Introduction to Riemannian Manifolds pp 263-282 | Cite as

# The Gauss–Bonnet Theorem

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## Abstract

In this chapter, we prove our first major local-to-global theorem in Riemannian geometry: the Gauss–Bonnet theorem. The grandfather of all such theorems in Riemannian geometry, it asserts the equality of two very differently defined quantities on a compact Riemannian 2-manifold: the integral of the Gaussian curvature, which is determined by the local geometry, and \(2\pi \) times the Euler characteristic, which is a global topological invariant.

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