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.
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