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 $M$.

Also see

• Results about tubular neighborhoods can be found here.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. Tubular Neighborhoods and Fermi Coordinates