The geometry of canal surfaces and the length of curves in de Sitter space

TL;DR

This paper investigates the minimal length of certain closed space-like curves in de Sitter space, linking them to canal surfaces and conformal geometry, and derives bounds for conformal torsion.

Contribution

It establishes a connection between closed space-like curves in de Sitter space and canal surfaces, providing a new geometric lower bound for conformal torsion.

Findings

Minimal length of closed space-like curves in de Sitter space identified

Lower bound for total conformal torsion derived

Relationship between curves and canal surfaces elucidated

Abstract

We find the minimal value of the length in de Sitter space of closed space-like curves with non-vanishing non-space-like geodesic curvature vector. These curves are in correspondence with closed almost-regular canal surfaces, and their length is a natural magnitude in conformal geometry. As an application, we get a lower bound for the total conformal torsion of closed space curves.