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?