Definition:One-Parameter Family of Curves on Riemannian Manifold

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

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

Let $\Gamma : I \times J \to M$ be a continuous map, where $\times$ denotes the cartesian product.

Then $\Gamma$ is called the one-parameter family of curves on $M$.