The Gauss-Bonnet Theorem

Abstract

We are finally in a position to prove our first major local-global theorem in Riemannian geometry: the Gauss-Bonnet theorem. This is a local-global theorem par excellence, because it asserts the equality of two very differently defined quantities on a compact, orientable Riemannian 2-manifold M: the integral of the Gaussian curvature, which is determined by the local geometry of M; and 2π times the Euler characteristic of M, which is a global topological invariant. Although it applies only in two dimensions, it has provided a model and an inspiration for innumerable local-global results in higher-dimensional geometry, some of which we will prove in Chapter 11.