The Gauss-Bonnet Theorem

Abstract

Smooth surfaces can be analyzed geometrically (as in Chapters 6 and 7) and topologically (since they are also topological surfaces). The Gauss-Bonnet Theorem, essentially the point toward which this entire book has been aimed, shows that these two approaches are deeply related. The simplicial Gauss-Bonnet Theorem (Theorem 3.7.2) has already shown us one connection between a topological invariant (the Euler characteristic) and a geometric quantity (the angle defect in simplicial surfaces). Although the statement of this theorem was somewhat surprising, the proof was not difficult, since both angle defect and the Euler characteristic were defined in terms of triangulations; indeed, the real surprise was that the Euler characteristic turns out to be a topological invariant, that is, it does not depend upon the choice of triangulation of a given surface.