Total torsion of three-dimensional lines of curvature

TL;DR

This paper investigates the total torsion of three-dimensional lines of curvature on hypersurfaces in Riemannian manifolds, establishing conditions under which the total torsion is quantized or vanishes, extending classical results.

Contribution

It introduces a new characterization of the total torsion for three-dimensional lines of curvature, linking it to the geometry of the hypersurface and generalizing classical theorems.

Findings

Total torsion of well-positioned lines of curvature is an integer multiple of 2π.

If total torsion is a multiple of 2π, such a curve exists on some hypersurface.

Total torsion vanishes when the hypersurface is convex.

Abstract

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