Definition:Geodesic Sphere 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 $\map {B_\epsilon} 0 \subseteq T_p M$ be the open ball in $T_p M$.

Let $\partial \map {B_\epsilon} 0$ be the boundary of $\map {B_\epsilon} 0$.

Then the image set $\map {\exp_p} {\map {\partial B_\epsilon} 0 }$ is called the geodesic sphere in $M$.