Definition:Normal Neighborhood
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Definition
Let $\struct {M, g}$ be a Riemannian or pseudo-Riemannian manifold without boundary.
Let $T_p M$ be the tangent space of $M$ at $p \in M$.
Let $\EE$ be the domain of the exponential map.
Let $\EE_p = \EE \cap T_p M$.
Let $\exp_p : \EE_p \to M$ be the restricted exponential map.
Let $U_p \subseteq M$ be a neighborhood of $p \in M$.
Suppose $U_p$ is the diffeomorphic image of $\exp_p$ of a star-shaped neighborhood of $0 \in T_p M$.
Then $U_p$ is called a normal neighborhood of $p$.
Also see
- Results about normal neighborhoods can be found here.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. Normal Neighborhoods and Normal Coordinates