Hyper-Kähler metrics conformal to left invariant metrics on four-dimensional Lie groups
type: Item
instance of: scholarly article
title: Hyper-Kähler metrics conformal to left invariant metrics on four-dimensional Lie groups
zbMATH Open document ID: 1031.53074
DOI: 10.1023/A:1022448007111
author: M. L. Barberis
published in: Mathematical Physics, Analysis and Geometry
publication date: 2003-05-11T00:00:00Z
review text: Let \(g\) be a hyper-Hermitian metric on a simply connected hypercomplex four-manifold \((M,{\mathcal H})\). We show that when the isometry group \(I(M,g)\) contains a subgroup \(G\) acting simply transitively on \(M\) by hypercomplex isometries, then the metric \(g\) is conformal to a hyper-Kähler metric. We describe explicitly the corresponding hyper-Kähler metrics, which are of cohomogeneity one with respect to a 3-dimensional normal subgroup of \(G\). It follows that, in four dimensions, these are the only hyper-Kähler metrics containing a homogeneous metric in its conformal class.
Mathematics Subject Classification ID: 53C26
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 1908552
zbMATH Keywords: hyper-Hermitian metric
zbMATH Keywords: hypercomplex manifold
zbMATH Keywords: conformally hyper-Kähler metric
reviewed by: Domenico Perrone
MaRDI profile type: Publication
full work available at URL: a:1022448007111
OpenAlex ID: W159590328
description: scientific article; zbMATH DE number 1908552