Homogeneous Einstein metrics on \(SU(n)\)
type: Item
instance of: scholarly article
title: Homogeneous Einstein metrics on \(SU(n)\)
zbMATH Open document ID: 1242.53049
author: A. H. Mujtaba
published in: Journal of Geometry and Physics
publication date: 2012-05-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1110.1978
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 6031295
zbMATH Keywords: Einstein metric
zbMATH Keywords: \(\mathrm{SU}(n)\)
zbMATH Keywords: left-invariant metric
reviewed by: Andreas Arvanitoyeorgos
MaRDI profile type: Publication
OpenAlex ID: W2986206861
arXiv ID: 1110.1978
cites work: Q4440310
cites work: Q3695986
cites work: Naturally reductive metrics and Einstein metrics on compact Lie groups
cites work: Einstein metrics on group manifolds and cosets
cites work: Invariant Einstein metrics on flag manifolds with four isotropy summands
cites work: Invariant Einstein Metrics on Some Homogeneous Spaces of Classical Lie Groups
cites work: Q3606340
cites work: Invariant Einstein metrics on certain Stiefel manifolds
cites work: Einstein metrics on compact Lie groups which are not naturally reductive
DOI: 10.1016/J.GEOMPHYS.2012.01.011
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review text: The author obtains explicitly left-invariant Einstein metrics on the compact Lie group \(SU(n)\) which are not bi-invariant, by imposing two kinds of ansatzes (named schemes 1 and 2). However, it seems that the metrics obtained are known by previous works of Jensen and D'Atri--Ziller.NEWLINENEWLINEMore precisely, in scheme 1 the left-invariant metrics depend on three parameters \(x_1, x_2, x_3\) and the Einstein metrics obtained are (up to scale) either bi-invariant \(x_1=x_2=x_3\) or \(x_1=x_3=\frac{3n+2}{n-2}\) \((n\geq 3)\), \(x_2=1\), which is known by \textit{G. R. Jensen}, [``Left invariant metrics on Lie groups'', J. Differ. Geom. 8, 599--614 (1973; Zbl 0284.53038)] and \textit{A. Besse}, Einstein Manifolds (Chapter 9). In scheme 2 the left-invariant metrics depend on four parameters \(x_1, x_2, x_3, x_4\) and are associated to the decomposition \(S(U(p)\times U(q))\subset SU(p+q)\), for any \(n\geq 2\), \(0\leq p\leq n\) and \(q=m-p\). The Einstein metrics obtained are (up to scale) \(x_1=x_2=x_3=1\), \(x_4=\frac2{pq(p+1)}\) and \(x_2=\frac{q}{p}x_1\), \(x_1=f(p,q)\), \(x_4=g(p,q)x_1\), where \(f(p,q), g(p,q)\) are rational functions of \(p,q\). These metrics are discussed in [\textit{J. E. D'Atri} and \textit{W. Ziller}, ``Naturally reductive metrics and Einstein metrics on compact Lie groups'', Mem. Am. Math. Soc. 215, 72 p. (1979; Zbl 0404.53044)].NEWLINENEWLINEIn both schemes the Einstein metrics obtained are naturally reductive.
description: scientific article; zbMATH DE number 6031295