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:

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 $S^{2}$ ?

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?

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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