Definition:Normal Neighborhood of Embedded Riemannian Submanifold
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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 with respect to $0$.
Suppose $U$ is the diffeomorphic image under $E$ of $V$.
Then $U$ is called the normal neighborhood of $P$ in $M$.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. Tubular Neighborhoods and Fermi Coordinates