The Gauss-Bonnet-Chern Theorem on Riemannian Manifolds

TL;DR

This paper provides a comprehensive introduction to the Gauss-Bonnet-Chern theorem, presenting four different proofs and exploring its connections to key developments in modern differential geometry.

Contribution

It offers a detailed exposition of four proofs of the Gauss-Bonnet-Chern theorem, highlighting its significance and relation to major geometric theories.

Findings

Four distinct proofs of the Gauss-Bonnet-Chern theorem are presented.

The theorem's connections to Chern-Weil theory, characteristic classes, and index theory are elucidated.

The paper demonstrates the theorem's importance in modern differential geometry.

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.