Definition:Transverse Curve

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

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

Let $\Gamma : J \times I \to M$ be a one-parameter family of curves, where $\times$ denotes the cartesian product.

Let $t \in I$ be constant.

Then for all $s \in J$ the map $\map {\Gamma^{\paren t}} s = \map \Gamma {s, t}$ is called the transverse curve.