Definition:Tubular 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 $U$ is the normal neighborhood of $P$ in $M$.

Let $\delta : P \to \R$ be a positive continuous function.

Suppose $U$ is the diffeomorphic image under $E$ of $V$ where:


 * $V = \set {\tuple {x, v} \in NP : \norm v_g < \map \delta x}$

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