Affine connections on homogeneous hypercomplex manifolds
type: Item
instance of: scholarly article
title: Affine connections on homogeneous hypercomplex manifolds
zbMATH Open document ID: 0957.53010
DOI: 10.1016/S0393-0440(99)00019-4
author: M. L. Barberis
published in: Journal of Geometry and Physics
publication date: 2001-03-09T00:00:00Z
review text: The paper deals with hypercomplex structures and related connections. The basic definitions and results on almost hypercomplex structures are recalled. An almost hypercomplex structure on a manifold \(M=M^{2n}\) is a triple \(H=(J_1,J_2,J_3)\) of anticommuting almost complex structures with \(J_3=J_1J_2\); the Obata connection of \(H=(J_\alpha)\) is the (uniquely defined) affine connection \(\nabla^H\) on \(M\) which preserves \(H\), i.e. \(\nabla^HJ_\alpha=0, \alpha =1,2,3\), and with torsion \(T^H=\frac{1}{6}\sum_\alpha N_\alpha\), where \(N_\alpha\) is the Nijenhuis tensor of \(J_\alpha, \alpha=1,2,3\). The condition of preserving \(H\) is equivalent to \(\text{Hol}(\nabla^H) \subset GL(n,\mathbb{H})\), where \(\text{Hol}(\nabla^H)\) is the holonomy group of \(\nabla^H\). \(H=(J_\alpha)\) is called a hypercomplex structure if the almost complex structures \(J_\alpha\) are integrable, i.e., \(N_\alpha =0\) on \(M\), or equivalently, the Obata connection is torsionless. A hypercomplex structure \(H\) is called integrable if the \(GL(n,\mathbb{H})\)-structure of \(M\) which is naturally associated to it is integrable. M. Obata proved that \(H\) is integrable if and only if \(\nabla^H\) is flat. The author studies in particular the homogeneous hypercomplex manifolds \((M,H)\), i.e., the homogeneous manifolds \(M=G/K\) where a \(G\)-invariant hypercomplex structure \(H\) is given. For this purpose, for any G-invariant almost hypercomplex structure \(H\) on \(M=G/K\), she introduces an affine connection which is closely related to the Obata connection and whose torsion vanishes if and only if \(H\) is an hypercomplex structure of a certain special type (an abelian hypercomplex structure). A classification of the 4-dimensional real Lie groups which admit an invariant almost hypercomplex structure is given and the hypercomplex structures which are integrable are determined. As a consequence, a self-dual, non-flat, Ricci flat affine connection on \({\mathbb{R}}^4\) admitt…
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 53C56
Mathematics Subject Classification ID: 53C26
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 53C29
zbMATH DE Number: 1369696
zbMATH Keywords: hypercomplex structure
zbMATH Keywords: hypercomplex homogeneous manifold
zbMATH Keywords: holonomy
zbMATH Keywords: almost hypercomplex structure
zbMATH Keywords: Obata connection
zbMATH Keywords: holonomy group
zbMATH Keywords: abelian hypercomplex structure
zbMATH Keywords: solvable Lie algebra
reviewed by: S. Marchiafava
MaRDI profile type: Publication
cites work: An affine connection in an almost quaternion manifold
cites work: Quaternionic structures on a manifold and subordinated structures
cites work: Abelian hypercomplex structures on central extensions of \(H\)-type Lie algebras.
cites work: Hypercomplex structures on four-dimensional Lie groups
cites work: Q5691428
cites work: A Note on Hyperhermitian Four-Manifolds
cites work: Non-existence of torsion free flat connections on reductive homogeneous spaces
cites work: Abelian hypercomplex 8-dimensional nilmanifolds
cites work: On homogeneous manifolds of negative curvature
cites work: Compact hypercomplex and quaternionic manifolds
cites work: Q5565587
cites work: Complex analytic coordinates in almost complex manifolds
cites work: Invariant Affine Connections on Homogeneous Spaces
cites work: Q3263884
cites work: Riemannian Submersions, Four-Manifolds and Einstein-Weyl Geometry
cites work: Complex structures on nilpotent Lie algebras
full work available at URL: s0393-0440(99)00019-4
OpenAlex ID: W2121018801
description: scientific article; zbMATH DE number 1369696