Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

TL;DR

This paper investigates the relationship between Einstein and geodesic orbit properties in Riemannian Lie groups, revealing that many compact simple Einstein groups are not geodesic orbit, and introduces methods to classify such manifolds.

Contribution

It establishes new structural results and classification techniques for geodesic orbit metrics on Lie groups, especially in relation to Einstein conditions.

Findings

Many compact simple Einstein Lie groups are not geodesic orbit.

Characterization of $G\times K$-invariant geodesic orbit metrics.

Development of classification tools for geodesic orbit manifolds.

Abstract

We study the relation between two special classes of Riemannian Lie groups $G$ with a left-invariant metric $g$: The Einstein Lie groups, defined by the condition $\operatorname{Ric}_g=cg$, and the geodesic orbit Lie groups, defined by the property that any geodesic is the integral curve of a Killing vector field. The main results imply that extensive classes of compact simple Einstein Lie groups $(G,g)$ are not geodesic orbit manifolds, thus providing large-scale answers to a relevant question of Y. Nikonorov. Our approach involves studying and characterizing the $G\times K$-invariant geodesic orbit metrics on Lie groups $G$ for a wide class of subgroups $K$ that we call (weakly) regular. By-products of our work are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds.