Stability of non-diagonal Einstein metrics on homogeneous spaces $H\times H/ \Delta K$

TL;DR

This paper investigates the stability of non-diagonal Einstein metrics on a specific class of homogeneous spaces, showing they are generally unstable, and provides formulas for scalar curvature in this context.

Contribution

It offers a formula for scalar curvature of certain invariant metrics and analyzes the stability of non-diagonal Einstein metrics on these spaces, extending previous classifications.

Findings

Non-diagonal Einstein metrics are unstable under the Hilbert action.

All studied non-diagonal Einstein metrics have different coindexes.

The work extends classification results of invariant Einstein metrics.

Abstract

We consider the homogeneous space $M=H\times H/\Delta K$, where $H/K$ is an irreducible symmetric space and $\Delta K$ denotes diagonal embedding. Recently, Lauret and Will provided a complete classification of $H\times H$-invariant Einstein metrics on M. They obtained that there is always at least one non-diagonal Einstein metric on $M$, and in some cases, diagonal Einstein metrics also exist. We give a formula for the scalar curvature of a subset of $H\times H$-invariant metrics and study the stability of non-diagonal Einstein metrics on $M$ with respect to the Hilbert action, obtaining that these metrics are unstable with different coindexes for all homogeneous spaces $M$.