Definition:Radial Geodesic in Normal Neighborhood

Definition

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

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

Let $\struct {U_p, \tuple {x^i}}$ be a normal coordinate chart.

Let $I \subseteq \R$ be a real interval such that $0 \in I$.

Let $\map \gamma t : I \to M$ be the geodesic such that:

$\map \gamma 0 = p$

Suppose:

$\map \gamma I \subseteq U$

Then $\gamma$ is called the radial geodesic.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. Normal Neighborhoods and Normal Coordinates