Radial Geodesic Connecting Two Points in Geodesic Ball is Unique Minimizing Curve

Theorem

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

Let $U = \map {\exp_p} {\map {B_\epsilon} 0 }$ be a geodesic ball centered at $p \in M$.

Suppose $q \in U$.

Suppose $\gamma$ is a radial geodesic from $p$ to $q$.

Then, up to reparametrization, $\gamma$ is the unique minimizing curve 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. Geodesics Are Locally Minimizing