Definition:Normal Neighborhood of Embedded Riemannian Submanifold

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 $\EE$ be the domain of the exponential map.

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

Let $U \subseteq M$, $V \subseteq \EE_P$ be open subsets.

Let $E$ be the normal exponential map.

Suppose $\forall x \in M$ the intersection of $V$ with fibers $N_x M$ is star-shapped $0$.

Suppose $U$ is the diffeomorphic image under $E$ of $V$.

Then $U$ is called the normal neighborhood of $P$ in $M$.