Equality of Radial Distance Function to Riemannian Distance

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

Let $U_p = \map {\exp_p} {\map {B_\epsilon} 0 }$ be an open or closed geodesic ball around $p \in M$.

Let $r : U_p \to \R$ be the radial distance function.

Let $d_g$ be the Riemannian distance.

Then:


 * $\forall p \in M : \forall x \in U_p : \map r x = \map {d_g} {p, x}$