Manifolds of Nonpositive Curvature pp 15-20 | Cite as

# The theorem of Hadamard-Cartan and complete simply connected manifolds of nonpositive curvature

## Abstract

Let V be a complete Riemannian manifold of nonpositive curvature and p ∈ V. Then exp_{p} is defined on the entire tangent space T_{p}V. The convexity of the norm of every Jacobi field (see 1.3) implies that no (non-zero) Jacobi field along a geodesic c: [0,1] → V vanishes at c(0) and c(1). Hence there are no conjugate points, and the differential of exp_{p} is everywhere nonsingular. Thus we can define a new metric on T_{p}V, such that exp_{p} is a local isometry. This metric is complete by the Hopf-Rinow theorem, because the lines through the origin in T_{p}V are geodesics in this metric. Now it is easy to prove that a local isometry *φ:* V_{1} → V_{2}, where V_{1} is complete, is a covering map (comp. [Cheeger-Ebin, 1975], p. 35).

## Keywords

Convex Subset Fundamental Group Universal Covering Complete Riemannian Manifold Nonpositive Curvature## Preview

Unable to display preview. Download preview PDF.