Quaternionic contact \(4n+3\)-manifolds and their \(4n\)-quotients
type: Item
instance of: scholarly article
title: Quaternionic contact \(4n+3\)-manifolds and their \(4n\)-quotients
zbMATH Open document ID: 1483.53070
author: Yoshinobu Kamishima
published in: Annals of Global Analysis and Geometry
publication date: 2021-06-14T00:00:00Z
review text: The author studies quaternionic contact structure (``qc-structure'' for short) on \((4n+3)\)-manifolds to construct quaternionic Hermitian \(4n\)-manifolds as their quotient. The qc-structure was first introduced by \textit{O. Biquard} [Métriques d'Einstein asymptotiquement symétriques. Paris: Société Mathématique de France (2000; Zbl 0967.53030)], together with a canonical connection (now known as ``Biquard connection''). A qc-structure on a \((4n+3)\)-manifold \(X\) is a distribution \(D\subset TX\) of codimension 3, such that \(D\) admits a \(n\)-dimensional quaternionic structure \(Q\). In a previous work of the author with \textit{D. Alekseevsky} [Ann. Mat. Pura Appl. (4) 187, No. 3, 487--529 (2008; Zbl 1223.53054)], a special case of qc-structures, called ``quaternionic CR-structure'' was studied, which corresponds to qc-Einstein manifolds with nonzero qc-scalar curvatures. In this paper, the author studies the complementing vanishing qc-scalar curvature case, which has an equivalent description as ``strict qc-structure''. In the first part of the paper, the author construct a family of simply connected strict qc solvable Lie groups \(\mathcal{M}_{k,l}\), starting with the standard quaternionic Heisenberg nilpotent Lie group \(\mathcal{M}\). These are then characterized as the only contractible unimodular strict qc-groups. Then, both compact and non-compact uniformizable strict qc-manifolds are classified modeling on \(\mathcal{M}\). In the second part, the author uses the quotients of a conformal deformation of the standard qc-structure on \(\mathcal{M}\) to construct a family of quaternionic Hermitian metrics on a domain of the standard quaternion space, one of which is a Bochner-flat Kähler metric.
reviewed by: Yalong Shi
Mathematics Subject Classification ID: 53C26
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 51M10
zbMATH DE Number: 7358590
zbMATH Keywords: quaternionic contact structure
zbMATH Keywords: quaternionic Hermitian structure
zbMATH Keywords: hyper-Kähler structure
zbMATH Keywords: quaternionic Heisenberg Lie group
zbMATH Keywords: spherical CR structure
zbMATH Keywords: Bochner geometry
MaRDI profile type: Publication
full work available at URL: s10455-021-09758-5
OpenAlex ID: W3134751747
cites work: Quaternionic and para-quaternionic CR structure on \((4n+3)\)-dimensional manifolds
cites work: Pseudo-conformal quaternionic \(CR\) structure on \((4n+3)\)-dimensional manifolds
cites work: Q4956010
cites work: Locally homogeneous aspherical Sasaki manifolds
cites work: Q4769015
cites work: Sur le groupe d'automorphismes des géométries paraboliques de rang 1
cites work: On Kaehlerian Homogeneous Spaces of Unimodular Lie Groups
cites work: Quaternionic contact Einstein manifolds
cites work: Quaternionic contact manifolds with a closed fundamental 4-form
cites work: Conformal quaternionic contact curvature and the local sphere theorem
cites work: HEISENBERG, SPHERICAL CR-GEOMETRY AND BOCHNER FLAT LOCALLY CONFORMAL KÄHLER MANIFOLDS
cites work: Geometric flows on compact manifolds and global rigidity
cites work: Three-dimensional Lie group actions on compact (4\(n\)+3)-dimensional geometric manifolds
cites work: On the conformal and CR automorphism groups
cites work: Q5544015
cites work: On the pseudo-conformal geometry of a Kähler manifold
DOI: 10.1007/S10455-021-09758-5
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description: scientific article; zbMATH DE number 7358590