Total torsion of three-dimensional lines of curvature

A curve $$ in a Riemannian manifold M is three-dimensional if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when $$ lies on an oriented hypersurface S of M, we say that $$ is well positioned if the curve's principal normal, its torsion vector, and the surface normal are everywhere coplanar. Suppose that $$ is three-dimensional and closed. We show that if $$ is a well-positioned line of curvature of S, then its total torsion is an integer multiple of $$; and that, conversely, if the total torsion of $$ is an integer multiple of $$, then there exists an oriented hypersurface of M in which $$ is a well-positioned line of curvature. Moreover, under the same assumptions, we prove that the total torsion of $$ vanishes when S is convex. This extends the classical total torsion theorem for spherical curves.