If I have $X$ a complete separable metric space, $x, y \in X$ arbitrary points, how can I define a constant speed geodesic, i.e. a continuous map $g : [0,1] \rightarrow X$ such that $$ d(g(t), g(s)) = |t-s| d(g(1), g(0)) \quad \forall t,s \in [0,1]$$ and $g(0) = x, g(1) = y$?
2026-03-25 19:03:23.1774465403
Existence of a geodesic in a complete separable metric space
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It is necessary that $X$ be a length space; this is much stronger than the topological conditions you're considering. If $X$ is additionally locally compact, then the Hopf-Rinow theorem says that constant speed geodesics exist. See Theorem 2.4 of these notes. The local compactness assumption cannot be dropped, according to Wikipedia; there are infinite-dimensional counterexamples.