On Legendre curves in 3-dimensional normal almost paracontact metric manifolds
type: Item
instance of: scholarly article
title: On Legendre curves in 3-dimensional normal almost paracontact metric manifolds
zbMATH Open document ID: 1180.53080
author: Joanna Wełyczko
published in: Results in Mathematics
publication date: 2009-11-02T00:00:00Z
review text: Let \(M\) be an almost paracontact manifold with a structure \(\Sigma=(\varphi,\xi,\eta)\), where a \(1\)-form \(\eta\), a vector field \(\xi\), and a field of endomorphisms \(\varphi\) satisfy \(\varphi^2=Id-\eta\otimes\xi\) and \(\eta(\xi)=1\). If there is a pseudo-Riemannian metric \(g\) satisfying \(g(\varphi X,\varphi Y)=-g(X,Y)+\eta(X)\eta(Y)\), then \((M,\Sigma)\) is called an almost paracontact metric manifold. If \(N_\varphi(X,Y)-2d\eta(X,Y)\xi=0\), where \(N_\varphi\) is the Nijenhuis tensor of \(\varphi\), then \((M,\Sigma)\) is a normal almost paracontact metric manifold. A normal almost paracontact metric manifold \(M\) is a para-Sasakian manifold if \(\Phi=d\eta\), where \(\Phi(X,Y)=g(X,\varphi Y)\). Let \((M,g)\) be a \(3\)-dimensional pseudo-Riemannian manifold. A curve \(\gamma:I\to M\) such that \(g(\dot\gamma,\dot\gamma)=\pm1\) is a Frenet curve on \(M\) if \(\gamma\) is of osculating order \(1\), \(2\), or \(3\). If \(M\) is almost paracontact metric, then \(\gamma\) is a Legendre curve if \(\eta(\dot\gamma)=0\). This interesting paper is devoted to study the curvature and torsion of Frenet Legendre curves in 3-dimensional normal almost paracontact metric manifolds. Some examples illustrating the results are presented.
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 53D15
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 5624113
zbMATH Keywords: Legendre curve
zbMATH Keywords: Frenet curve
zbMATH Keywords: normal almost paracontact metric manifold
zbMATH Keywords: quasi-para-Sasakian manifold
MaRDI profile type: Publication
full work available at URL: s00025-009-0364-2
OpenAlex ID: W2010414756
DOI: 10.1007/S00025-009-0364-2
description: scientific article; zbMATH DE number 5624113