The initial value problem for cohomogeneity one Einstein metrics
type: Item
instance of: scholarly article
title: The initial value problem for cohomogeneity one Einstein metrics
zbMATH Open document ID: 0992.53033
DOI: 10.1007/BF02921808
published in: The Journal of Geometric Analysis
publication date: 2000-12-11T00:00:00Z
review text: If a compact Lie group \(G\) acts with cohomogeneity one on a manifold \(M\) (i.e. an action whose principal orbits are hypersurfaces in \(M)\), then a smooth \(G\)-invariant Einstein metric \(g(\text{Ric} (g)=\lambda g)\) with any Einstein constant \(\lambda\) is obtained in a tubular neighborhood around a singular orbit \(Q\) for any prescribed \(G\)-invariant metric \(g_Q\) and second fundamental form \(L_Q\) on \(Q\), provided the representations of the principal isotropy group on the tangent and normal space of \(Q\) have no common subrepresentations. Examples show that \(g\) is not uniquely determined by \(g_Q\) and \(L_Q\). The proof involves ODE techniques and representation theory of the principal and singular isotropy group.
reviewed by: Cornelia-Livia Bejan
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 34C60
zbMATH DE Number: 1542070
zbMATH Keywords: Einstein metrics with symmetry
zbMATH Keywords: singular orbits
zbMATH Keywords: singularities
zbMATH Keywords: initial value problem
zbMATH Keywords: power series solution
author: J.-H. Eschenburg
author: McKenzie Y. Wang
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02921808
OpenAlex ID: W2044968419
description: scientific article; zbMATH DE number 1542070