Sectional Curvature Comparison I

Abstract

We shall first classify spaces with constant curvature. The real subject of this chapter is how one can compare manifolds to spaces with constant curvature. We shall for instance prove the Hadamard-Cartan theorem, which says that a simply connected manifold with sec ≤ 0 is diffeomorphic to ℝⁿ. There are also some interesting restrictions on the topology in positive curvature that we shall inves­tigate, notably, Synge’s theorem, which says that an orientable even-dimensional manifold with positive curvature is simply connected. In Chapter 11 we shall deal with some more advanced topics in the theory of manifolds with lower sectional curvature bounds.