Classification of 3-dimensional left-invariant almost paracontact metric structures
type: Item
instance of: scholarly article
title: Classification of 3-dimensional left-invariant almost paracontact metric structures
zbMATH Open document ID: 1391.53035
author: Giovanni Calvaruso
author: Antonella Perrone
publication date: 2018-03-29T00:00:00Z
review text: Recall that an almost paracontact structure on a differential manifold \(M\) of dimension \(2n+1\) is a triple \((\varphi,\xi,\eta)\), where \(\varphi\) is a \((1,1)\)-tensor, \(\xi\) is a vector field and \(\eta\) is a \(1\)-form satisfying the following compatibility conditions \[ \varphi(\xi) = 0, \quad \eta\circ\varphi = 0, \quad \eta(\xi)= 1, \quad \varphi^2 = id - \eta\otimes\xi, \] and the restriction of \(\varphi\) on the horizontal distribution \(\ker \eta\) is an almost paracomplex structure, that is, the eigensubbundles \(T^+\), \(T^-\) corresponding to the eigenvalues \(1,-1\) of \(\varphi\), respectively, have equal dimension \(n\). An almost paracontact metric in an almost paracontact manifold is a pseudo-Riemannian metric \(g\) compatible with the almost paracontact structur \(a\), that is, \[ g(\varphi X,\varphi Y)=-g(X,Y)+\eta(X)\eta(Y), \quad X,Y \in \Gamma(TM). \] In this case \((\varphi,\xi,\eta,g)\) is said to be an almost paracontact metric structure. The authors study left-invariant almost paracontact metric structures on Lie groups, or equivalently on its Lie algebras. In particular they are interested in almost paracontact metric structure satisfying an extra condition, that is, \(\xi\in\ker\eta\). An almost paracontact structure \((\varphi,\xi,\eta)\) with this property is said to be natural. They obtain a complete classification of 3-dimensional simply connected Lie groups admitting a left-invariant natural almost paracontact metric structure, up to isomorphism (see Theorem 4.3 for the unimodular case and Theorem 4.5 for the non-unimodular case). They also study special cases of natural almost paracontact metric structure, namely normal structures and almost paracosymplectic structures. And they classify 3-dimensional simply connected Lie groups admitting these kind of structures in Theorem 5.1 and Theorem 5.4 respectively.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53B05
Mathematics Subject Classification ID: 53D15
zbMATH DE Number: 6854795
zbMATH Keywords: almost paracontact metric structures
zbMATH Keywords: normal structures
zbMATH Keywords: almost para-cosymplectic structures
reviewed by: Marcos Origlia
MaRDI profile type: Publication
full work available at URL: advgeom-2017-0025
OpenAlex ID: W2736420550
cites work: Maximally homogeneous para-CR manifolds
cites work: Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds
cites work: Homogeneous structures on three-dimensional Lorentzian manifolds
cites work: Homogeneous paracontact metric three-manifolds
cites work: Paracontact metric structures on the unit tangent sphere bundle
cites work: Ricci solitons in three-dimensional paracontact geometry
cites work: Geometry of $H$-paracontact metric manifolds
cites work: BI-LEGENDRIAN STRUCTURES AND PARACONTACT GEOMETRY
cites work: Nullity conditions in paracontact geometry
cites work: Three-dimensional Lorentz manifolds admitting a parallel null vector field
cites work: On weakly para-cosymplectic manifolds of dimension 3
cites work: Differential geometry and analysis on CR manifolds
cites work: Differential equations and para-CR structures
cites work: Conformal paracontact curvature and the local flatness theorem
cites work: Almost paracontact and parahodge structures on manifolds
cites work: Curvatures of left invariant metrics on Lie groups
cites work: Some results on almost paracontact metric manifolds
cites work: Lorentz metrics on unimodular Lie groups of dimension three
cites work: On Legendre curves in 3-dimensional normal almost paracontact metric manifolds
cites work: Para-CR structures on almost paracontact metric manifolds
cites work: Slant curves in 3-dimensional normal almost paracontact metric manifolds
cites work: Canonical connections on paracontact manifolds
cites work: Geometry of paraquaternionic Kähler manifolds with torsion
DOI: 10.1515/ADVGEOM-2017-0025
published in: advg
description: scientific article; zbMATH DE number 6854795