the Hopf-Rinow theorem for length spaces
/ HOPF REE-noh /
When is a space 'finished' enough that you can always find the shortest route between any two points, and always keep walking along a geodesic as far as you like? The Hopf-Rinow theorem answers this for length spaces. It identifies the mild, checkable hypotheses under which a length space is as well-behaved as a complete surface: every pair of points is joined by a shortest path, and bounded regions are compact. It is the metric-geometry generalization of the classical Riemannian Hopf-Rinow theorem, dropping all smoothness.
The statement: let X be a length space that is complete (Cauchy sequences converge) and locally compact (every point has a compact neighborhood). Then X is proper — every closed metric ball is compact — and X is a geodesic space, so any two points are joined by a minimizing geodesic. The proof skeleton is a continuity-and-compactness argument: local compactness lets you extend a shortest path step by step, completeness keeps the limit inside the space, and together they upgrade the length-space infimum into an attained minimum via the Arzela-Ascoli theorem applied to a sequence of near-shortest paths. The four conditions — complete, locally compact, proper, geodesic — become tightly linked once you are a length space.
This theorem is what licenses treating complete Riemannian manifolds, Cayley graphs, and many singular spaces as honest geodesic spaces, and it is the workhorse behind existence of geodesics in comparison geometry. The honest caveat: local compactness cannot be dropped. Infinite-dimensional examples — like an infinite-dimensional Hilbert space with the round-sphere metric, or certain infinite-dimensional CAT(0) spaces — can be complete length spaces that are not proper and where some pairs have no shortest path. So 'complete length space' alone does not give you geodesics; you genuinely need the local compactness.
The Cayley graph of a finitely generated group, with each edge of length 1, is a complete, locally compact length space (it is locally finite if the generating set is finite). Hopf-Rinow then makes it a proper geodesic space: every closed ball contains finitely many vertices and is compact, and any two vertices are joined by a shortest edge-path realizing the word metric. This is exactly the property that lets geometric group theory treat the graph as a genuine geometric object.
On a locally finite Cayley graph, Hopf-Rinow guarantees proper, geodesic structure realizing the word metric.
The metric version is strictly weaker than asserting smooth geodesic completeness — it does not give a smooth exponential map. Also it genuinely needs local compactness: complete-but-not-locally-compact length spaces exist where no shortest path joins some pairs.