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 $M$.

Sources

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