Points of Riemannian Manifold are Contained in Geodesically Convex Geodesic Balls

Theorem

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

Let $\epsilon_0 \in \R_{> 0}$.

Then for all $p \in M$ there exists a closed geodesic ball or open geodesic ball centered at $p \in M$ of radius $\epsilon \le \epsilon_0$ which is also geodesically convex.

Proof

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Sources

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