Riemannian Manifolds

Abstract

AbstractThe necessary foundations for the study of the following chapters are provided. We introduce the main notions, and prove, discuss or simply mention the basic results on smooth manifolds, manifolds with covariant derivatives, and Riemannian manifolds that will be needed later. In particular, we consider the Lie algebra of smooth vector fields on smooth manifolds, the Levi-Civita connection on Riemannian manifolds, their curvature tensors, sectional and other curvatures, geodesics, the intrinsic metric, and shortest curves. As a rule, we prove the main results. In particular, we prove the most important results of this chapter, namely the Hopf–Rinow theorem, the Myers theorem on Riemannian manifolds with strictly positive Ricci curvature, the Synge theorem, the Rauch comparison theorem, the Hadamard–Cartan theorem, and the O’Neill formulas for Riemannian submersions. We also give or indicate some geometric applications of the main results.