On generalized Mannheim curves in Euclidean 4-space
type: Item
instance of: scholarly article
title: On generalized Mannheim curves in Euclidean 4-space
zbMATH Open document ID: 1184.53004
author: Hiroo Matsuda
author: Sinsuke Yorozu
published in: Nihonkai Mathematical Journal
publication date: 2009-11-30T00:00:00Z
review text: Two curves \(C\), \(\hat{C}\) in Euclidean three-space are called ``Mannheim mates'' if there exists a bijection between their points such that the principal normal of \(C\) is the binormal at the corresponding point of \(\hat{C}\). The authors generalize this concept to curves in Euclidean four-space in the sense that the first axis of the Frenet frame of \(C\) is required to be contained in the plane spanned by the second and third axis of the Frenet frame of \(\hat{C}\). The authors prove that the first curvature \(k_1\) and the second curvature \(k_2\) of \(C\) necessarily satisfy a relation of the form \(k_1 = \alpha(k_1^2 + k_2^2)\) with a positive constant \(\alpha\), thus generalizing a well-known result on Mannheim curves in three-space. Under certain additional assumptions the converse is also shown to be true. Finally, the authors provide an explicit parametrization of a family of Mannheim curves in four-space that depends on a constant \(\alpha\) and two arbitrary smooth functions.
reviewed by: Hans-Peter Schröcker
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 5640682
zbMATH Keywords: Mannheim mates
zbMATH Keywords: Frenet frame
zbMATH Keywords: curvature
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5640682