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Deformations of curves with constant curvature

Mohammad Ghomi, Matteo Raffaelli

TL;DR

The paper establishes a parametric $C^1$-dense relative $h$-principle for curves of constant curvature in $\mathbb{R}^n$ ($n\ge 3$), showing that unit-speed curves with nonvanishing curvature can be deformed through constant-curvature representatives in a way that tightly preserves boundary data. The authors reduce the global problem to a local one, then develop a suite of tools—nonflat perturbations via jet transversality, control of the center of mass on spherical tantrices, and parametric convex-geometry results (Carathéodory–Steinitz)—to construct the required homotopies. Their main consequence for knots in $\mathbb{R}^3$ is a complete isotopy/homotopy classification through constant-curvature curves governed by self-linking invariants, aligning the constant-curvature deformation theory with the classical nonflat curvature theory. The approach leverages convex integration, explicit geodesic perturbations, and careful parametric convex-combination arguments to realize deformations to constant-curvature curves while maintaining $C^1$-control. This advances understanding of when geometric constraints (constant curvature) can be reconciled with topological constraints (isotopy/homotopy) in space curves, with potential implications for the manipulation and classification of knots under curvature-preserving deformations.

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.

Deformations of curves with constant curvature

TL;DR

The paper establishes a parametric -dense relative -principle for curves of constant curvature in (), showing that unit-speed curves with nonvanishing curvature can be deformed through constant-curvature representatives in a way that tightly preserves boundary data. The authors reduce the global problem to a local one, then develop a suite of tools—nonflat perturbations via jet transversality, control of the center of mass on spherical tantrices, and parametric convex-geometry results (Carathéodory–Steinitz)—to construct the required homotopies. Their main consequence for knots in is a complete isotopy/homotopy classification through constant-curvature curves governed by self-linking invariants, aligning the constant-curvature deformation theory with the classical nonflat curvature theory. The approach leverages convex integration, explicit geodesic perturbations, and careful parametric convex-combination arguments to realize deformations to constant-curvature curves while maintaining -control. This advances understanding of when geometric constraints (constant curvature) can be reconciled with topological constraints (isotopy/homotopy) in space curves, with potential implications for the manipulation and classification of knots under curvature-preserving deformations.

Abstract

We prove that curves of constant curvature satisfy the parametric -dense relative -principle in the space of immersed curves with nonvanishing curvature in Euclidean space . It follows that two knots of constant curvature in 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 , are equal.
Paper Structure (7 sections, 11 theorems, 25 equations, 2 figures)

This paper contains 7 sections, 11 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Reduction to a Local Problem
  3. Approximation with Nonflat Curves
  4. The Length and Center of Mass of Spherical Curves
  5. Perturbation of Geodesics
  6. Parametric Carathéodory and Steinitz Theorems
  7. Proof of the Main Results

Key Result

Theorem 1.1

Let $f_t\in \mathop{\mathrm{Imm}}\nolimits^{k\geq 2}(\Gamma,\mathbf{R}^{n\geq 3})$ be a $\mathcal{C}^2$-homotopy of curves with nonvanishing curvature. Suppose that $f_0$ and $f_1$ have constant curvature. Then, for any $\varepsilon>0$, there exists a $\mathcal{C}^2$-homotopy $\widetilde{f}_t\in\mat

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (17)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • Corollary 4.2
  • ...and 7 more