Structures on Riemannian Manifolds

Abstract

Let M be an n-dimensional connected differentiable manifold of class C^∞ covered by a system of coordinate neighborhoods {U; x^h}, where U denotes a neighborhood and x^h local coordinates in U. If, from any system of coordinate neighborhoods covering the manifold M, we can choose a finite number of coordinate neighborhoods which cover the whole manifold, then M is said to be compact.