Definition:Normal Exponential Map
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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 $TM$ be the tangent bundle of $M$.
Let $\exp$ be the exponential map.
Let $\EE \subseteq TM$ be the domain of $\exp$ of $M$.
Let $\EE_P = \EE \cap NP$.
Suppose $E : \EE_P \to M$ is a restriction of $\exp$ to $\EE_P$.
Then $E$ is called the normal exponential map (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