On left-invariant Einstein Riemannian metrics that are not geodesic orbit

TL;DR

This paper demonstrates that the compact Lie group G2 admits a left-invariant Einstein metric that is not geodesic orbit, introducing new tools for analyzing such manifolds and building on recent discoveries.

Contribution

It provides the first example of a non-geodesic orbit Einstein metric on G2 and develops specialized methods for studying geodesic orbit Riemannian manifolds.

Findings

G2 admits a non-geodesic orbit Einstein metric

Developed new tools for geodesic orbit manifold analysis

Connected to recent work on non-naturally reductive metrics

Abstract

In this paper we prove that the compact Lie group $G_2$ admits a left-invariant Einstein metric that is not geodesic orbit. In order to prove the required assertion, we develop some special tools for geodesic orbit Riemannian manifolds. It should be noted that a suitable metric is discovered in a recent paper by I. Chrysikos and Y. Sakane, where the authors proved also that this metric is not naturally reductive.