Curvature

Abstract

In this chapter, we begin our study of the local invariants of Riemannian metrics. Starting with the question of whether all Riemannian metrics are locally isometric, we are led to a definition of the Riemannian curvature tensor as a measure of the failure of second covariant derivatives to commute. Then we prove the main result of this chapter: a Riemannian manifold has zero curvature if and only if it is flat, or locally isometric to Euclidean space. At the end of the chapter, we explore how the curvature can be used to detect conformal flatness.