On Left-Invariant Einstein Riemannian Metrics That Are Not Geodesic Orbit

Abstract

In this paper we prove that the compact Lie group G2 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.2010 Mathematical Subject Classification: 53C20, 53C25, 53C35.

Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

“…As a result of theorems 1.4 and 1.5, we obtain the following answers to Question 1.2, which largely extend the results of the papers [14], [28] and [37].…”

“…The above condition was firstly presented in [28] for compact simple Lie groups and under the assumption that k is an adapted subalgebra. Here we provide a proof that bypasses both the simplicity assumption and the condition for k to be adapted.…”

Geodesic orbit metrics on compact simple Lie groups arising from flag manifolds

“…In [10], Nikonorov consider the isometry group of compact simple Lie group G as G × K, where K is a closed subgroup of G. Then he obtained the equivalent algebraic description of g.o. metrics g on compact simple Lie groups G: Let B denote the minus of Killing form of g, the Lie algebra of G.…”

Geodesic orbit metrics in a class of homogeneous bundles over real and complex Stiefel manifolds

“…Another related topic of recent interest is the study of Einstein metrics that are not g.o. metrics ( [11], [22]). For a review about g.o.…”