On 4-flattening theorems and the curves of Carathéodory, Barner and Segre
type: Item
instance of: scholarly article
title: On 4-flattening theorems and the curves of Carathéodory, Barner and Segre
zbMATH Open document ID: 1029.51018
author: Ricardo Uribe-Vargas
published in: Journal of Geometry
publication date: 2003-09-16T00:00:00Z
review text: Three classes of closed space curves, called Barner, Segre and Carathéodory class, resp., are studied. It is proved: (A) There is a non-empty open set of Carathéodory curves in \(\mathbb{R}^3\) which are not Segre curves. (B) There is a non-empty open set of Segre curves in \(\mathbb{R}^3\) which are not Carathéodory curves. (C) Any Barner curve in \(\mathbb{R}^3\) is a Carathéodory curve and a Segre curve at the same time. A new class of closed curves containing the Segre class is also introduced. It is conjectured that any curve of the new class has at least 4 flattenings (points at which the torsion vanishes).
Mathematics Subject Classification ID: 51L15
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53C99
Mathematics Subject Classification ID: 53D10
zbMATH DE Number: 1982553
zbMATH Keywords: tangent indicatrix
zbMATH Keywords: Legendrian knot
zbMATH Keywords: space curves
zbMATH Keywords: Carathéodory curves
zbMATH Keywords: Segre curves
zbMATH Keywords: Barner curve
zbMATH Keywords: flattenings
reviewed by: József Szilasi
MaRDI profile type: Publication
full work available at URL: s00022-003-1641-y
OpenAlex ID: W2055516729
DOI: 10.1007/S00022-003-1641-Y
description: scientific article; zbMATH DE number 1982553