Total torsion of curves in three-dimensional manifolds

A classical result in differential geometry assures that the total torsion of a closed spherical curve in the three-dimensional space vanishes. Besides, if a surface is such that the total torsion vanishes for all closed curves, it is part of a sphere or a plane. Here we extend these results to closed curves in three dimensional Riemannian manifolds with constant curvature. We also extend an interesting companion for the total torsion theorem, which was proved for surfaces in by L. A. Santaló, and some results involving the total torsion of lines of curvature. Dedicated to Professor Manfredo P. do Carmo on his 80th birthday.