In differential geometry of surfaces, the Gauss-Bonnet Theorem was one of the first and most important local-to-global Theorems: it provides a surprising relation between a local, geometric invariant (the (Gaussian) curvature) and a global, topological invariant (the Euler characteristic). As such, it was an important foundational theorem for the study of higher dimensional spaces (manifolds), and the development of a generalised Gauss-Bonnet Theorem led to many of the concepts of modern differential geometry (and algebraic topology), in particular, the theory of characteristic classes. As such, some of the first "proofs" of the generalised Gauss-Bonnet theorem depended on then not yet proved theorems such as the Nash embedding theorem (which states than any Riemannian manifold can be isometrically embedded in for some ), allowing us to consider manifolds as embedded manifolds in Euclidean space. This first proof also made use of a similarly striking local-to-global result, relating the Euler characteristic of a manifold with the zeroes of any (continuous, tangent) vector field on it.

In this talk, we will go through the classical version of the Gauss-Bonnet Theorem (for surfaces), and some of the history of the development of the generalised Gauss-Bonnet theorem (also called the Gauss-Bonnet-Chern Theorem for Shiing-Shen Chern that was the first to give a complete proof not depending on unproved results), including the first proof using the Nash embedding theorem and a sketch of the modern theory of characteristic classes and of Chern's proof. Then we relocalise to the pub.