Six-dimensional Lie-Einstein metrics
type: Item
instance of: scholarly article
title: Six-dimensional Lie-Einstein metrics
zbMATH Open document ID: 1490.22007
author: Rishi Raj Subedi
author: Gerard Thompson
published in: Journal of Geometry
publication date: 2021-06-24T00:00:00Z
review text: On an Einstein manifold the Ricci tensor is a multiple of the metric. In the present paper the authors study Einstein manifolds that arise for right-invariant Riemannian metrics on a six-dimensional Lie group \(G\). They impose severe restrictions on the Lie algebra it-self: indecomposable, six-dimensional and solvable. They have five-dimensional and four-dimensional nilradical algebras. In those cases they assume very specific forms of the metric. The metric was reduced by using the following method: the space of derivations was found and then exponentiated so as to obtain the connected component of the identity of the automorphism group of \(\mathfrak g\), the Lie algebra of \(G\). This automorphism group was used to reduce the matrix of the metric. At that point it became feasible to find the Ricci tensor of g and set it equal to a multiple of the metric. Remark that any constant non-zero multiple of a Lie-Einstein metric is again Lie-Einstein.
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 53B21
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 7362842
zbMATH Keywords: Lie group
zbMATH Keywords: Lie algebra
zbMATH Keywords: invariant metric
zbMATH Keywords: Ricci tensor
zbMATH Keywords: Einstein space
reviewed by: Gabriela Paola Ovando
MaRDI profile type: Publication
full work available at URL: s00022-021-00573-8
OpenAlex ID: W3127666858
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DOI: 10.1007/S00022-021-00573-8
description: scientific article; zbMATH DE number 7362842