Definition:Radial Distance Function

Definition

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

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

Let $\tuple {x^i}$ be the local coordinates on $U_p$ centered at $p \in M$.

Then the mapping $r : U_p \to \R$ defined by

$\ds \map r x = \sqrt {\sum_{i \mathop = 1}^n x_i^2}$

is called the radial distance function.

Sources

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