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I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book of do Carmo.

"Let $\gamma$ be a unit speed closed curve on $S^{2}$. Then $\int_{\gamma} \tau(s)ds=0$ where $\tau$ is the torsion of $\gamma$"

Regarding the above theorem, I have the folloing three questions:

  1. Is there any paper or a reference which uses this fact as a criterion for existence or non-existence of periodic orbit for a vector field on $S^{2}$ ?

  2. In the above theorem the "torsion" is a universal quantity whose integral along any unit speed (Frenet) closed curve is equal to zero. Now let's replace the sphere $S^{2}$ with the torus or another compact Riemann surface $S$. Is there a universal quantity $Q$ whose integral along every unit speed Frenet closed curve contained in $S$ is zero? More generally, what is a generalization of this theorem in Riemannian geometry?

  3. In the last part of the exercise it is written in the parentheses that this integral condition is also a sufficient condition for a closed non-planar curve to lie in a sphere. The reference is not in English. But I have a misunderstanding on this statement. Because it seems that every planar curve can be perturbed such that the resulting curve satisfies this integral but it does not lie in any sphere. Could you please help me to remove this misunderstanding?

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    $\begingroup$ It is one of the main problems in integral geometry to find out which functions (or vector fields or tensor fields) on a Riemannian manifold integrate to zero over all closed geodesics. Results on this problem might not answer exactly the question you ask but they seem similar. $\endgroup$
    – Joonas Ilmavirta
    Commented Dec 19, 2014 at 18:05
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    $\begingroup$ Here are two references to get you started: arxiv.org/abs/1303.6114 and arxiv.org/abs/1410.2114 They both contain references to results in this direction. The first paper also discusses the case of manifolds with boundary, but this is somewhat different. The answer is known on some closed manifolds, like Lie groups and Anosov manifolds. (The problem of recovering a function from its integrals over lines or similar objects has been studied a lot in Euclidean spaces. Google "Radon transform" if you want to learn more. The literature on it is vast.) $\endgroup$
    – Joonas Ilmavirta
    Commented Dec 21, 2014 at 17:32
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    $\begingroup$ I just want to remark that this is a mix of intrinsic and extrinsic geometry. You have to view the sphere as a specific submanifold of $\mathbb R^3$, since the torsion of $\gamma$ refers to viewing $\gamma$ as a curve in $\mathbb R^3$. So a priori if you look for generalizations to other surfaces or to Riemannian geometry, you are looking for extrinsic invariants of a curve, whose integral vaishes provided that the curve lies in a given hypersurface. $\endgroup$
    – Andreas Cap
    Commented Apr 17, 2016 at 13:18
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    $\begingroup$ One needs to double check this information: I ran in 1974/5 into a pre-WWII paper in German authored by Karol Borsuk that stated and proved the mentioned theorem about the integral of torsion over a closed curve contained in $\ S^2\ $ being $\ 0$. $\endgroup$
    – Wlod AA
    Commented Jan 28, 2023 at 13:34
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    $\begingroup$ @WlodAA Thank you very much for your very helpful comment $\endgroup$
    – Ali Taghavi
    Commented Jan 28, 2023 at 13:40

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For #3: the comment in the parenthesis is wrong. By the way it disappeared in the latter edition.

Yes one can perturb a given curve to make it total torsion zero. The following statement should be right:

If any closed smooth curve on a smooth nonflat surface has vanishing total curvature, then the surface is a part of round sphere.

I guess that do Carmo wanted to say. By the way, the book is translated and there is a chance that this mistake is a result of translation.

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  • $\begingroup$ Thank you very much for your very interesting answer $\endgroup$
    – Ali Taghavi
    Commented Jan 28, 2023 at 12:45
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Here is a paper published in 2017, which proves a more general result by considering $\displaystyle\int_{\gamma} f(s)\tau(s) ds$, where $f$ is a function satisfying certain condition.

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