The Gauss–Bonnet Theorem

  • John M. Lee
Part of the Graduate Texts in Mathematics book series (GTM, volume 176)


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.

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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