On curves with nonnegative torsion

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.