Geodesic Sphere as Metric Sphere in Riemannian Manifold

Theorem

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

Let $U$ be geodesic sphere with radius $R$ in $M$.

Then $U$ is a metric sphere with radius $\epsilon = R$.

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