A Characterization of Flat Pseudo-Riemannian Manifolds

Abstract

This thesis is concerned with the curvature of pseudo-Riemannian manifolds. A pseudo-Riemannian manifold is called flat when it can be covered by charts that intertwine the pseudo-metrics of the manifold and psuedo-Euclidian space. In general, the curvature of a manifold is described by an operator, called the Riemann curvature. In the main theorem of this thesis, we will prove that a pseudo-Riemannian manifold is flat if and only if the Riemann curvature vanishes identically.