Geodesic Ball as Metric Ball in Riemannian Manifold

Theorem

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

Let $U^c$ and $U^o$ be closed and open geodesic balls with radii $R_c$ and $R_o$ respectively in $M$.

Then $U^c$ is a closed metric ball with radius $\epsilon_c = R_c$, and $U^o$ is an open metric ball with radius $\epsilon_o = R_o$.

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