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 $\int_{\gamma} \tau(s)ds=0$."

I have three questions on this subject:

1. Is there any paper or a reference which used 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 which integral along any unit speed (Frenette) closed curve is equal to zero. Now Lets replace the sphere $S^{2}$ with the torus or another compact Riemann surface $S$. Is there a universal quantity $Q$ which integral along every unit speed Frenette 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?