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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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    $\begingroup$ 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 $\Vert c^\prime (t) \Vert$. $\endgroup$ – pseudocydonia Aug 5 at 3:17
  • $\begingroup$ @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 $c$? $\endgroup$ – Selene Auckland Aug 5 at 3:26
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    $\begingroup$ 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. $\endgroup$ – pseudocydonia Aug 5 at 3:28
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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.

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    $\begingroup$ Sorry, I misspoke again. Absolute continuity should be sufficient to define arc length and the arc length function - no smoothness is required. $\endgroup$ – pseudocydonia Aug 5 at 3:49
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    $\begingroup$ This is because the definition of absolute continuity directly gives you an arc length function if you look carefully. $\endgroup$ – pseudocydonia Aug 5 at 3:50
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    $\begingroup$ 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. $\endgroup$ – pseudocydonia Aug 5 at 4:27
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    $\begingroup$ Sorry, for 7.2: the point is that one can give various, more general definitons of length/speed which do not require the curve to be an immersion, but which nonetheless coincide with the definition you have given, for every curve which actually is an immersion. $\endgroup$ – pseudocydonia Aug 5 at 4:32
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    $\begingroup$ And I misread, you are right for (7.3). It does not work for non-immersive curves, simply because the derivative tangent vector does not generally exist at every point. $\endgroup$ – pseudocydonia Aug 5 at 4:33

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