# Can speed be defined for a parametrized curve that is not regular/not an immersion?

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (Volume 2) and An Introduction to Manifolds by Loring W. Tu (Volume 1).

Use $$t$$ to denote the standard coordinate on $$[a,b]$$, and use $$t_0$$ to denote a point in $$[a,b]$$. Let $$x$$ be the standard coordinate on $$[0,l]$$. The speed of a curve $$c: [a,b] \to M$$ into a Riemannian manifold $$M$$ at a point $$t_0 \in [a,b]$$ is defined $$\|c'(t_0)\| := \sqrt{\langle c'(t_0), c'(t_0) \rangle_{t_0}}$$. Then we can define speed as a map by $$\|c'\|: [a,b] \to [0, \infty), (\|c'\|)(t_0) := \|c'(t_0)\|$$. Here, it seems to be claimed that this map $$\|c'\|$$ is the derivative of arc length function $$s$$ of $$c$$.

Question: In the first place, is $$c$$ supposed to be assumed regular/an immersion for the definition of speed $$\|c'\|$$, arc length $$l$$ or arc length function $$s$$, and why/why not?

My thoughts:

1. If $$c$$ is regular/an immersion, then $$\|c'\|$$ is smooth by this, but I think it's possible to define $$\|c'\|$$, $$l$$ and $$s$$ for continuous $$\|c'\|$$. I can't think of a condition on $$c$$ to make $$\|c'\|$$ continuous but not necessarily smooth (see thought (2) below).

• 1.1. Edit: I actually didn't mention earlier: Observe that in the paragraph before Proposition 2.3, Tu uses the fundamental theorem of calculus. Based on the version of FTC on Wikipedia, I think the rule behind FTC is something like
• "continuous $$\mathbb R$$-valued functions defined on a closed interval $$[a,b]$$ of $$\mathbb R$$ are Riemann integrable on $$[a,t]$$ for any $$a"
• Without such rule, I don't think we can define the "$$F$$" in the version of FTC on Wikipedia. With such rule, if $$\|c'\|$$ (the "f") were continuous, then we could define $$s$$ (the "F") and thus define $$l$$. If $$c$$ is regular/an immersion, then $$\|c'\|$$ is smooth and thus continuous. If $$c$$ were irregular/not an immersion, then $$\|c'\|$$ is not necessarily smooth, I think (see thought (2) below). But we can still define $$s$$ (and thus define $$l$$) by the rule if $$\|c'\|$$ is somehow at least continuous.
2. It could be possible $$\|c'\|$$ is actually continuous or even smooth for an irregular/a non-immersion, but still smooth, $$c$$ because in this question, Paulo Mourão can prove the smoothness part without immersion.

3. Update: I think we can still define $$\|c'\|$$, $$l$$ and $$s$$ for an irregular/a non-immersion $$c$$ because there's this exercise: Exercise 2.6, which asks for the arc length of a parametrized curve that was shown in Example 2.2 (see here) to be irregular/not an immersion. At the very least $$l$$ and $$\|c'\|$$ are defined. Not sure if $$s$$ is.

Context:

• One generalization you might find interesting is for absolutely continuous curves. I believe you should be able to define $c^\prime$ in the sense of the metric derivative (en.wikipedia.org/wiki/Metric_derivative) using local coordinates, and then you can use the Riemannian inner product to compute $\Vert c^\prime (t) \Vert$. – pseudocydonia Aug 5 at 3:17
• @pseudocydonia Thanks. I assume you mean define $\|c'\|$ (with manifold implies Riemannian manifold implies metric space) and not $c'$ (asked about here). Anyway, so indeed, $\|c'\|$, arc length $l$ or arc length function $s$ can be defined for irregular/non-immersion $c$? – Selene Auckland Aug 5 at 3:26
• Yes you're right, I was mistaken, the metric derivative should be able to give you the speed directly - it does not of course actually give you a vector $c^\prime$ in the tangent space. – pseudocydonia Aug 5 at 3:28