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arXiv · 2407.01729

Deformations of curves with constant curvature

Mohammad Ghomi·Matteo Raffaelli
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Abstract

We prove that curves of constant curvature satisfy the parametric $C^1$-dense relative $h$-principle in the space of immersed curves with nonvanishing curvature in Euclidean space $R^{n\geq 3}$. It follows that two knots of constant curvature in $R^3$ are isotopic, resp. homotopic, through curves of constant curvature if and only if they are isotopic, resp. homotopic, and their self-linking numbers, resp. self-linking numbers mod $2$, are equal. The proofs are based on convex integration techniques, utilizing parametric versions of classical theorems by Carath\'{e}odory and Steinitz in convex geometry.

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Mohammad Ghomi, Matteo Raffaelli. 2024-07-01. Deformations of curves with constant curvature. https://arxiv.org/abs/2407.01729

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