Coindex and rigidity of Einstein metrics on homogeneous Gray manifolds

TL;DR

This paper investigates the stability and deformation properties of Einstein metrics on 6-dimensional homogeneous Gray manifolds, revealing their coindex and rigidity characteristics through explicit computations.

Contribution

It computes the coindex of Einstein metrics on compact homogeneous Gray manifolds and demonstrates the non-integrability of certain infinitesimal Einstein deformations.

Findings

The Einstein metrics on these manifolds have specific coindex values.

Infinitesimal Einstein deformations on F_{1,2} are not integrable.

The results provide insights into the stability and rigidity of these Einstein metrics.

Abstract

Any $6$-dimensional strict nearly K\"ahler manifold is Einstein with positive scalar curvature. We compute the coindex of the metric with respect to the Einstein-Hilbert functional on each of the compact homogeneous examples. Moreover, we show that the infinitesimal Einstein deformations on $F_{1,2}=\mathrm{SU}(3)/T^2$ are not integrable into a curve of Einstein metrics.