# Definition:Piecewise Regular Curve Segment

## Definition

Let $M$ be a smooth manifold.

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

Let $\gamma : I \to M$ be a curve segment.

Let $\tuple {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ be a finite subdivision of $I$.

For all $i \in \N : i < n$ let $\closedint {x_i} {x_{i + 1} }$ be a subinterval of subdivision of $I$.

Suppose for all $i \in \N : i < n$ the curve segment $\bigvalueat \gamma {\closedint {x_i} {x_{i + 1}} }$ is regular.

Then $\gamma$ is said to be a piecewise regular curve segment.

## Also known as

A piecewise regular curve segment is also known as an admissible curve.