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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