Chapter 4 Metric Riemannian geometry

Abstract

This chapter is a survey of (a part of) Riemannian geometry. Riemannian geometry is a huge area that occupies at least 1/3rd of the entire differential geometry. It presents the part of Riemannian geometry that describes relations of curvature (sectional or Ricci curvature) to topology of the underlying manifold. Manifolds of nonnegative curvature and more generally the class of manifolds with curvature bounded from below are described. The study of such Riemannian manifolds started with sphere theorems in the 1950s where comparison theorems are introduced by Rauch as an important tool of study. The chapter reviews several basic facts on global Riemannian geometry, such as Rauch's comparison theorem, cut points, conjugate points, and injectivity radius. One of the main tools of global Riemannian geometry is the Gromov–Hausdorff distance, which is discussed and Gromov's precompactness theorem is proved.