ABSTRACT
Historians of mathematics often point to Euclid as the inventor, or at least the father in some metaphorical sense, of the synthetic methods of mathematical proofs. This chapter discusses applications of the Gauss-Bonnet Theorem to non-Euclidean geometry. No course on classical differential geometry is complete without the Gauss-Bonnet Theorem, arguably the most profound theorem in the differential geometry study of surfaces. The Gauss-Bonnet Theorem simultaneously encompasses a total curvature theorem for surfaces, the total geodesic curvature formula for plane curves, and other famous results, such as the sum of angles formula for a triangle in plane, spherical, or hyperbolic geometry. In intuitive terms, a triangulation of a surface consists of a network of a finite number of regular curve segments on the surface such that any point on the surface either lies on one of the curves or lies in a region that is bounded by precisely three curve segments.
