Definition:Velocity of Main Curve

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

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

Let $\map {\Gamma_s} t : J \times I \to M$ be the main curve, where $\times$ denotes the cartesian product.

Suppose $\map {\Gamma_s} t$ is differentiable on $I$.

Then the velocity of the main curve is denoted by:


 * $\partial_t \map {\Gamma} {s, t} = \map {\paren {\Gamma_s}'} t$

Here $\partial_t \map {\Gamma} {s, t} \in T_{\map \Gamma {s, t} } M$ 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 $M$.