Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups (Q6634052)

The author studies some relations between two special classes of Riemannian Lie groups \(G\) with a left-invariant metric \(g\): the Einstein Lie groups and the geodesic orbit Lie groups. Recall that the Einstein Lie groups are defined by the condition \(\operatorname{Ric}_g = cg\), where \(\operatorname{Ric}_g\) is the Ricci curvature of the metric \(g\), and the geodesic orbit Lie groups are defined by the property that any geodesic is the integral curve of a complete Killing vector field, that is, the orbit of some \(1\)-parameter isometry group. \N\N[\textit{Yu. G. Nikonorov}, Transform. Groups 24, No. 2, 511--530 (2019; Zbl 1427.53064)] posed the following question: Which compact Einstein Lie groups \((G,g)\) are not geodesic orbit? The first example of such groups was obtain in the same article (it is the compact Lie group \(G_2\) with a suitable left-invariant Einstein metric).\N\NSince any naturally reductive Riemannian metric is geodesic orbit, then the above question is a more difficult version of the following one: Which compact Einstein Lie groups \((G, g)\) are not naturally reductive? The question was initiated by \textit{J. E. D'Atri} and \textit{W. Ziller} [Naturally reductive metrics and Einstein metrics on compact Lie groups. Providence, RI: American Mathematical Society (AMS) (1979; Zbl 0404.53044)] and served as the basis for numerous studies.\N\NNew examples of compact Einstein Lie groups \((G,g)\) that are not geodesic orbit were obtained in [\textit{H. Chen} et al., C. R., Math., Acad. Sci. Paris 356, No. 1, 81--84 (2018; Zbl 1390.53039)] and [\textit{N. Xu} and \textit{J. Tan}, C. R., Math., Acad. Sci. Paris 357, No. 7, 624--628 (2019; Zbl 1423.53062)].\N\NThe paper under review gives a large number of examples of compact Einstein Lie groups \((G,g)\) that are not geodesic orbit. The main tool is Theorem 1.4: Let \((G, g)\) be a connected compact Riemannian simple Lie group and assume that the metric \(g\) is \(G\times K\)-invariant, where \(K\) is a closed regular subgroup of \(G\). If \((G, g)\) is a \(G\times K\)-geodesic orbit manifold then \((G,g)\) is a naturally reductive manifold.\N\NWith the help of this result it is possible to prove that many Einstein Lie groups that are not naturally reductive are also not geodesic orbit (Theorem 1.6, Theorem 1.7). It should be noted that additional examples of this kind were obtained by considering the case of weakly regular subgroups \(K \subset G\) instead of regular ones. \N\NBy-products of this paper are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds.