Corollary of Gauss Lemma for Riemannian Manifolds

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.