Definition:Closed Geodesic Ball in Riemannian Manifold
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Definition
Let $\struct {M, g}$ be a Riemannian manifold.
Let $T_p M$ be the tangent space at $p \in M$.
Let $\exp_p$ be the restricted exponential map.
Let $\exp_p$ on $V$ be a diffeomorphism onto its image.
Let $V \subseteq T_p M$ be an open set.
Let $\map {B_\epsilon^-} 0 \subseteq V$ be the closed ball in $T_p M$ with $\epsilon \in \R_{>0}$ such that $\exp_p$ is a diffeomorphism from $\map {B_\epsilon} 0$ to its image.
Then the image set $\map {\exp_p} {\map {B_\epsilon^-} 0 }$ is called the closed geodesic ball in $M$.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Geodesics Are Locally Minimizing