6 Geometry and topology
Differential geometry is a fine, quantitative geometry, in which relationships between lengths and angles are important. Topology, by contrast, is of a much coarser and more qualitative nature. Here only those quantities that are preserved under distortions are studied. In order to obtain a topological description of the total Gauss curvature, we triangulate the surfaces, i.e. we cut them into triangles. The theorem of Gauss–Bonnet now tells us that we can determine the total curvature by counting vertices, edges and triangles.
In the last sections of this book we want to study global properties of surfaces. For example, we want be able to decide whether two given surfaces are homeomorphic or not. For ...