Connected Riemannian Manifold with Restricted Exponential Map defined on Whole Tangent Space admits Minimizing Geodesic Segment

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 defined on the whole $T_p M$.

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Then for all $q \in M$ there is a minimizing geodesic segment from $p$ to $q$.

Proof

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Sources

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