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<p>Let <i>S</i> be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on <i>S</i> which start and end at given points in given directions and whose curvatures are constrained to lie in a given open interval, in terms of all parameters involved. Any connected component of such a space is either contractible or homotopy equivalent to an <i>n</i>-sphere, and every integer <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n\ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is realizable. Explicit homotopy equivalences between the components and the corresponding spheres are constructed.</p>

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Homotopy Type of Spaces of Curves with Constrained Curvature on Flat Surfaces

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月13日

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  • Nicolau C. Saldanha,
  • Pedro Zühlke

摘要

Let S be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on S which start and end at given points in given directions and whose curvatures are constrained to lie in a given open interval, in terms of all parameters involved. Any connected component of such a space is either contractible or homotopy equivalent to an n-sphere, and every integer \(n\ge 1\) n ≥ 1 is realizable. Explicit homotopy equivalences between the components and the corresponding spheres are constructed.