Definition:Regular Curve

Definition

Let $M$ be a smooth manifold.

Let $I \subseteq \R$ be a real interval.

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

For all $t \in I$ let $T_{\map \gamma t} M$ be the tangent space of $M$ at the point $\map \gamma t$.

Then $\gamma$ is called a regular curve if and only if:

$\forall t \in I : \map {\gamma'} t \ne 0$

This article, or a section of it, needs explaining. In particular: Where is $\gamma'$ defined? It should be defined similarly to Definition:Tangent Vector/Definition 2 Do you mean Definition:Velocity of Smooth Curve? Unfortunately, this explicit form is given 2 chapters later. 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.

$3$-Dimensional Real Vector Space

Let $\alpha : I \to \R^3$ be a smooth curve.

$\alpha$ is said to be regular if and only if:

$\forall t \in I : \map {\alpha'} t \ne \bszero_{\R^3}$

Also see

• Results about regular curves can be found here.

Sources

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