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.

This article, or a section of it, needs explaining. In particular: How is the regularity defined for a closed interval? The book does not state it clearly, but from what I gather, for closed intervals we are taking one-sided limits. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Lengths and Distances