Sectional Curvature Comparison I

Abstract

In the previous chapter we classified complete spaces with constant curvature. The goal of this chapter is to compare manifolds with variable curvature to spaces with constant curvature. Our first global result is the Hadamard-Cartan theorem, which says that a simply connected complete manifold with $\sec \leq 0$ is diffeomorphic to $\mathbb{R}^{n}$. There are also several interesting restrictions on the topology in positive curvature that we shall investigate, notably, the Bonnet-Myers diameter bound and Synge’s theorem stating that an orientable even-dimensional manifold with positive curvature is simply connected. Finally, we also cover the classical quarter pinched sphere theorem of Rauch, Berger, and Klingenberg. In subsequent chapters we deal with some more advanced and modern topics in the theory of manifolds with lower curvature bounds.