Corollary of 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 $r$ be the radial distance function.
Let $\partial_r$ be the radial vector field.
Suppose $\grad$ is the gradient operator.
Then:
- $\forall x \in U \setminus \set p : \valueat {\grad r}x = \valueat {\partial_r} x$
where $\setminus$ denotes the set difference.
Proof
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Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Geodesics Are Locally Minimizing