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.

Statement

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

1. The closed and bounded subsets of $M$ are compact;
2. $M$ is a complete metric space;
3. $M$ is geodesically complete; that is, for every $p\in M,$ the exponential map expp is defined on the entire tangent space $\operatorname {T} _{p}M.$ Furthermore, any one of the above implies that given any two points $p,q\in M,$ 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).

Variations and generalizations

• The Hopf–Rinow theorem is generalized to length-metric spaces the following way:
• If a length-metric space $(M,d)$ is complete and locally compact then any two points in $M$ can be connected by a minimizing geodesic, and any bounded closed set in $M$ is compact.
• The theorem does not hold in infinite dimensions: (Atkin 1975) showed that two points in an infinite dimensional complete Hilbert manifold need not be connected by a geodesic.
• The theorem also does not generalize to Lorentzian manifolds: the Clifton–Pohl torus provides an example that is compact but not complete.