The Gauss-Bonnet Theorem An Introduction to Index Theory Gianmarco - - PowerPoint PPT Presentation

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 balls in Rn 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 χ =

n

• i=0

(−1)ibi 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 structure of Rn 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 2-manifolds isometrically embedded in R3, and take an historical perspective. Given a smooth curve γ : [0, 1] → M we can define its curvature kγ(s) = |γ′′(s)| This is an extrinsic definition; the derivatives are taken in R3 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

γ (kγ),

k2 = sup

γ (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 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
• M

K = 2πχ 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

• R

K = −

• π−1(R)
• Ev

2 √ EG

• v

+

• Gu

2 √ EG

• u
• dudv

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

• R

K +

• ∂R

kg +

3

• i=1

θi = 2π 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

• M

K = 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 conclude that

• M

K = 2π(F − E + V ) = 2πχ

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,

• M

Pf(Ω) = (2π)nχ 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

• f 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

• M

Pf(Ω) 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 χ =

n

• i=0

(−1)ibi that the bi can be determined as the Betti numbers βi defined as βi = dim Hi

dR(M)

Where Hi

dR(M) = closed i-forms

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 ker ∆i

∼ =

− → Hi

dR(M)

ω → [ω] and so dim ker ∆i = 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 )e−t∆ = 0

e−t∆|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 e−i∆α(x) =

• M

e(t, x, y)α(y) dvol(y)

Gianmarco Molino (SIGMA Seminar) The Gauss-Bonnet Theorem 1 Februrary, 2019 18 / 23

Heat kernel proof of the Chern-Gauss-Bonnet Theorem

Defining E i

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

• i

(−1)i dim E i

λ = 0

and as a result χ =

• i

(−1)i dim ker ∆i =

• i

(−1)i

j

e−tλi

j =

• i

(−1)i Tr e−t∆i We can conclude from this that χ =

• i

(−1)i

• M

tr ei(t, x, x)dvol(x)

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) σ(M) =

• M

Lk(Ω)4k 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 dim ker D − dim ker D∗ =

• M

ch(D)Td(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 dim ker D − dim ker D∗ =

• M

Pf(Ω) and

• M

ch(D)Td(M) = χ(M)

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