Math @ Duke

Publications [#287065] of Hubert Bray
Papers Published
 Bray, HL; Jauregui, JL, On curves with nonnegative torsion,
Archiv Der Mathematik, vol. 104 no. 6
(June, 2015),
pp. 561575, Springer Nature, ISSN 0003889X [arXiv:1312.5171 [math.DG]], [c8d239381b86496b96d95ff26f1061eb], [doi]
(last updated on 2019/07/22)
Abstract: © 2015, Springer Basel. We provide new results and new proofs of results about the torsion of curves in $${\mathbb{R}^3}$$ R3 . Let $${\gamma}$$γ be a smooth curve in $${\mathbb{R}^3}$$ R3 that is the graph over a simple closed curve in $${\mathbb{R}^2}$$ R2 with positive curvature. We give a new proof that if $${\gamma}$$γ has nonnegative (or nonpositive) torsion, then $${\gamma}$$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 $${\mathbb{R}^{2,1}}$$ R2,1 which are related to important open questions about time flat surfaces in spacetimes and mass in general relativity.


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