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 $p \in M$.

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

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 $U_p$.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Geodesics Are Locally Minimizing