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Let $M$ be a connected Riemannian manifold. Hopf-Rinow states that if $\exp_p$ is defined on all of $T_p M$ for all $p \in M$, then $M$ is geodesically complete.

I'm wondering whether it is sufficient for $\exp_p$ to be defined on the whole tangent space for one $p \in M$. (If you go with the interpretation that geodesic incompleteness comes from "cuts" or "holes" in your manifold, then the exponential map should be able to "see" these.)

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Yes it is sufficient, as explains Jack Lee in the comment I posted. You can find in his book Riemannian Manifolds: An Introduction to Curvature the following corollary of Hopf-Rinow theorem.

Corollary 6.14: If there exists one point $p ∈ M$ such that the restricted exponential map $\exp_p$ is defined on all of $T_p M$, then $M$ is complete.

It is more a corollary of the proof of Hopf-Rinow, not really of the theorem itself.

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    $\begingroup$ There are also some formulations of Hopf-Rinow that make this explicit. I googled and found Thomas_Franzinetti_38996.pdf in which Hopf-Rinow (theorem 3.35) contains four equivalent statement, and the one asked here, is called (C1). $\endgroup$ – Jeppe Stig Nielsen Jul 12 at 23:40

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