Definition:Admissible Subdivision of Family of Curves

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

Let $I = \closedint a b$ is a closed real interval.

Let $J$ is an open real interval.

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

Let $\tuple {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ be a finite subdivision of $I$ such that for all $i \in \N_{> 0} : i \le n$ the mapping $\Gamma$ is smooth on $J \times \closedint {a_{i - 1}} {a_i}$.

Suppose that for all $s \in J$ the mapping $\map {\Gamma_s} t = \map \Gamma {s, t}$ is an admissible curve.

Then $\tuple {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ is called the admissible subdivision of family of curves.