# Does an exponential map defined on all of $T_p M$ for one $p$ imply completeness?

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.)

## 1 Answer

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.

• 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). – Jeppe Stig Nielsen Jul 12 at 23:40