Hopf-Rinow Theorem/Corollary 1

Theorem

Let $\struct {M, g}$ be a connected Riemannian manifold.

Let $T_p M$ be the tangent space at $p \in M$.

Let $\exp_p$ be the restricted exponential map.

Suppose there is $p \in M$ such that $\exp_p$ is defined on all of $T_p M$.

Then $M$ is complete.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Completeness