# Hopf–Rinow theorem

Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow, who published it in 1931.[1]

## Statement

Let ${\displaystyle (M,g)}$ be a connected Riemannian manifold. Then the following statements are equivalent:

1. The closed and bounded subsets of ${\displaystyle M}$ are compact;
2. ${\displaystyle M}$ is a complete metric space;
3. ${\displaystyle M}$ is geodesically complete; that is, for every ${\displaystyle p\in M,}$ the exponential map expp is defined on the entire tangent space ${\displaystyle \operatorname {T} _{p}M.}$

Furthermore, any one of the above implies that given any two points ${\displaystyle p,q\in M,}$ there exists a length minimizing geodesic connecting these two points (geodesics are in general critical points for the length functional, and may or may not be minima).

## Notes

1. ^ Hopf, H.; Rinow, W. (1931). "Ueber den Begriff der vollständigen differentialgeometrischen Fläche". Commentarii Mathematici Helvetici. 3 (1): 209–225. doi:10.1007/BF01601813. hdl:10338.dmlcz/101427.
2. ^ Atkin, C. J. (1975), "The Hopf–Rinow theorem is false in infinite dimensions" (PDF), The Bulletin of the London Mathematical Society, 7 (3): 261–266, doi:10.1112/blms/7.3.261, MR 0400283.
3. ^ O'Neill, Barrett (1983), Semi-Riemannian Geometry With Applications to Relativity, Pure and Applied Mathematics, 103, Academic Press, p. 193, ISBN 9780080570570.