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 R n 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 b i = # { 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 � ( − 1) i b i χ = 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 structure of R n 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 R 3 , and take an historical perspective. Given a smooth curve γ : [0 , 1] → M we can define its curvature k γ ( s ) = | γ ′′ ( s ) | This is an extrinsic definition; the derivatives are taken in R 3 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’ k 1 = inf γ ( k γ ) , k 2 = sup γ ( k γ ) Gauss defined the Gaussian curvature of a surface M to be K = k 1 k 2 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 � K = 2 πχ 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 �� � � � � � �� E v G u K = − √ + √ dudv 2 2 EG EG π − 1 ( R ) R v 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 � � � K + k g + θ i = 2 π R ∂ R i =1 where the θ i 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 � � K = 2 π F − θ ij M i , j where θ 1 j , θ 2 j , θ 3 j are the external angles to triangle j . Rewriting this in terms of interior angles, we will be able to conclude that � K = 2 π ( F − E + V ) = 2 πχ 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, 2 n -dimensional Riemannian manifold M , � Pf(Ω) = (2 π ) n χ M Here Ω is a so (2 n ) 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 � 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 � ( − 1) i b i χ = i =0 that the b i can be determined as the Betti numbers β i defined as β i = dim H i dR ( M ) Where dR ( M ) = closed i-forms H i 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 ∆ = d δ + δ d 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 → H i − dR ( M ) ω �→ [ ω ] and so dim ker ∆ i = β i 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 ) = 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 Defining E i λ to be the λ -eigenspace of ∆ i , we can show that for λ > 0 ( − 1) i dim E i � λ = 0 i and as a result ( − 1) i dim ker ∆ i = ( − 1) i Tr e − t ∆ i e − t λ i � � ( − 1) i � j = � χ = i i j i We can conclude from this that � � ( − 1) i tr e i ( t , x , x ) dvol ( x ) χ = M i 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 ) = L k (Ω) 4 k 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 � 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