The theorem of Hadamard-Cartan and complete simply connected manifolds of nonpositive curvature
Let V be a complete Riemannian manifold of nonpositive curvature and p ∈ V. Then expp is defined on the entire tangent space TpV. 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 expp is everywhere nonsingular. Thus we can define a new metric on TpV, such that expp is a local isometry. This metric is complete by the Hopf-Rinow theorem, because the lines through the origin in TpV are geodesics in this metric. Now it is easy to prove that a local isometry φ: V1 → V2, where V1 is complete, is a covering map (comp. [Cheeger-Ebin, 1975], p. 35).
KeywordsConvex Subset Fundamental Group Universal Covering Complete Riemannian Manifold Nonpositive Curvature
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