I am having some confusion about geodesic completeness. Is the interval [0,1] geodesically complete? As metric space it is complete hence Hopf Rinow theorem implies it is geodesically complete but a geodesic cannot be extended indefinitely on [0,1]. So what am i getting wrong here?
2026-04-29 20:18:58.1777493938
Is the interval [0,1] geodesically complete?
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Hopf-Rinow applies to connected boundaryless Riemannian manifolds. The closed interval $[0,1]$ can only be given its usual topology as a manifold with boundary.
That is: the closed interval is not geodesically complete, despite being Cauchy complete.