The Gauss-Bonnet Theorem An 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 of representing space: Formally it is a set of points M, a collection of ‘open sets’ T , and a set of continuous bijections of neighborhoods of each point with open n balls in R called charts. Topological manifolds don’t really have a sense of ‘distance’; that’s the key difference between the study of topology and geometry.

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 2 / 23 Topological manifolds

Topological manifolds are considered equivalent (homeomorphic) if they 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 a topological property.

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 3 / 23 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 dimensional simplices. Given a triangulation, we define the constants

bi = #{i-simplices}

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 4 / 23 Euler Characteristic

We then define the Euler Characteristic of a manifold M with a given triangulation as n X i χ = (−1) bi i=0 The Euler characteristic can be shown to be independent of the triangulation, and is thus a property of the manifold. It’s moreover invariant under homeomorphism, and even more than that it’s invariant under homotopy equivalence.

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 5 / 23 Riemannian Manifolds

We can add to topological manifolds more structure; A topological manifold equipped with charts that preserve the smooth n structure of R are called smooth manifolds. A smooth manifold equipped with a smoothly varying inner product g(·, ·) on its tangent bundle is called a Riemannian manifold. Riemannian manifolds have well defined notions of distance and volume, and can be naturally equipped with a notion of derivative (Levi-Civita connection).

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 6 / 23 Surfaces and Gaussian Curvature

We’ll begin by only considering surfaces, that is Riemannian 3 2-manifolds isometrically embedded in R , and take an historical perspective. Given a smooth curve γ : [0, 1] → M we can define its curvature

00 kγ(s) = |γ (s)|

3 This is an extrinsic definition; the derivatives are taken in R and depend on the embedding of M.

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 7 / 23 Surfaces and Gaussian Curvature

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

k1 = inf(kγ), k2 = sup(kγ) γ γ

Gauss defined the Gaussian curvature of a surface M to be

K = k1k2

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

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 8 / 23 Gauss-Bonnet Theorem

O. Bonnet (1848) showed that for a closed, compact surface M Z K = 2πχ M This is remarkable, relating a global, topological quantity χ to a local, analytical property K.

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 9 / 23 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 local coordinates as Z ZZ  E   G   K = − √v + √ u dudv R π−1(R) 2 EG v 2 EG u

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 10 / 23 Proof of the Gauss-Bonnet Theorem

By an application of the Gauss-Green theorem, this is equivalent to the integral over the boundary of the curvatures of the triangular arcs plus a correction term at each vertex; This correction measures what total angle the ‘direction vector’ of the boundary traverses in one loop, and so

3 Z Z X K + kg + θi = 2π R ∂R i=1

where the θi are the external angles at each vertex.

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 11 / 23 Proof of the Gauss-Bonnet Theorem

Now consider an arbitrary triangulization of M. Applying the above result repeatedly and accounting for the cancellation of the boundary integrals because of orientation, we will see that

Z X K = 2πF − θij M i,j

where θ1j , θ2j , θ3j are the external angles to triangle j. Rewriting this in terms of interior angles, we will be able to conclude that Z K = 2π(F − E + V ) = 2πχ M

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 12 / 23 Chern-Gauss-Bonnet Theorem

In 1945 Shiing-Shen Chern proved that for a closed, 2n-dimensional Riemannian manifold M, Z Pf(Ω) = (2π)nχ M Here Ω is a so(2n) valued differential 2-form called the curvature form associated to the Levi-Civita connection on M and Pf denotes the Pfaffian, which is roughly the square root of the determinant.

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 13 / 23 Chern-Gauss-Bonnet Theorem

This theorem is again remarkable; it implies that the possible notions of curvature (and by extension smooth and Riemannian structures) on a topological manifold are strongly limited by the topology. It also implies a strong integrality condition; a priori χ is an integer, but Z Pf(Ω) M is only necessarily rational. One nice immediate corollary of the theorem is a topological restriction on the existence of flat metrics; specifically, if M admits a flat metric, then χ = 0.

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 14 / 23 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 the Euler characteristic n X i χ = (−1) bi i=0

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

i βi = dim HdR (M)

Where closed i-forms Hi ( ) = dR M exact i-forms are the deRham cohomology groups.

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 15 / 23 Heat kernel proof of the Chern-Gauss-Bonnet Theorem

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 adjoint under the Riemannian metric.

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 16 / 23 Heat kernel proof of the Chern-Gauss-Bonnet Theorem

Then, we use the famous Hodge Isomorphism which asserts that

i =∼ i ker ∆ −→ HdR (M) ω 7→ [ω]

and so i dim ker ∆ = βi

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 17 / 23 Heat kernel proof of the Chern-Gauss-Bonnet Theorem

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

( ∂ −t∆ (∆ + ∂t )e = 0 −t∆ e |t=0 = Id

With some work it can be shown that the heat operator exists on closed compact Riemannian manifolds, and that it has an integral kernel e(t, x, y), that is Z e−i∆α(x) = e(t, x, y)α(y) dvol(y) M

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 18 / 23 Heat kernel proof of the Chern-Gauss-Bonnet Theorem

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

X i i (−1) dim Eλ = 0 i and as a result

X X X −tλi X i χ = (−1)i dim ker ∆i = (−1)i e j = (−1)i Tr e−t∆ i i j i

We can conclude from this that X Z χ = (−1)i tr ei (t, x, x)dvol(x) i M

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 19 / 23 Heat kernel proof of the Chern-Gauss-Bonnet Theorem

Unfortunately, for most manifolds the computation of the heat kernel is impossible, but we can approximate it using a parametrix (an approximation close to the diagonal). Using this approximation and making repeated use of the fact that χ is independent of t we will be able to conclude the theorem.

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 20 / 23 Further Generalizations

Hirzebruch Signature Theorem (1954) Z σ(M) = Lk (Ω)4k M Riemann-Roch Theorem (1954)

l(D) − l(K − D) = deg(D) − g + 1

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 21 / 23 Atiyah-Singer Index Theorem

(Atiyah-Singer, 1963) On a compact smooth manifold M with empty boundary equipped with an elliptic differential operator D between vector bundles over M it holds that Z ∗ dim ker D − dim ker D = ch(D)Td(M) M The Gauss-Bonnet theorem and all of the previously mentioned extensions are specific instances of this theorem.

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 22 / 23 In particular, recall that the heat kernel proof of the Chern-Gauss-Bonnet theorem used the properties of the Hodge-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 Z dim ker D − dim ker D∗ = Pf(Ω) M

and Z ch(D)Td(M) = χ(M) M

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