The Gauss-Bonnet Theorem

Abstract

The Gauss—Bonnet theorem is the most beautiful and profound result in the theory of surfaces. Its most important version relates the average over a surface of its gaussian curvature to a property of the surface called its ‘Euler number’ which is ‘topological’, i.e. it is unchanged by any continuous deformation of the surface. Such deformations will in general change the value of the gaussian curvature, but the theorem says that its average over the surface does not change. The real importance of the Gauss—Bonnet theorem is as a prototype of analogous results which apply in higher dimensional situations, and which relate geometrical properties to topological ones. The study of such relations is one of the most important themes of 20th century Mathematics.