Total torsion of three-dimensional lines of curvature

Abstract

A curve γ 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 γ lies on an oriented hypersurface S of M, we say that γ is well positioned if the curve's principal normal, its torsion vector, and the surface normal are everywhere coplanar. Suppose that γ is three-dimensional and closed. We show that if γ is a well-positioned line of curvature of S, then its total torsion is an integer multiple of 2π; and that, conversely, if the total torsion of γ is an integer multiple of 2π, then there exists an oriented hypersurface of M in which γ is a well-positioned line of curvature. Moreover, under the same assumptions, we prove that the total torsion of γ vanishes when S is convex. This extends the classical total torsion theorem for spherical curves.