The Cartan-Hadamard Theorem

Abstract

Requiring a Riemannian manifold to have non-positive sectional curvature is a restriction on the infinitesimal geometry of the space. Much of the power and elegance of the theory of such manifolds stems from the fact that one can use this infinitesimal condition to make deductions about the global geometry and topology of the manifold. The result which underpins this passage from the local to the global context is a fundamental theorem that is due to Hadamard [H18981 in the case of surfaces and to Cartan [Car28] in the case of arbitrary Riemannian manifolds of non-positive curvature. The main purpose of this chapter is to show that the Cartan-Hadamard Theorem can be generalized to the context of complete geodesic metric spaces. We shall see in subsequent chapters that this generalization is of fundamental importance in the study of complete (1-connected) metric spaces of curvature ≤ к,where к ≤ 0. Related results concerning non-simply connected spaces and CAT(к) spaces with к > 0 will also be presented in this chapter. For a more complete treatment of the case к > 0, see Bowditch [Bow95c].