Bour's theorem for helicoidal surfaces with singularities

2023/10/25 by Yuki Hattori, Hattori, Yuki, Atsufumi Honda +3 · 1 citation
Mathematics · Physics and Astronomy · #53A04 #53B25 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Primary 53A05 #Secondary 57R45

paper · doi:10.48550/arxiv.2310.16418

openalex publication_date 2023/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, generalizing the techniques of Bour's theorem, we prove that every generic cuspidal edge, more generally, generic n-type edge, which is invariant under a helicoidal motion in Euclidean 3-space admits non-trivial isometric deformations. As a corollary, several geometric invariants, such as the limiting normal curvature, the cusp-directional torsion, the higher order cuspidal curvature and the bias, are proved to be extrinsic invariants.

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• Curvature criteria of A-simple singularities R,0-> R²,0 and their parallel curves

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