Einstein metrics on aligned homogeneous spaces with two factors

TL;DR

This paper investigates the existence of G_1xG_2-invariant Einstein metrics on certain homogeneous spaces formed from compact simple Lie groups, providing a classification based on polynomial root conditions.

Contribution

It introduces a criterion linking Einstein metric existence to roots of a quartic polynomial, enabling classification for a broad class of spaces.

Findings

Existence of Einstein metrics characterized by roots of a specific quartic polynomial.

Full classification achieved for a large subclass with three isotropy irreducible summands.

Identified conditions for existence and non-existence of Einstein metrics in the studied spaces.

Abstract

Given two homogeneous spaces of the form G_1/K and G_2/K, where G_1 and G_2 are compact simple Lie groups, we study the existence problem for G_1xG_2-invariant Einstein metrics on the homogeneous space M=G_1xG_2/K. For the large subclass C of spaces having three pairwise inequivalent isotropy irreducible summands (12 infinite families and 70 sporadic examples), we obtain that existence is equivalent to the existence of a real root for certain quartic polynomial depending on the dimensions and two Killing constants, which allows a full classification and the possibility to weigh the existence and non-existence pieces of C.