The Global Geometry of Surfaces

Abstract

In this chapter, we will consider some problems in the global differential geometry of surfaces. A “global” problem can be described as one which in general cannot be stated locally in terms of one coordinate system on a surface with a Riemannian metric, but must necessarily involve the total behavior of the surface. Most often, this total behavior is related to the topology of the surface. For example, Theorem (6.3.5) equates the integral of the curvature function K(p) over a compact surface M with a topological invariant of M (the Euler characteristic). Neither of these two quantities can be described completely in terms of a single coordinate system.