The curvature and torsion of curves in a surface

Abstract

Let f be a function with certain properties and $$\gamma $$ be a closed curve with the torsion $$\tau $$ . We prove that $$\oint _{\gamma }f\tau ds=0$$ if $$\gamma $$ is a spherical curve, and conversely, if a surface makes the integral equal to zero for all closed curves, it is part of a sphere or a plane. This generalizes a known theorem on the total torsion for a closed curve.