Gauss Lemma for Riemannian Manifolds
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Theorem
Let $\struct {M, g}$ be a Riemannian manifold.
Let $U = \map {\exp_p} {\map {B_\epsilon} 0 }$ be a geodesic ball centered at $p \in M$.
Let $\partial_r$ be the radial vector field on $U \setminus \set p$, where $\setminus$ denotes the set difference.
Then $\partial_r$ is a unit vector field orthogonal to the geodesic spheres in $U \setminus \set p$.
Proof
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Source of Name
This entry was named for Carl Friedrich Gauss.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Geodesics Are Locally Minimizing