Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

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 Ricg = 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 open question of Y. Nikonorov. Our approach involves studying and characterizing the G × K-invariant geodesic orbit metrics on Lie groups G, where K are certain closed subgroups of G. A key ingredient of our study is the favorable representation-theoretic behaviour of 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.