On curves with nonnegative torsion

Hubert L. Bray; Jeffrey L. Jauregui

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arXiv · 1312.5171

On curves with nonnegative torsion

Hubert L. Bray·Jeffrey L. Jauregui

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Abstract

We provide new results and new proofs of results about the torsion of curves in $\mathbb{R}^3$. Let $γ$ 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 $γ$ has nonnegative (or nonpositive) torsion, then $γ$ 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.

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BibTeX RIS

Hubert L. Bray, Jeffrey L. Jauregui. 2014-01-31. On curves with nonnegative torsion. https://arxiv.org/abs/1312.5171

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