the Hopf-Rinow theorem

Imagine a world map with a hole punched in it: some pairs of cities have no shortest road because every route has to detour around the gap, and you can shave the trip a little forever without ever reaching a true minimum. The Hopf-Rinow theorem rules out such pathologies. It says that for a connected Riemannian manifold, three very different-looking notions of 'no holes' are in fact the same, and that when they hold, every two points really are joined by a shortest path.

Precisely: for a connected Riemannian manifold M, the following are equivalent. (1) M is complete as a metric space — Cauchy sequences in the distance d converge. (2) M is geodesically complete — every geodesic extends to all values of its parameter, i.e. the exponential map exp_p is defined on all of T_p M, for one (hence every) point p; you can keep walking straight forever and never fall off an edge. (3) Closed and bounded subsets of M are compact (Heine-Borel holds). Moreover — and this is the payoff — any one of these implies that for every pair p, q there exists a minimizing geodesic from p to q realizing the distance d(p,q). The proof of the last part walks outward along the geodesic emanating in the 'best' direction and shows the set of points it minimizes to is both open and closed.

Hopf-Rinow is the foundational hypothesis of global Riemannian geometry: it is why comparison theorems may freely say 'let M be complete' and then produce honest minimizing geodesics to compare. Two honest caveats. First, all clauses require connectedness and finite dimension; in infinite-dimensional Riemannian (Hilbert) manifolds the equivalence and the existence-of-minimizers conclusion can both fail. Second, compactness is sufficient but not necessary for completeness — Euclidean R^n is complete and non-compact. The metric version, valid for length spaces without any smoothness, lives in metric geometry, not here.