Source: wikibot/hopf-rinow-theorem
= Hopf–Rinow theorem
{wiki=Hopf–Rinow_theorem}
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.