Reference about Gauss-Bonnet-Chern theorem.

Question

I would like to get some references which explains Gauss-Bonnet-Chern theorem and its original proof by Chern. I tried to read his paper published in 1944 "A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds", but this was too difficult for me. Thanks to your help!

Answer 1

You might be interested in Yin Li's expository paper The Gauss-Bonnet-Chern Theorem on Riemannian Manifolds which can be found on the arXiv. Here is the abstract:

This expository paper contains a detailed introduction to some important works concerning the Gauss-Bonnet-Chern theorem. The study of this theorem has a long history dating back to Gauss's Theorema Egregium (Latin: Remarkable Theorem) and culminated in Chern's groundbreaking work [$14$] in $1944$, which is a deep and wonderful application of Elie Cartan's formalism. The idea and tools in [$14$] have a great generalization and continue to produce important results till today.

In this paper, we give four different proofs of the Gauss-Bonnet-Chern theorem on Riemannian manifolds, namely Chern's simple intrinsic proof, a topological proof, Mathai-Quillen's Thom form proof and McKean-Singer-Patodi's heat equation proof. These proofs are related with remarkable developments in differential geometry such as the Chern-Weil theory, theory of characteristic classes, Mathai-Quillen's formalism and the Atiyah-Singer index theorem. It is through these brilliant achievements the great importance and influence of Chern's insights and ideas are shown. Our purpose here is to use the Gauss- Bonnet-Chern theorem as a guide to expose the reader to some advanced topics in modern differential geometry.