I found the following $*$ problem in final pages of the book of De Carmo "Differential geometry of curves and surface". This problem is a classical theorem which is referenced in the book of De Carmo.

"Let $\gamma$ be a unit speed closed curve on $S^{2}$ . Then $\int_{\gamma} Tor(s)ds=0$

I have two 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 term of Riemannian geometry?