Hopf–Rinow theorem

Remarks

The key step is to show that if $\exp_p$ is defined on all of $T_pM$, then every $q$ is reached by a minimizing geodesic from $p$. Choose a point $x$ nearest to $q$ on a small sphere around $p$, and follow the geodesic from $p$ through $x$. One shows that it decreases the distance to $q$ at unit rate until it reaches $q$. From this, bounded closed sets are closed subsets of images of compact balls under $\exp_p$, which gives the other conditions. Example: $\R^n$ and every compact Riemannian manifold satisfy the conditions. The open disc $\set{x \in \R^2 : \abs{x} < 1}$ with the Euclidean metric fails them: a radial geodesic leaves the disc in finite time, and the sequence $(1 - 1/k, 0)$ is Cauchy with no limit. The last conclusion can hold without completeness, as in the open disc, so it is not equivalent to the conditions.

Going deeper

For a student

The theorem is used constantly in global Riemannian geometry. Completeness is a standing hypothesis in the theorems of Myers, Cartan–Hadamard and others, and one can check it in whichever form is easiest. For example, compact manifolds and homogeneous Riemannian manifolds are complete. It also shows that $\exp_p$ is surjective for complete $M$, which is the starting point for the study of the cut locus. The theorem fails for pseudo-Riemannian manifolds, where geodesic completeness and connection by geodesics behave differently.