Einstein metrics on \(S^ 3\), \(R^ 3\) and \(R^ 4\) bundles
type: Item
instance of: scholarly article
title: Einstein metrics on \(S^ 3\), \(R^ 3\) and \(R^ 4\) bundles
zbMATH Open document ID: 0699.53053
DOI: 10.1007/BF02104500
author: Don N. Page
author: Gary W. Gibbons
author: Christopher N. Pope
published in: Communications in Mathematical Physics
publication date: 1990-01-01T00:00:00Z
review text: Let M be a quaternionic Kähler manifold, i.e. a Riemannian manifold with holonomy group \(Hol\subset Sp(I)\cdot Sp(n),\) and \[ Z=\{J\in Sp(I)\quad \subset \quad GL(_{x}),\quad j^ 2=-id,\quad x\in M\} \] its twistor space. The authors consider the natural Riemannian metric \(g_{\lambda}\) on Z, depending on a real parameter \(\lambda\) and look for an Einstein metric in the form \(g=\alpha^ 2dr^ 2+\beta^ 2g_{\lambda},\) where \(\alpha\), \(\beta\), \(\lambda\) are functions of a new coordinate r. Then the Einstein equation reduces to a system of ODE. A numerical analysis indicates the existence of solutions of this system that can be extended to Einstein metrics on \(S^ 3\) bundles over a compact manifold M. Some explicit solutions are also obtained and their asymptotic properties are examined. These solutions define Ricci-flat cohomogeneity one metrics with holonomy groups \(G_ 2\) and Spin(7) on vector bundles over \(M=R^ 4\) or \(S^ 4\). The dimension of the moduli space of Ricci-flat metrics near a metric with holonomy \(G_ 2\) is calculated. It is proved also that on a manifold with holonomy \(G_ 2\) there is covariantly-constant spinor field and that any Riemannian manifold with covariantly-constant spinor field is Ricci-flat.
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 83C05
zbMATH DE Number: 4147030
zbMATH Keywords: quaternionic Kähler manifold
zbMATH Keywords: holonomy group
zbMATH Keywords: Einstein metric
zbMATH Keywords: Ricci- flat cohomogeneity one metrics
Wikidata QID: Q59348100
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02104500
OpenAlex ID: W2914503121
description: scientific article; zbMATH DE number 4147030