Definition:Velocity of Transverse Curve

Definition

Let $\struct {M, g}$ be a Riemannian manifold.

Let $I, J \subseteq \R$ be real intervals.

Let $\map {\Gamma^{\paren t} } s : J \times I \to M$ be the transverse curve, where $\times$ denotes the cartesian product.

Suppose $\map {\Gamma^{\paren t}} s$ is differentiable on $J$.

Then the velocity of the main curve is denoted by:

$\partial_s \map {\Gamma} {s, t} = \map {\paren {\Gamma^{\paren t}}'} s$

Here $\partial_s \map {\Gamma} {s, t} \in T_{\map \Gamma {s, t} } M$ where $T_p M$ is the tangent space of $M$ at $p \in M$, and $\map \Gamma {s, t}$ is the one-parameter family of curves on $M$.

Sources

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