Definition:Uniform Tubular Neighborhood

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

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

Let $U \subseteq M$ be the tubular neighborhood of $P$ in $M$ with $\map \delta x = \epsilon$ where $\epsilon \in \R_{> 0}$.

Then $U$ is called the uniform tubular neighborhood (or $\epsilon$-tubular neighborhood) of $P$ in $M$.