On curves with nonnegative torsion

TL;DR

This paper investigates the torsion properties of space curves, providing new proofs that nonnegative torsion implies planarity and showing that closed plane curves cannot be perturbed into space curves with constant nonzero torsion, with extensions to Lorentzian geometry.

Contribution

It offers novel proofs regarding torsion conditions for space curves and establishes new limitations on perturbing plane curves into space curves with constant torsion, including Lorentzian cases.

Findings

Nonnegative torsion implies zero torsion and planarity.

Closed plane curves cannot be perturbed into space curves with constant nonzero torsion.

Results extend to Lorentzian $ ext{R}^{2,1}$} and relate to open questions in general relativity.

Abstract

We provide new results and new proofs of results about the torsion of curves in $\mathbb{R}^3$. Let $\gamma$ be a smooth curve in $\mathbb{R}^3$ that is the graph over a simple closed curve in $\mathbb{R}^2$ with positive curvature. We give a new proof that if $\gamma$ has nonnegative (or nonpositive) torsion, then $\gamma$ has zero torsion and hence lies in a plane. Additionally, we prove the new result that a simple closed plane curve, without any assumption on its curvature, cannot be perturbed to a closed space curve of constant nonzero torsion. We also prove similar statements for curves in Lorentzian $\mathbb{R}^{2,1}$ which are related to important open questions about time flat surfaces in spacetimes and mass in general relativity.