The Gauss-Bonnet Theorem - Connectiongianmarcomolino.com/wp-content/uploads/2019/02/presentation.pdf · The Gauss-Bonnet theorem and all of the previously mentioned extensions are

The Gauss-Bonnet TheoremAn Introduction to Index Theory

Gianmarco Molino

SIGMA Seminar

1 Februrary, 2019

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 1 / 23

Topological Manifolds

An n-dimensional topological manifold M is an abstract way ofrepresenting space:

Formally it is a set of points M, a collection of ‘open sets’ T , and aset of continuous bijections of neighborhoods of each point with openballs in Rn called charts.

Topological manifolds don’t really have a sense of ‘distance’; that’sthe key difference between the study of topology and geometry.


Topological manifolds

Topological manifolds are considered equivalent (homeomorphic) ifthey can be ‘stretched’ to look like one another without being ‘cut’ or‘glued’.

A homeomorphism is a continuous bijection.

Any property that is invariant under homeomorphisms is considered atopological property.


Euler Characteristic

It’s possible to decompose topological manifolds into ‘triangulations’.

In the context of surfaces, this will be a combination of vertices,edges, and faces; in higher dimensions we use higher dimensionalsimplices.

Given a triangulation, we define the constants

bi = #i-simplices


Euler Characteristic

We then define the Euler Characteristic of a manifold M with a giventriangulation as

χ =n∑

i=0

(−1)ibi

The Euler characteristic can be shown to be independent of thetriangulation, and is thus a property of the manifold.

It’s moreover invariant under homeomorphism, and even more thanthat it’s invariant under homotopy equivalence.


Riemannian Manifolds

We can add to topological manifolds more structure;

A topological manifold equipped with charts that preserve the smoothstructure of Rn are called smooth manifolds.

A smooth manifold equipped with a smoothly varying inner productg(·, ·) on its tangent bundle is called a Riemannian manifold.

Riemannian manifolds have well defined notions of distance andvolume, and can be naturally equipped with a notion of derivative(Levi-Civita connection).


Surfaces and Gaussian Curvature

We’ll begin by only considering surfaces, that is Riemannian2-manifolds isometrically embedded in R3, and take an historicalperspective.

Given a smooth curveγ : [0, 1]→M

we can define its curvature

kγ(s) = |γ′′(s)|

This is an extrinsic definition; the derivatives are taken in R3 anddepend on the embedding of M.


Surfaces and Gaussian Curvature

For each point x ∈M we consider the collection of all smooth curvespassing through x and define the ‘principal curvatures’

k1 = infγ

(kγ), k2 = supγ

(kγ)

Gauss defined the Gaussian curvature of a surface M to be

K = k1k2

and proved in his famous Theorema Egregium (1827) that it is anintrinisic property; that is it is independent of the embedding.


Gauss-Bonnet Theorem

O. Bonnet (1848) showed that for a closed, compact surface M∫MK = 2πχ

This is remarkable, relating a global, topological quantity χ to a local,analytical property K .


Proof of the Gauss-Bonnet Theorem

Consider first a triangular region R of a surface.

Using a parameterization (u, v) we can write the curvature in localcoordinates as∫

RK = −

∫∫π−1(R)

((Ev

2√EG

)v

+

(Gu

2√EG

)u

)dudv



By an application of the Gauss-Green theorem, this is equivalent tothe integral over the boundary of the curvatures of the triangular arcsplus a correction term at each vertex;

This correction measures what total angle the ‘direction vector’ of theboundary traverses in one loop, and so∫

RK +

∫∂R

kg +3∑

i=1

θi = 2π

where the θi are the external angles at each vertex.



Now consider an arbitrary triangulization of M. Applying the aboveresult repeatedly and accounting for the cancellation of the boundaryintegrals because of orientation, we will see that∫

MK = 2πF −

∑i ,j

θij

where θ1j , θ2j , θ3j are the external angles to triangle j .

Rewriting this in terms of interior angles, we will be able to concludethat ∫

MK = 2π(F − E + V ) = 2πχ


Chern-Gauss-Bonnet Theorem

In 1945 Shiing-Shen Chern proved that for a closed, 2n-dimensionalRiemannian manifold M,∫

MPf(Ω) = (2π)nχ

Here Ω is a so(2n) valued differential 2-form called the curvatureform associated to the Levi-Civita connection on M and

Pf denotes the Pfaffian, which is roughly the square root of thedeterminant.


Chern-Gauss-Bonnet Theorem

This theorem is again remarkable; it implies that the possible notionsof curvature (and by extension smooth and Riemannian structures) ona topological manifold are strongly limited by the topology.

It also implies a strong integrality condition; a priori χ is an integer,but ∫

MPf(Ω)

is only necessarily rational.

One nice immediate corollary of the theorem is a topologicalrestriction on the existence of flat metrics; specifically, if M admits aflat metric, then χ = 0.


Heat kernel proof of the Chern-Gauss-Bonnet Theorem

We will consider a proof due to Parker (1985).

First, a series of results in algebraic topology indicates that for theEuler characteristic

χ =n∑

i=0

(−1)ibi

that the bi can be determined as the Betti numbers βi defined as

βi = dimH idR(M)

Where

H idR(M) =

closed i-forms

exact i-forms

are the deRham cohomology groups.



We define the Hodge Laplacian on a Riemannian manifold

∆ = dδ + δd

which is an operator on the space of differential forms.

Here d is the exterior derivative, and δ = d∗ is its formal adjointunder the Riemannian metric.



Then, we use the famous Hodge Isomorphism which asserts that

ker ∆i ∼=−→ H idR(M) ω 7→ [ω]

and sodim ker ∆i = βi



We can define the heat operator e−t∆ acting on differential forms asthe solution to the heat equation

(∆ + ∂∂t )e−t∆ = 0

e−t∆|t=0 = Id

With some work it can be shown that the heat operator exists onclosed compact Riemannian manifolds, and that it has an integralkernel e(t, x , y), that is

e−i∆α(x) =

∫Me(t, x , y)α(y) dvol(y)



Defining E iλ to be the λ-eigenspace of ∆i , we can show that for λ > 0∑

i

(−1)i dimE iλ = 0

and as a result

χ =∑i

(−1)i dim ker ∆i =∑i

(−1)i∑j

e−tλij =

∑i

(−1)i Tr e−t∆i

We can conclude from this that

χ =∑i

(−1)i∫M

tr e i (t, x , x)dvol(x)



Unfortunately, for most manifolds the computation of the heat kernelis impossible, but we can approximate it using a parametrix (anapproximation close to the diagonal).

Using this approximation and making repeated use of the fact that χis independent of t we will be able to conclude the theorem.


Further Generalizations

Hirzebruch Signature Theorem (1954)

σ(M) =

∫MLk(Ω)4k

Riemann-Roch Theorem (1954)

l(D)− l(K − D) = deg(D)− g + 1


Atiyah-Singer Index Theorem

(Atiyah-Singer, 1963) On a compact smooth manifold M with emptyboundary equipped with an elliptic differential operator D betweenvector bundles over M it holds that

dim kerD − dim kerD∗ =

∫Mch(D)Td(M)

The Gauss-Bonnet theorem and all of the previously mentionedextensions are specific instances of this theorem.


In particular, recall that the heat kernel proof of theChern-Gauss-Bonnet theorem used the properties of theHodge-Laplacian

∆ = dδ + δd = (d + δ)2

Defining the Dirac operator

D = d + δ

we will find that D is an elliptic differential operator and that

dim kerD − dim kerD∗ =

∫M

Pf(Ω)

and ∫Mch(D)Td(M) = χ(M)


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The Gauss-Bonnet Theorem - Connectiongianmarcomolino.com/wp-content/uploads/2019/02/presentation.pdf · The Gauss-Bonnet theorem and all of the previously mentioned extensions are