Some characterizations of Mannheim partner curves in the Minkowski 3-space \(E_{1}^{3}\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1247.53005
author: Tanju Kahraman
author: Mehmet Önder
author: Mustafa Kazaz
author: H. Hüseyin Uğurlu
published in: Proceedings of the Estonian Academy of Sciences
publication date: 2012-06-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/1108.4570
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 53B30
zbMATH DE Number: 6049745
zbMATH Keywords: Minkowski 3-space
zbMATH Keywords: time-like curve
zbMATH Keywords: space-like curve
zbMATH Keywords: Mannheim partner curves
MaRDI profile type: Publication
OpenAlex ID: W2963018484
arXiv ID: 1108.4570
DOI: 10.3176/PROC.2011.4.02
title: Some characterizations of Mannheim partner curves in the Minkowski 3-space \(E_{1}^{3}\)
review text: Whereas a pair of Bertrand curves has common principal normals in corresponding points, for Mannheim partner curves the principal normals of one of the curves coincide with the binormals of the other one in corresponding points. Mannheim partner curves are studied in Minkowski space \(E^3_1\). It is shown that Mannheim's theorem is not valid, that means that a certain cross ratio is not constant. The reviewer could not resolve the following contradiction:NEWLINENEWLINE In Theorem 3.2 it is claimed that for any curve there is another curve so that these curves form a Mannheim pair. But in Theorem 3.4 it is shown that curvature and torsion of a curve belonging to a Mannheim pair obey a linear relation.
reviewed by: Erhard Heil
description: scientific article; zbMATH DE number 6049745