Total Torsion of Curves in Three-Dimensional Riemannian 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. We here extend these results to closed curves in three dimensional manifolds with constant curvature. We also extend an interesting companion to the total torsion theorem, which was proved for surfaces in $\mathds{R}^3$ by L. A. Santal\'o, and some results involving the total torsion of lines of curvature.