Total torsion of three-dimensional lines of curvature

Abstract

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