Invariant geodesic orbit metrics on certain compact homogeneous spaces

In this paper, we study G-g.o. metrics on a class of compact homogeneous spaces G/H constructed from homogeneous spaces G/(H x K) with K a compact Lie subgroup of G. G/H can be viewed as a total space over G/(H x K) with a fiber K. We establish the relation between G-g.o. metrics on G/H and G-g.o. metrics on G/(H x K). Furthermore, we provide a method to determine G-g.o. metrics on G/H based on the classification of G-g.o. metrics on G/(H x K). We also consider G x H-invariant g.o. metrics on compact simple Lie groups arising from the derived homogeneous spaces G/H. As an application, based on the classification of compact strongly isotropy irreducible spaces G/(H x K), we give a complete classification of G-g.o. metrics on derived homogeneous spaces G/H and prove that all G x H-invariant g.o. metrics on compact simple Lie groups G arising from G/H are naturally reductive with respect to G x H.