On a class of geodesic orbit spaces with abelian isotropy subgroup (Q2049522)

We say that a Riemannian manifold \((M,g)\) is a geodesic orbit (g.o) manifold if any geodesic of \(M\) is an orbit of a one-parameter subgroup of the entire isometry group of \(M\). The author considers g.o. spaces of the form \((G/S, g)\) such that \(G\) is a compact, connected, semisimple Lie group and the isotropy subgroup \(S\) is abelian. He states (Theorem 1.1) that \(G/S\) is a g.o space if and only if \(g\) is a \(G\)-naturally reductive metric, i.e., a suitable \(G\)-invariant metric (resp., an induced from a bi-invariant Riemannian metric on \(G\)).