On one class of homogeneous compact Einstein manifolds
type: Item
instance of: scholarly article
title: On one class of homogeneous compact Einstein manifolds
zbMATH Open document ID: 0949.53032
DOI: 10.1007/BF02674006
published in: Siberian Mathematical Journal
publication date: 2000-08-30T00:00:00Z
full work available at URL: https://eudml.org/doc/120642
review text: It is well known that Wallach spaces (that are the flag manifolds over complex, quaternionic, and Cayley projective planes) admit Einstein metrics. All these spaces are compact homogeneous manifolds of the form \(G/H\). Consider the orthogonal decomposition with respect to the Killing form \(B\): \(g=h \oplus p\). In the case of Wallach spaces, there is an additional property: \(p=p_1 \oplus p_2 \oplus p_3\), where \(p_i\) are \(\text{ad}_h\)-irreducible and pairwise \(B\)-orthogonal modules, and \([p_i, p_i] \subset h\). The author proves that, for every compact homogeneous space \(G/H\) with semisimple Lie group \(G\), this property implies the existence of an Einstein metric of the form \(x_1 B|_{p_1} +x_2 B|_{p_2} +x_3 B|_{p_3}\) for some \(x_i < 0\). In particular, the spaces \(SO(a+b+c)/SO(a) \times SO(b) \times SO(c)\) and \(Sp(a+b+c)/Sp(a) \times Sp(b) \times Sp(c)\) admit homogeneous Einstein metrics.
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 1498342
zbMATH Keywords: homogeneous space
zbMATH Keywords: Einstein manifold
zbMATH Keywords: Wallach space
author: Yurii G. Nikonorov
reviewed by: Yaroslav V. Bazaikin
MaRDI profile type: Publication
cites work: Q4002725
description: scientific article; zbMATH DE number 1498342