Definition:Normal Exponential Map

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

Let $P \subseteq M$ be an embedded submanifold.

Let $\pi : NP \to P$ be the normal bundle of $P$ in $M$.

Let $TM$ be the tangent bundle of $M$.

Let $\exp$ be the exponential map.

Let $\EE \subseteq TM$ be the domain of $\exp$ of $M$.

Let $\EE_P = \EE \cap NP$.

Suppose $E : \EE_P \to M$ is a restriction of $\exp$ to $\EE_P$.

Then $E$ is called the normal exponential map (of $P$ in $M$).