Definition:Geodesically Convex Neighborhood

Definition

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

Let $U \subseteq M$ be a subset.

Suppose, for all $p, q \in U$ there is a unique minimizing geodesic segment $\gamma$ from $p$ to $q$.

Suppose the image of all such $\gamma$ is contained in $U$.

Then $U$ is said to be geodesically convex.

Sources

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