A new class of quaternionic manifolds
type: Item
instance of: scholarly article
title: A new class of quaternionic manifolds
zbMATH Open document ID: 0661.53019
DOI: 10.1007/BF02844895
author: B. R. Moreiras
author: Marisa Fernández
publication date: 1987-01-01T00:00:00Z
review text: Let M be a quaternionic manifold with 4-form \(\Omega\) and almost complex structures I, J, K. Define a differential 1-form on M by \[ L_ 1(\nabla \Omega)=\sum^{4n}_{r,s}\nabla_{E_ r}(\Omega)(IE_ r,JE_ s,KE_ s,X). \] A new class of quaternionic manifolds called \(L_ 1KQ\) is introduced by the condition \(L_ 1(\nabla \Omega)=0\). The structure group of an \(L_ 1KQ\)-manifold is Sp(n); conformal but non-homothetic changes of the metric do not preserve this class. Let AKQ denote the class of almost Kähler quaternionic manifolds; the inclusion \(AKQ\subseteq L_ 1KQ\) is shown. The example \(M^{4r+4}=H\times R^{4r+1},\) where H is the Heisenberg group, constructed in the final part of the paper, shows that the inclusion is strict. Moreover it is proved that \(M^{4r+4}\) has no Kählerian quaternionic structure.
Mathematics Subject Classification ID: 53B35
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 4080305
zbMATH Keywords: nilmanifold
zbMATH Keywords: quaternionic manifold
zbMATH Keywords: \(L_ 1KQ\)-manifold
zbMATH Keywords: almost Kähler quaternionic manifolds
zbMATH Keywords: Heisenberg group
zbMATH Keywords: no Kählerian quaternionic structure
reviewed by: Liviu Ornea
MaRDI profile type: Publication
cites work: Q5185636
cites work: Sur les groupes d'holonomie homogènes de variétés à connexion affine et des variétés riemanniennes
cites work: Q5557714
cites work: Symplectic manifolds with no Kähler structure
cites work: Examples of Compact Non-Kahler almost Kahler Manifolds
cites work: Q3767968
cites work: Some examples of almost Hermitian manifolds
cites work: A note on manifolds whose holonomy group is a subgroup of Sp(n) Sp(1)
cites work: Weak holonomy groups
cites work: Topology of Quaternionic Manifolds
cites work: Shorter Notes: Some Simple Examples of Symplectic Manifolds
cites work: Q4196994
cites work: Almost Hermitian submersions
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 4080305