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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).

I refer to Section 2.1, Section 2.2, Volume 1 Section 8.6 (Part 1) and Volume 1 Section 8.6 (Part 2).

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 $a<t\le b$"
    • 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:

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    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 (https://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 05 '19 at 03:17
  • @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 $c$? –  Aug 05 '19 at 03:26
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    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 05 '19 at 03:28

1 Answers1

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Here is a good reference that goes into some detail, of how to work with absolutely continuous curves on Riemannian manifolds: http://nyjm.albany.edu/j/2015/21-12v.pdf In other words, there is a reasonable extension of notions like the speed of a curve on a Riemannian manifold so that the answer to your question is negative.

Notably, a similar strategy sometimes allows you to work even with curves which are defined in an abstract metric space, with no manifold structure at all. For this, a good reference is the first half of the book by Ambrosio, Gigli, and Savaré.

pseudocydonia
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  • Thanks. Wait, the issue I had in mind is using the assumption behind the fundamental theorem of calculus. I've forgotten the exact rule, but I think it's like "a continuous $\mathbb R$-valued function on a closed interval is Riemann integrable". If $|c'|$ is continuous, then we can define $s$ (and thus $l$). If $c$ is regular, then $|c'|$ is smooth and thus continuous. 1. Now you're saying that if $c$ is still smooth but now also absolutely continuous (TIL smooth does not imply absolutely continuous), then $|c'|$ is at least continuous? 2. 2. If not... –  Aug 05 '19 at 03:41
  • If not, then what do you mean? If so, then to strengthen as much as possible, what is the highest $k$ for $|c'| \in C^k[a,b]$ (just $k=0$, actually $k=\infty$, etc)?
    • –  Aug 05 '19 at 03:41
  • You mean $c$ non-regular and $c$ smooth and absolutely continuous implies $|c'|$ is absolutely continuous? (Edit: pseudocydonia seems to have deleted the previous comment) –  Aug 05 '19 at 03:45
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    Sorry, I misspoke again. Absolute continuity should be sufficient to define arc length and the arc length function - no smoothness is required. – pseudocydonia Aug 05 '19 at 03:49
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    This is because the definition of absolute continuity directly gives you an arc length function if you look carefully. – pseudocydonia Aug 05 '19 at 03:50
  • Thanks. You don't have to be sorry. You mean 3. $c$ non-regular and $c$ absolutely continuous and smooth implies $|c′|$ is absolutely continuous? 4. And $c$ non-regular and $c$ absolutely continuous and smooth implies $|c′|$ is absolutely continuous, even if $c$ is not smooth, but with a different definition of $|c′|$ or $c'$? (I think I might think about (4) another day. For today, please just stick to (3).) –  Aug 05 '19 at 04:00
  • I don't know about that. I just meant that $c$ non-regular, absolutely continuous is enough for every section of the curve to have a well-defined arc length. – pseudocydonia Aug 05 '19 at 04:04
  • Oh wait I think I get you. Arc length here is defined as Riemann integral over domain of $||c'||$ (which is $[a,b]$) assuming we can Riemann integrate $||c'||$ over $[a,b]$. Now you're defining arc length in a way that doesn't involve $||c'||$, where your way doesn't even involve smoothness of $c$, so long as $c$ is absolutely continuous? –  Aug 05 '19 at 04:06
  • "In other words, there is a reasonable extension of notions like the speed of a curve on a Riemannian manifold so that the answer to your question is negative." 5. Why negative and not affirmative? (My question is defining speed for irregular/not an immersion) I suspect negative and not affirmative because of not assuming "absolutely continuous" 6. For this particular notion of speed: can it be done without assuming $c$ regular/an immersion and without additional assumptions (e.g. absolute continuity)? –  Aug 05 '19 at 04:08
  • Yes and no - if you look at Section 3 of the paper I linked, it shows that your particular notion of speed is actually the same as the way one usually defines length for AC curves. – pseudocydonia Aug 05 '19 at 04:22
  • And another point is that $\Vert c^\prime \Vert$ need not be continuous, just Riemann/Lebesgue integrable. – pseudocydonia Aug 05 '19 at 04:25
  • Are all of these correct please? 7.1 This notion of speed works for $c$ regular/an immersion. 7.2 This notion of speed works for $c$ not an immersion/irregular if $c$ is absolutely continuous. 7.3. This notion of speed does not work for $c$ not an immersion/irregular and $c$ not absolutely continuous.
    • –  Aug 05 '19 at 04:25
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    7.1 correct, 7.2 see comment below, 7.3 probably not correct. One issue, which I'm sure you've seen, is that non-immersive curves can cross themselves. At these points, the derivative is no longer well-defined. Therefore the notion of speed no longer makes any sense. – pseudocydonia Aug 05 '19 at 04:27
  • Re "∥c′∥ need not be continuous, just Riemann/Lebesgue integrable": Maybe for your definition this is true, but for this definition, Tu explicitly uses fundamental theorem of calculus. Is there a version of FTC where instead of assuming $f$ continuous we assume $f$ Riemann integrable (the conclusions maybe different then, so we need not have $F$ uniformly continuous, but as long as $F$ is differentiable, then that's ok I guess; my version is based on wikipedia)?
    • –  Aug 05 '19 at 04:28