For a connected Riemannian manifold, metric completeness, geodesic completeness, compactness of every closed bounded set, and the existence of a minimizing geodesic between every pair of points are equivalent.
The Hopf-Rinow theorem is a fundamental result in differential geometry and the study of Riemannian manifolds. It connects concepts of completeness, compactness, and geodesics in the context of Riemannian geometry. The theorem states the following: 1. **For a complete Riemannian manifold**: If \( M \) is a complete Riemannian manifold, then it is compact if and only if it is geodesically complete.
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