Homogeneous Einstein metrics on G_2/T

TL;DR

This paper constructs and analyzes invariant Einstein metrics on the exceptional full flag manifold G_2/T, discovering exactly two non-Kähler Einstein metrics, marking a significant example in the study of Einstein metrics on homogeneous spaces.

Contribution

It provides the first known example of an exceptional full flag manifold with at least one non-Kähler, non-normal homogeneous Einstein metric, using algebraic and computational methods.

Findings

G_2/T admits exactly two non-Kähler invariant Einstein metrics

First example of an exceptional full flag manifold with such metrics

Utilized Gr"obner basis computations to prove the results

Abstract

We construct the Einstein equation for an invariant Riemannian metric on the exceptional full flag manifold $M=G_2/T$. By computing a Gr\"obner basis for a system of polynomials of multi-variables we prove that this manifold admits exactly two non-K\"ahler invariant Einstein metrics. Thus $G_2/T$ turns out to be the first known example of an exceptional full flag manifold which admits at least one non-K\"ahler and not normal homogeneous Einstein metric.