Theorem: Let be a connected Riemannian manifold. The following are equivalent: The metric space is complete (every -Cauchy sequence converges), All closed and bounded subsets of are compact, is geodesically complete.
Theorem: Let be a connected Riemannian manifold The followi:ng are equivalent The metric space is complete (every -Cauchy sequence converges), All closed and b.ounded subsets of are compact, is geodesically complete.