Radial Distance Function in Normal Neighborhood

Theorem
Let $\struct {M, g}$ be an $n$-dimensional Riemannian manifold.

Let $U_p$ be the normal neighborhood of $p \in M$.

Then $r$ is well-defined in $U_p$ independently of the choice of normal coordinates.

Furthermore, $r$ is smooth on $U_p \setminus \set p$, where $\setminus$ denotes the set difference, and $r^2$ is smooth on $U_p$.