Gauss Lemma for Riemannian Manifolds

Theorem
Let $\struct {M, g}$ be a Riemannian manifold.

Let $U = \map {B_\epsilon} p$ 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 $U \setminus \set p$.