On \(K\)-left invariant almost contact 3-structures
type: Item
instance of: scholarly article
title: On \(K\)-left invariant almost contact 3-structures
zbMATH Open document ID: 0832.53025
DOI: 10.1007/BF03322276
author: R. Rosca
published in: Results in Mathematics
publication date: 1996-02-26T00:00:00Z
review text: Let \(M_{2m + 3}\) be a Riemannian manifold carrying 3 vector fields \(\xi_r\) (\(r \in \{2m + 1,2m + 2, 2m + 3\})\) whose covariant derivatives \(\nabla \xi_r\) satisfy \(\nabla \xi_r = \xi_s \wedge \xi_t\), where \(\wedge\) denotes the wedge product and \((r,s,t)\) is a cyclic order of \((2m + 1, 2m + 2, 2m + 3)\). Such vector fields define a perfect group of left invariant skew symmetric Killing vector fields. If \(\eta^r(X) = g(X,\xi_r)\), then \(M_{2m + 3}\) with the structure \((\eta^r, \xi_r, g)\) is called a \(K\)-left invariant almost 3- contact manifold. If \(M_{2m + 3}\) is endowed with an \(f\)-structure \(\phi\), that is \(\phi\) is a (1,1)-tensor satisfying \(\phi^3 + \phi = 0\), then \(M_{2m + 3}\) is called a framed \(f\)-manifold. Among other interesting things, the author proves that any framed \(f\)-manifold \(M_{2m + 3}\) endowed with a \(K\)-left invariant almost contact 3- structure is a framed \(f\)-CR-manifold and may be viewed as the local product \(M_{2m + 3} = M \times M^\perp\) of two totally geodesic submanifolds such that \(M\) is a \(2n\)-dimensional Kähler manifold tangent to the horizontal distribution of \(M_{2m + 3}\) and \(M^\perp\) is a 3-dimensional manifold of constant curvature 1, tangent to the vertical distribution of \(M_{2m + 3}\), and any invariant submanifold \(\overline {M}\) of \(M_{2m +3}\) is minimal.
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 785203
zbMATH Keywords: almost contact 3-structures
zbMATH Keywords: Godbillon-Vey form
zbMATH Keywords: Hamiltonian vector field
zbMATH Keywords: left invariant skew symmetric Killing vector fields
MaRDI profile type: Publication
cites work: Q3743977
cites work: Q3694201
cites work: Q4276576
cites work: Q3479199
cites work: Q3339693
cites work: Q5565587
cites work: Q4693505
cites work: Q3664006
cites work: Q3034464
cites work: Q3738190
cites work: On a Riemannian space admitting more than one Sasakian structures
cites work: Q3973967
cites work: Q3222791
cites work: Totally real submanifolds of complex space forms. I. II
full work available at URL: https://doi.org/10.1007/bf03322276
OpenAlex ID: W1967959035
Recommended article: Left-invariant hypercontact structures on three-dimensional Lie groups
Recommended article: On almost contact 3-structure
Recommended article: On 3-killing almost contact structure on a Riemannian manifold
Recommended article: Remarks on a triple of \(K\)-contact structures
Recommended article: Classification of 3-dimensional left-invariant almost paracontact metric structures
Recommended article: Q4844929
Recommended article: Q4005260
Recommended article: On some affine connections on manifolds with almost contact 3-structure
Recommended article: Q4847320
Recommended article: Q4881903
description: scientific article; zbMATH DE number 785203