Definition:Closed Geodesic Ball in Riemannian Manifold

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 $M$.