Einstein metrics on homogeneous spaces $H\times H/\Delta K$

TL;DR

This paper investigates the existence, classification, and stability of Einstein metrics on a new class of homogeneous spaces formed from compact simple Lie groups, revealing unstable Einstein metrics and a complete classification in certain cases.

Contribution

It introduces and analyzes Einstein metrics on spaces constructed as quotients of product groups, providing new classifications and stability results in the context of non-simple Lie groups.

Findings

Unstable Einstein metrics found on most spaces where the standard metric is Einstein.

Complete classification achieved for irreducible symmetric spaces with simple isotropy subgroup.

Standard metric is a global minimum of scalar curvature among all normal metrics.

Abstract

Given any compact homogeneous space $H/K$ with $H$ simple, we consider the new space $M=H\times H/\Delta K$, where $\Delta K$ denotes diagonal embedding, and study the existence, classification and stability of $H\times H$-invariant Einstein metrics on $M$, as a first step into the largely unexplored case of homogeneous spaces of compact non-simple Lie groups. We find unstable Einstein metrics on $M$ for most spaces $H/K$ such that their standard metric is Einstein (e.g., isotropy irreducible) and the Killing form of $\mathfrak{k}$ is a multiple of the Killing form of $\mathfrak{h}$ (e.g., $K$ simple), a class which contains $17$ families and $50$ individual examples. A complete classification is obtained in the case when $H/K$ is an irreducible symmetric space with $K$ simple. We also study the behavior of the scalar curvature function on the space of all normal metrics on $M=H\times H/\Delta K$ (none of which is Einstein), obtaining that the standard metric is a global minimum.