) = ∗(d(ϕep+1 ∧ · · · ∧ en))

= ∗(n∑j=1

ej ∧∇ej (ϕep+1 ∧ · · · ∧ en))

=

n∑j=1

∗(ej ∧

[(ejϕ)ep+1 ∧ · · · ∧ en + ϕ∇ejep+1 ∧ · · · ∧ en

])=

p∑j=1

(ejϕ) ∗ (ej ∧ ep+1 ∧ · · · ∧ en)

=

p∑j=1

(ejϕ)(−1)n(p+1)+j−1e1 ∧ · · · ∧ ej ∧ · · · ∧ ep

= (−1)n(p+1)n∑j=1

ejx((ejϕ)e1 ∧ · · · ∧ ep)

= (−1)n(p+1)n∑j=1

ejx∇ej (e1 ∧ · · · ∧ ep)

and so multiplying by (−1)n(p+1)+1 on both sides gives

d∗ = −n∑j=1

ejx∇ej

as required.

Now Lemmas 1.3 and 1.22 together give us the following corollary

Corollary 1.23. The Dirac operator for the Levi-Civita connection over the exterior bundle Λ∗M with the

exterior representation of Cl(M) can be written as

D = d+ d∗.

The right Dirac operator D for the same bundle can be written as

D = (−1)p(d− d∗).

We will refer to the operator above more succinctly as the Hodge-de Rham Dirac operator for M . This

32

gives us the nice property that D2 = ∆ where ∆ = dd∗ + d∗d is the Hodge Laplacian on M .

Proposition 1.24. If D is the Hodge-de Rham operator for M and M has no boundary, then

ker(D) = ker(∆)

Proof. Recall from proposition 1.19 that when M has no boundary D is symmetric. Clearly since ∆ = D2,

ker(D) ⊆ ker(∆), so suppose ∆α = 0. Then

0 = 〈∆α, α〉 = 〈D2α, α〉

= 〈Dα,Dα〉 =⇒ Dα = 0

and we are done.

Lemma 1.25. For {ei} a local orthonormal frame field we have the formulas

∑i<j

eiejRei,ek(ϕ) =∑i<j

Rei,ej (ϕ)eiej (27)

∑i<j

eiRei,ej (ϕ)ej = 0 (28)

(29)

and as a consequence of (27) we have

∑i<j

eiejRei,ej (ϕ) =1

2

∑i<j

[eiej , Rei,ej ]. (30)

33

Proof. We choose local parallel coordinates so that ∇eiej = 0 and calculate

D2ϕ =

n∑i,j=1

ei∇ei(ej∇ej (ϕ)) =

n∑i,j=1

ei[∇eiej · (∇ej (ϕ)) + ej∇ei∇ej (ϕ)

]=

n∑i,j=1

eiej∇ei∇ejϕ

= −n∑i=1

∇ei∇eiϕ+∑i<j

eiej(∇ei∇ej −∇ej∇ei)ϕ

= −n∑i=1

∇ei∇eiϕ+∑i<j

eiejRei,ejϕ

where in the last line we have used the torsion free property of the Levi-Civita connection

[ei, ej ] = ∇eiej −∇ejei = 0.

Similarly we obtain for D the right Dirac operator that

D2 = −n∑i=1

∇ei∇ej −∑i<j

Rei,ej (ϕ)eiej

and so since D2 = dd∗ + d∗d = D2 (note that when squaring D the (−1)p becomes a (−1)p+1 in the second

iteration) we obtain (27). Simlarly (28) follows from the equations

DDϕ =∑i,j=1

ei(∇ei∇ejϕ)ej

DDϕ =∑i,j=1

ei(∇ej∇eiϕ)ei

and the fact that DD = (−1)p(dd∗ − d∗d) = DD.

34

2 Topology

2.1 de Rham Cohomology

The definition of the singular homology groups of a topological space requires a fair bit of background and

will not be discussed here, our starting point will be the theorem of de Rham that the singular and de Rham

cohomology groups for a compact manifold are the same, or in the language of cohomology,

H∗sing(M ;Z)⊗Z R ∼= H∗dR(M).

This section will mostly be quoting a list of facts that we will need in the rest of this report.

Definition 2.1. Let M be a manifold and dp denote the the exterior derivative on ΛpM . The p-th de Rham

cohomology group is the space

Hp(M) = Ker(dp)/ Im(dp−1).

The p-th Betti number bp(M) is the dimension of the p-th singular cohomology group Hp(M ;R). The

space of ‘harmonic p-forms’ is the vector space Hp of φ ∈ ΛpM such that ∆φ = 0. We call the direct sum

⊕dim(M)p=0 Hp of each space of harmonic p-forms H.

Remark. We make the convention that d−1 ≡ 0 so that H0 = Ker(d0). Since d0f can be written as a sum

of partial derivatives of f with respect to coordinates, we see that H0 consists of functions constant on each

connected component of M , and so that for M compact b0(M) is the number of connected components of

M . In particular, for a compact and connected manifold b0(M) = 1. Since dp ≡ 0 for p > dim(M), the only

non-trivial de Rham cohomology groups are when 0 ≤ p ≤ dim(M).

[8, proposition 5.3.1] tells us that when M is compact, Hp is finite dimensional.

Theorem 2.2 (Hodge Decomposition theorem). Let M be a compact manifold. Then there is an orthogonal

decomposition

Γ(Λ∗M) = H⊕ Im(d)⊕ Im(d∗),

in particular there is an isomorphism

Hp ∼= Hp(M).

35

Theorem 2.3 (Poincare Duality). If M is a compact manifold of dimension n then

Hp(M) ∼= Hn−p(M)

for all p.

Theorem 2.3 is the main focus of chapter 1 of [8] where it is proved in detail.

2.2 Euler Characteristic

Definition 2.4. The Euler characteristic χ(M) of a topological space is the sum

χ(M) =

n∑p=0

(−1)pbp(M).

The Euler characteristic is a homotopy invariant. One can see straight away from Poincare duality that if

M is a compact manifold of odd dimension that χ(M) = 0 and if M has even dimension n that χ(M) =

2b0(M) + (−1)n/2bn/2(M).

Now we look at our first result concerning links between geometry and topology. Let S →M be a Dirac

bundle. A Z2-grading for S means a parallel decomposition

S = S0 ⊕ S1

so that Cli(M) · Sj ⊆ Si+j for all i, j ∈ Z2. Since ∇ preserves these bundles and multiplication by elements

in Cl1(M) reverses the bundles, we may write

D =

0 D1

D0 0

with respect to the grading on S where Di : Γ(Si) → Γ(Si+1) (indices are mod 2). Since D is self-adjoint

we can see that D0 and D1 are adjoints of one another.

Definition 2.5. A bounded linear operator T is Fredholm if its kernel and and cokernel are both finite

dimensional. The index of T is the integer

index(T ) = dim(ker(T ))− dim coker(T ).

36

To speak of D being a Fredholm operator, we first must restrict its domain to an appropriate Sobolev

space on which it is bounded, then appeal to elliptic theory to relate this back to L2(M, g), however this is

not something of particular concern in this document.

We can then compute the index of D0 as

index(D0) = dim(ker(D0))− dim(ker(D1))

and similarly for D1. From the decomposition of D into D0 and D1 we can see that ker(D) = ker(D0) ⊕

ker(D1) and so if D is injective then index(Di) = 0.

Example 2.6. Consider the Hodge-de Rham operator d + d∗ for M on Λ∗M , with the usual grading

Λ∗M = ΛevenM ⊕ ΛoddM := Λ0M ⊕ Λ1M . It’s evident from the definition of ∇ on Λ∗M that this grading

is parallel, and since e· = e ∧ · − ex· maps elements of Λ∗M up and down one exterior power, products of

odd elements of Cl(M) map Λ0M into Λ1M and vice versa, while products of even elements map Λ1M and

Λ0M into themselves. That is, Cli(M) · ΛjM ⊆ Λj+i, and so we have a Z2 grading. Using Proposition 1.24

the kernel of (d+ d∗)0 is (dd∗ + d∗d)0 = Heven and so the index index(D0) is

index(D0) = ker(D0)− ker(D1) = Heven −Hodd = χ(M).

This gives us an interesting relationship between the geometry of M (D depends on the metric) and the

topology of M . Through some development of the above result, one can deduce as a special case the

Gauss-Bonnet theorem:

Theorem 2.7. If M is a compact 2-manifold then

∫M

κdg = χ(M)

for any metric g on M where K is the scalar curvature.

We can deduce then that if M has positive scalar curvature that then the Euler characteristic must be

positive. Our main results will concern some theorems in this spirit that link topology to positive curvature.

37

3 Bochner’s method and Betti numbers

Consider M equipped with a connection ∇ on a Riemannian vector bundle E. Let ∇LC be the Levi-Civita

connection on TM . We may define the map ∇2V,W : Γ(E)→ Γ(E) by

∇2V,W = ∇V∇W −∇∇TM

V W

and think about the map ∇2.,. point-wise as a bilinear form on the tangent bundle with values in E. The

Connection Laplacian ∇∗∇ : Γ(E)→ Γ(E) is defined by taking the trace as

∇∗∇ϕ = −trace(∇2.,.ϕ),

or in local coordinates

∇∗∇ϕ = −n∑j=1

∇2ej ,ejϕ

for {ei} a local orthonormal frame field. Note that we have used the rather suggestive notation ∇∗∇. We

will now prove that this operator really is ∇ composed with it’s adjoint.

Proposition 3.1. The operator ∇∗∇ is non-negative and symmetric, and hence is essentially self-adjoint.

In particular,

〈∇∗∇ϕ,ψ〉 = 〈∇ϕ,∇ψ〉

for all ϕ,ψ ∈ Γ(E). If M has boundary then the statement is true provided that one of ϕ or ψ has compact

support.

Proof. Fix x ∈ M and choose local parallel and orthonormal coordinates {ei} around x. We have then

around x that

〈∇∗∇ϕ,ψ〉L2 =

∫M

〈∇∗∇ϕ,ψ〉x = −∫M

n∑j=1

〈∇ej∇ejϕ,ψ〉

= −∫M

n∑j=1

[ej〈∇ejϕ,ψ〉 − 〈∇ejϕ,∇ejψ〉

]= −

∫M

div(V ) +

∫M

〈∇ϕ,∇ψ〉

38

where V is the vector field defined implicitly by 〈V,W 〉 = 〈∇WV ϕ, ψ〉. This last line is established by

div(V ) =

n∑j=1

〈∇ejV, ej〉

=

n∑j=1

ej〈V, ej〉 − 〈V,∇ejej〉

=

n∑j=1

ej〈V, ej〉 =

n∑j=1

ej〈∇ejϕ,ψ〉.

Now since one of ϕ and ψ has compact support, one of φ and ψ is zero on ∂M and so 〈V,W 〉 = 〈∇WV ϕ, ψ〉 = 0

for all W defines V as zero on ∂M and so∫M

div(V ) =∫∂M

V = 0. Now that ∇∗∇ is self-adjoint follows

again from [4, Corollary 10.2.6] as in Corollary 1.20.

Now suppose that S →M is a Dirac bundle. We define the operator R : Γ(S)→ End(S) by

R(ϕ) =1

2

n∑j,k=1

ejekRej ,ek(ϕ)

where {ei} is any local orthonormal frame around the point in question and R.,. is the curvature tensor on

S. Note that we have the property

∇2V,W −∇2

W,V = ∇V∇W −∇W∇V −∇∇TMV W +∇∇TM

W V

= ∇V∇W −∇W∇W +∇∇TMW V−∇TM

V W

= ∇V∇W −∇W∇V −∇[V,W ] = RV,W (31)

where we have used the torsion free property of the Levi-Civita connection ∇TMV W − ∇TMW V = [V,W ] in

39

the last line. We have the alternate form of R:

R =1

2

n∑j,k=1

ejekRej ,ek =1

2

∑j<k

ejekRej ,ek +1

2

∑j>k

ejekRej ,ek

=1

2

∑j<k

ejekRej ,ek +1

2

∑j<k

ekejRek,ej

=1

2

∑j<k

ejekRej ,ek −−1

2

∑j<k

ejekRej ,ek

=∑j<k

ejekRej ,ek .

Theorem 3.2 (The general Bochner identity). Let D be the Dirac operator for a Dirac bundle S over M

and ∇∗∇ the connection Laplacian. Then

D2 = ∇∗∇+ R (32)

Proof. Fix x ∈ M and choose parallel and orthonormal coordinates {ei}. In these coordinates we have

40

∇2ei,ej = ∇ei∇ej , and we calculate

D2 =

n∑j,k=1

ej∇ej (ek∇)

=

n∑j,k=1

ej((∇ejek)∇ek + ek(∇ej∇ek))

=

n∑j,k=1

ejek∇ej∇ek =

n∑j,k=1

ejek∇2ej ,ek

= −n∑j=1

∇2ej ,ej +

n∑j<k

ejek∇2ej ,ek

+

n∑j>k

ejek∇2ej ,ek

= ∇∗∇+

n∑j<k

ejek∇2ej ,ek

+

n∑j<k

ekej∇2ek,ej

= ∇∗∇+

n∑j<k

ejek∇2ej ,ek

−n∑j<k

ejek∇2ek,ej

= ∇∗∇+

n∑j<k

ejek

(∇2ej ,ek

−∇2ek,ej

)= ∇∗∇+

n∑j<k

ejekRek,ej by (31)

= ∇∗∇+ R

3.1 Positive curvature and the cotangent bundle

Recall that the Ricci curvature of M is defined by

Ric(ϕ) = −n∑j=1

Rej ,ϕ(ej)

where {ej} is an orthonormal local frame field. The following corollary says that in the case of the Hodge-de

Rham operator for M , R is actually the Ricci curvature.

Corollary 3.3. Let D = d + d∗ be the Hodge-de Rham operator on M . Then on the cotangent bundle

T ∗M ⊂ Λ∗M we have

D2 = ∆ = ∇∗∇+ Ric .

Proof. It is clear from the definition of ∇ on Λ∗M and the definition ∆ = dd∗ + d∗d that ∇∗∇∗ and ∆

41

preserve the sub-bundle T ∗M ⊂ Λ∗M whence R must preserve T ∗M too. Using the relations e· = e∧·−ex·,

it is immediate that whenever i 6= j 6= k 6= i, eiejek = ei ∧ ej ∧ ek ∈ Λ3M , so in what follows we isolate these

terms and conclude, since R preserves T ∗M = Λ1M , that they are all zero. We calculate for ϕ ∈ T ∗M

Rϕ =1

2

n∑i,j=1

eiejRei,ej (ϕ)

=1

2

n∑i,j,k=1

eiej〈Rei,ej (ϕ), ek〉ek

=1

2

∑i 6=j 6=k 6=i

〈Rei,ej (ϕ), ek〉eiejek +1

2

∑i 6=k

〈Rei,ei(ϕ), ek〉eieiek +1

2

n∑i,j=1

〈Rei,ej (ϕ), ei〉eiejei +

n∑i,j=1

〈Rei,ej (ϕ), ej〉eiejej

=1

2

n∑i,j=1

〈Rei,ej (ϕ), ei〉ej −1

2

n∑i,j=1

〈Rei,ej (ϕ), ej〉ei

=

n∑i,j=1

〈Rei,ej (ϕ), ei〉ej by swapping indices and using (10) on the right sum

= −n∑

i,j=1

〈Rei,ϕ(ei), ej〉ej by (11) and (10)

= −n∑i=1

Rei,ϕei = Ric(ϕ).

Corollary 3.4. Let M be a compact manifold. If Ric ≥ 0 on M then the first Betti number b1(M) is equal

to the number of linearly independent parallel 1 forms on M . If in addition Ric > 0 at at least one point,

then b1(M) is zero.

Proof. From Theorem 2.2 we know that b1(M) is the number of linearly independent harmonic 1-forms, that

is, φ ∈ Γ(Λ1M) such that ∆φ = 0. So suppose that φ ∈ Λ1M is harmonic. Then

∆2φ = D2φ = ∇∗∇φ+ Ric(φ) =⇒ Ric(φ) = −∇∗∇φ

and so

−||∇φ||2 = −〈∇∗∇φ, φ〉 =

∫M

〈Ric(φ), φ)〉 ≥ 0

and we conclude that ∇φ = 0. On the other hand since Ric and ∇∗∇ are both defined in terms of the

connection, if φ is parallel then we may conclude ∆φ = ∇∗∇φ + Ric(φ) = 0, and so ∆φ = 0 ⇐⇒ ∇φ = 0

42

and we determine that the first cohomology group is equal to the space of parallel 1-forms. If we also have

Ric > 0 at some point then∫M〈Ric(φ), φ〉 > 0 giving a contradiction and we determine that there are no

harmonic 1-forms and b1(M) = 0.

Example 3.5. For any compact manifold with constant positive sectional curvature K we can see that

〈Ric(X), X〉 = −n∑i=1

〈Rei,Xei, X〉

=

n∑i=1

〈Rei,XX, ei〉

=

n∑i=1

K|ei ∧X|2

= K

n∑i=1

〈ei, ei〉〈X,X〉 − 〈ei, X〉2

= Kn|X|2 −K|X|2

= K(n− 1)|X2| ≥ 0

with 〈Ric(X), X〉 = 0 iff X = 0. Hence there are no compact manifolds of constant positive sectional

curvature with b1(M) 6= 0. An example of a manifold with this property is Sn for n > 1.

In fact, the above is an example of an Einstein manifold. An Einstein manifold is a Riemannian manifold

such that

〈Ric(X), Y 〉 = λ〈X,Y 〉

for all vector fields X,Y . Clearly since 〈 , 〉 is positive definite, if M is a compact Einstein manifold with

λ > 0 then b1(M) = 0.

Example 3.6. The de Rham cohomology groups for the n-torus Tn are

Hp(M ;R) ∼= Z(np).

Thus the first Betti number for Tn is b1(Tn) = n, and so Tn cannot support a metric of non-negative Ricci

curvature that is not identically zero. It is interesting to note though that the torus can support a metric

which has Ricci (and all other) curvature identically zero, and so the hypothesis that Ric(X) > 0 at some

point is cruicial in our theorem.

43

Corollary 3.4 forms an interesting link betweent the topology and geometry of a compact orientable

manifold. The above theorem requires only that there exists some metric from which positive Ricci curvature

arises, and then we obtain information about the topology of the manifold. In contrast, [9] and [10] show

us that negative Ricci curvature has no effect on the topology of the manifold in the sense of the following

theorem

Theorem 3.7 (Lohkamp). Let M be a manifold of dimension n ≥ 3. Then there exists positive constants

a and b depending only on n and a complete metric g such that

−a < 〈Ric(ϕ), ϕ〉 < −b

on all of M .

3.2 Positive curvature and the exterior bundle

We now generalise corollary 3.4 to all betti numbers bi(M), 0 < i < dim(M). When proving b1(M) = 0

we required a positivity hypothesis on Ric which operates on Λ1M , so one might expect that to determine

similar results for bi(M) we would require positivity hypotheses on some form of curvature operator acting

on ΛiM . Surprisingly however, we need only to impose positivity for an operator on Λ2M to be able to

conlclude results about vanishing higher cohomology groups. This operator is R : Λ2M → Λ2M given by

R(a ∧ b) = −∑i<j

〈Rei,ej (a), b〉ei ∧ ej

where {ei} is a local orthonormal frame field. Before we state this formally we need to know the following

fact about R.

Lemma 3.8. The operator R is a self-adjoint operator on Λ2M .

Proof. Take {ei} a local orthonormal frame field. Recall that the inner product on Λ2M is given by

〈a ∧ b, c ∧ d〉 = 〈a, c〉〈b, d〉 − 〈a, d〉〈b, c〉.

44

Using property (11) of R, this gives

〈R(a ∧ b), c ∧ d〉 = −∑i<j

〈Rei,eja, b〉〈ei ∧ ej , c ∧ d〉

= −n∑

k,l=1

∑i<j

〈a, ek〉〈b, el〉〈Rei,ejek, el〉 [〈c, ei〉〈d, ej〉+ 〈c, ej〉〈d, ei〉]

= −n∑

k,l=l

∑i<j

〈Rei,ejek, el〉〈a, ek〉〈b, el〉〈c, ei〉〈d, ej〉+

n∑k,l=l

∑i<j

〈Rei,ejek, el〉〈a, ek〉〈b, el〉〈c, ej〉〈d, ei〉

= −n∑

k,l=l

∑j<i

〈Rei,ejek, el〉〈a, ek〉〈b, el〉〈c, ei〉〈d, ej〉 −n∑

l,k=l

∑i<j

〈Rei,ejek, el〉〈a, el〉〈b, ek〉〈c, ei〉〈d, ej〉

= −n∑

i,j,k,l=1

〈Rei,ejek, el〉〈a, el〉〈b, ek〉〈c, ei〉〈d, ej〉 since Rei,ei = 0

= −n∑

k,l=1

〈Rc,dek, el〉〈a, el〉〈b, ek〉

= −∑k<l

〈Rc,dek, el〉〈a, el〉〈b, ek〉 −∑l<k

〈Rc,dek, el〉〈a, el〉〈b, ek〉

= −∑k<l

〈Rc,dek, el〉〈a, el〉〈b, ek〉+∑k<l

〈Rc,dek, el〉〈a, ek〉〈b, el〉

= −∑k<l

〈Rc,dek, el〉〈a ∧ b, ek ∧ el〉

= −∑k<l

〈Rek,elc, d〉〈a ∧ b, ek ∧ el〉

= 〈a ∧ b,R(c ∧ d)〉

as required.

Since we know now that R is self-adjoint, the spectral theorem says we have an orthonormal basis of

eigenvectors {ξij}i<j with corresponding real eigenvectors λij . We say that the curvature operator R is

positive if all of its eigenvalues are positive.

Theorem 3.9 (Gallot and Meyer). Let M be a compact Riemannian manifold of dimension n. If R is a

positive operator, then Hp(M ;R) = 0 for all 0 < p < n.

Proof. We need only show that R is a positive operator when R is positive, then the proof will follow the

45

same as the proof of corollary 3.4. An identity we will need is

∑m<n

〈R(eiej), ekel〉 = −∑m<n

〈Rem,en(ei), ej〉〈emen, ekel〉

= −∑m<n

〈Rem,en(ei), ej〉δmkδnl

= −〈Rek,el(ei), ej〉 = −〈Rei,ej (ek), el〉 (33)

A quick calculation shows that

〈[A,B], C〉 = 〈AB,C〉 − 〈BA,C〉 = 〈B,A∗C〉 − 〈B,CA∗〉 = 〈B, [A∗C]〉

which we then use in conjunction with (27) to calculate

〈Rϕ,ϕ〉 =∑i<j

〈eiejRei,ej (ϕ), ϕ〉

=1

2

∑i<j

〈[eiejRei,ej (ϕ)], ϕ〉

= −1

2

∑i<j

〈Rei,ej (ϕ), [eiej , ϕ]〉.

Lemma 1.14 then tells us that

〈Rϕ,ϕ〉 = −1

4

∑i<j,k<l

〈Rei,ej (ek), el〉||[eiej , ϕ]||2

=1

4

∑i<j,k<l

〈R(eiej), ekel〉||[eiej , ϕ]||2 using (33).

Now standard arguments show the above expression is independent of orthonormal basis of Λ2M , so we may

use the basis {ξij} in which R is diagonal and we obtain

〈Rϕ,ϕ〉 =1

4

∑i<j

λij ||[ξiξj , ϕ]||2 ≥ 0.

46

Now all that is left is to show that if the above sum is zero then ϕ = 0. If

1

4

∑i<j

λij ||[eiej , ϕ]||2 = 0

then since λij > 0 for all i, j, we would have [eiej , ϕ] = 0 for all i, j. Expressing ϕ =∑I aIeI , we obtain

[eiej , aIeI ] =

0 i, j ∈ I

0 i, j /∈ I

2aIeiejeI i ∈ I, j /∈ I or i /∈ I, j ∈ I

so by choosing i in I and j not in I we see that aI = 0 whence ϕ = 0. This is always possible unless eI = 1

or eI = e1 . . . en, thus finishing the proof.

Corollary 3.10. Any compact orientable manifold with χ(M) 6= 0 does not admit a metric with positive

curvature operator R.

3.3 Positive curvature and the spinor bundle

We have so far looked at the Bochner identity on Dirac operators on T ∗M and Λ∗M . We would next like to

do the same on the spinor bundle for M which will give us information about the scalar curvature. Having

positive scalar curvature is the weakest positivity assumption of all curvatures, in the sense that positive

sectional curvature implies positive Ricci curvature which in turn implies positive scalar curvature. We have

the following theorem about the existence of positive scalar curvature on manifolds from [11].

Theorem 3.11. Let Mn be a compact connected manifold of dimension n ≥ 3.

1 If M admits a metric whose scalar curvature is non-negative and not identically zero, then every smooth

function is realized as the scalar curvature function of some Riemannian metric on M .

2 If M admits a metric whose scalar curvature is identically zero, then a function f is the scalar curvature

of some metric if and only if either f(x) < 0 for some point x ∈M , or else f ≡ 0. If the scalar curvature

of some metric g vanishes identically, then so too does the Ricci curvature.

3 If M does not satisfy the hypotheses of (1) or (2), then f ∈ C∞(M) is the scalar curvature of some

metric if and only if f(x) < 0 for some point x ∈M .

47

It is interesting to note in theorem ?? above that there are wild differences between compact manifolds

which admit metrics of non-negative scalar curvature that do not vanish everywhere, and compact manifolds

that admit metrics with scalar curvature that do vanish everywhere. In the following we will prove that

the first cohomology group of M is trivial when the Ricci curvature is non-negative and does not vanish

everywhere, but give an example where this is not true if the hypothesis is relaxed to allow the Ricci curvature

to vanish identically.

We now construct the spinor bundle.

Recall that Spinn(M) is the sub-algebra

Spinn(M) = {v ∈ Cl0(M) : 〈v, v〉 = 1} ⊂ Cl0(M)

and we have a 2:1 covering map from Spinn into SOn. Any n−manifold admits a principal GLn(R) called

PGL whose fibres are the set of all possible bases of TxM at the point x. The bundle PGL is a principal

GLn(R) bundle since any invertible matrix can be thought of as a change of basis matrix, and so acts as a

group on the space of all bases. Given an orientable manifold with a Riemannian metric we may restrict to

the bundle PSO whose fibres are orthonormal bases of a particular orientation of the tangent bundle. We

may similarly talk about the bundle PSO(E) for any oriented vector bundle with a smoothly varying family

of inner products by taking the fibres to be all orthonormal bases of the fibre of E in some orientation. In

the same way that Spinn is a double cover for SOn, we may ask if we can find a principal Spinn bundle

PSpin(E) that double covers PSO(E). This leads us to the definition of a Spin structure.

Definition 3.12. A Spin structure for an oriented vector bundle with a smoothly varying family of inner

products is a principal Spinn bundle PSpin(E) together with an equivariant 2 sheeted covering map

ξ : PSpin(E)→ PSO(E).

Let ξ0 be the 2:1 covering map of spinn over SOn. By equivariant we mean that ξ(pg) = ξ(p)ξ0(g) for all

p ∈ PSpin(E) and g ∈Spinn.

Remark. It can be shown that E admits a spin structure iff the second and first Stiefel-Whitney classes of E

vanish. Thus admitting a spin structure can be thought of as being slightly stronger than being orientable.

We call a manifold that admits a spin structure a ‘spin manifold’.

Definition 3.13. If ρ : G→ V is a representation of a group G on a vector space V and P is a principal G

48

bundle over a manifold M (that is, a fibre bundle P → M with a free and transitive left G action in each

fibre), then we may form a vector bundle

P ×ρ V →M

called the associated bundle to P by ρ. The associated bundle is defined to be the quotient space of the

G-action on P × V defined by

g(p, v) = (pg−1, ρ(g)v).

A spinor bundle for a vector bundle over M is the associated bundle

S(E) = PSpin(E)×µ X

where X is a vector space on which Cl(M) is represented, and µ is the restriction of the representation to

Spinn(M) ⊂ Cl0(M).

It is shown in [3, II, Theorem 4.15, 4.17] that there is a unique connection on PSO(TM) with vanishing

torsion tensor which can be lifted to a unique connection on PSpin(TM). This thens descends to a connection

on the spinor bundle S(TM). The Riemann curvature tensor for this connection can be expressed in a similar

formula to that of the Riemann curvature tensor on Cl(M) as

RS(TM)V,W σ =

1

4

∑i,j

〈RTMV,W (ei), ej〉eiejσ (34)

where {ei} is an orthonormal frame field for TM (equivalently a section of PSO(TM)) and eiejσ denotes

clifford multiplication. Note that the induced representation of Cl(M) on the spinor bundle and the canonical

connection give rise to a Dirac operator. It can also be shown that the curvature operator satisfies similar

properties to the Levi-Civita connection, namely that

〈RU,VW,Y 〉 = 〈RW,Y U, V 〉 and RU,VW +RV,WU +RW,UV = 0.

We are now ready to prove the following theorem.

Theorem 3.14. Let M be a spin manifold and D the Dirac operator on the spinor bundle S(T ∗M). Then

D2 = ∇∗∇+1

4κ

49

where κ is the scalar curvature

κ = −n∑

i,j=1

〈Rei,ej (ei), ej〉.

Proof. We calculate

R =1

2

n∑i,j=1

eiejRSei,ej

=1

8

n∑i,j,k,l=1

〈Rei,ej (ek), el〉eiejekel

=1

8

n∑l=1

∑i6=j 6=k

〈Rei,ejek, el〉eiejek +

n∑i,j=1

〈Rei,ejei, el〉eiejei +

n∑i,j=1

〈Rei,ejej , el〉eie2j

el since Rei,ei = 0

=1

8

n∑l=1

1

3

∑i 6=j 6=k

〈Rei,ejek +Rek,eiej +Rej ,ekei, el〉eiejek + 2

n∑i,j=1

〈Rei,ejei, el〉ej

el=

1

4

n∑i,j,l=1

〈Rei,ejei, el〉ejel from (3.3)

=1

4

n∑i,j,l=1

〈Rei,ejei, el〉ejel

=1

4

n∑j,l=1

〈Ric(ej), el〉ejel

=1

4

1

2

n∑j,l=1

〈Ric(ej), el〉ejel + 〈Ric(el), ej〉elej

=

1

4

1

2

n∑j,l=1

〈Ric(ej), el〉(ejel + elej)

Since Ric is a symmetric operator

= −1

4

n∑j,l=1

〈Ric(ej), el〉δjl

= −1

4

n∑j=1

〈Ric(ej), ej〉

=1

4κ.

As a consequence of this, we can see from the same method as the proof of corollary 3.4 that the kernel

of D2 is empty when M supports a metric with positive (or non-negative and non-vanishing). Recalling

50

proposition 1.24 we then see the kernel of D itself is empty and so the index of D is zero. We then finish this

paper with a conjecture that the converse is true. The following conjecture is for dimension 0 and 4 (mod

8) only because of torsion that cannot be seen by de Rham cohomology.

Conjecture 3.15 (Gromov-Lawson-Rosenberg Conjecture). Let M be a compact, Spin manifold of dimen-

sion 4k and let D be the Dirac operator on a spinor bundle of the tangent bundle. If index(D) = 0 then M

admits a metric of non-vanishing and non-negative scalar curvature.

Conclusion

We have seen how the group structure of Clifford bundles can be used to make computations quite tractible

involving the Dirac operator and the Laplacian, and we have seen from Bochner’s identity how these operators

allow us to access geometric information about our manifold. Future directions we may be able to take our

results are perhaps extending theorem 3.9 to not only compact manifolds, but open manifolds with some

restraints on the ‘behaviour at infinity’. This could be quite believable, since for example a paraboloid is

non-compact and asymptotically flat, with non-negative Gauss curvature everywhere, and itself has trivial

cohomology. It would have also been good to look further into the spinor bundle and its Dirac operator,

perhaps looking at the Atiyah-Singer index theorem and the A genus.

References

[1] R.W.R Darling, Differential Forms and Connections (1994)

[2] Jurgen Jost, Riemannian Geometry and Geometric Analysis 7th Edition (2017)

[3] H. Blaine Lawson, Marie-Louise Michelsohn, Spin Geometry (1990)

[4] Nigel Higson, John Roe, Analytic K-Homology (2000)

[5] Adam Rennie, Topics in Algebra and Topology (2017)

[6] Adam Rennie Spectral theory for differential operators (2017)

[7] Manfredo do Carmo, Reimannian Geometry (1992)

[8] Raoul Bott, Loring W.Tu, Differential Forms in Algebraic Topology (1982)

51

[9] J.Lohkamp, Metrics of negative Ricci curvature (1994)

[10] J.Lohkamp, Negative Ricci curved manifolds (1992)

[11] Jerry L. Kadan, F.W.Warner, Scalar curvature and conformal deformation of Riemannian structure

(1975)

[12] Jonathan Rosenberg, Manifolds of Positive Scalar Curvature: a Progress Report (2007)

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