On some types of slant curves in contact pseudo-Hermitian 3-manifolds
type: Item
instance of: scholarly article
title: On some types of slant curves in contact pseudo-Hermitian 3-manifolds
zbMATH Open document ID: 1257.53029
author: Cihan Özgür
author: Şaban Güvenç
published in: Annales Polonici Mathematici
publication date: 2012-04-17T00:00:00Z
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53D10
Mathematics Subject Classification ID: 53B30
Mathematics Subject Classification ID: 53B35
zbMATH DE Number: 6024874
zbMATH Keywords: pseudo-Hermitian geometry
zbMATH Keywords: slant curves
MaRDI profile type: Publication
full work available at URL: ap104-3-1
OpenAlex ID: W2327620868
DOI: 10.4064/AP104-3-1
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review text: Following the introduction, the authors review contact Riemannian geometry and the related contact pseudo-convex pseudo-Hermitian structure including a discussion of the ``Tanaka-Webster connection'' or ``pseudo-Hermitian connection''.NEWLINENEWLINEFor a curve \(\gamma\) in a contact Riemannian 3-manifold, one can define a Frenet frame field \(\{T, N, B\}\) along \(\gamma\) using the pseudo-Hermitian connection \(\widehat\nabla\). In this setting, we have the Frenet-Serret equations: \(\widehat\nabla_TT=\widehat\kappa N\), \(\widehat\nabla_TN=-\widehat\kappa T+\widehat\tau B\), \(\widehat\nabla_TB=-\widehat\tau N\), where \(\widehat\kappa=\|\widehat\nabla_TT\|\) is the ``pseudo-Hermitian curvature'' of \(\gamma\) and \(\widehat\tau\) its ``pseudo-Hermitian torsion''. The authors assume throughout that \(\widehat\kappa\neq 0\). The curve \(\gamma\) is said to be a ``slant curve'' if it makes a constant angle with the Reeb vector field of the contact form. A slant curve for which the angle is \(\pi/2\) is traditionally called a ``Legendre curve''. The ``pseudo-Hermitian mean curvature vector field'' of \(\gamma\) is defined by \(\widehat H=\widehat\nabla_TT=\widehat\kappa N\). Denoting the corresponding normal connection by \(\widehat\nabla^\perp\), one can speak of normal vector fields being ``pseudo-Hermitian parallel''.NEWLINENEWLINEThe authors first show that \(\gamma\) is a curve with pseudo-Hermitian parallel mean curvature vector field if and only if it is a pseudo-Hermitian circle (\(\widehat\kappa\) constant). Moreover, if in addition \(\gamma\) is slant, it is a Legendre circle.NEWLINENEWLINEThe curve \(\gamma\) is said to be a curve with ``pseudo-Hermitian proper mean curvature vector field'' if \(\widehat\Delta\widehat H=\lambda\widehat H\) where \(\lambda\) is a non-zero \(C^\infty\) function and \(\widehat\Delta\) is the Laplacian. The authors prove that such a curve has pseudo-Hermitian proper mean curvature vector field if and only if it is a pseudo-He…
reviewed by: David E. Blair
description: scientific article; zbMATH DE number 6024874