The Gauss-Bonnet theorem

Abstract

[en] The Gauss-Bonnet Theorem was first published by Gauss in 1827 for the case of a geodesic triangle on a surface. Since then, the theorem has progressively increased in generality. The purpose of this work is to prove it for the case of 2-dimensional Riemannian manifolds, while discussing the historic development of its other versions. For that, the necessary concepts of differential geometry are introduced, such as smooth manifolds, their tangent spaces, and the measurement of areas and angles via Riemannian metrics. The concepts of curves, lifts, orientability, and curvature are also adapted to the nature of manifolds. With that toolbox, the Rotation Index Theorem is proved, subsequently the Gauss-Bonnet Formula, and finally the Gauss-Bonnet theorem for orientable and non-orientable manifolds. The latter employs combinatorial arguments, combining local results to yield a global one. The most remarkable aspect of this theorem is precisely that it connects local properties of differential geometry, specifically the integral of the curvature, with a global topological invariant, the Euler characteristic.