Definition:Radial Vector Field

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 $U$ centered at $p \in M$.

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

Then the vector field $\partial_r$ on $U \setminus \set p$, where $\setminus$ denotes the set difference, defined by

$\ds \partial_r = \frac {x^i}{\map r x} \frac {\partial}{\partial x^i}$

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