Geodesic orbit metrics in compact homogeneous manifolds with equivalent isotropy submodules

Abstract

A geodesic orbit manifold (GO manifold) is a Riemannian manifold (M,g) with the property that any geodesic in M is an orbit of a one-parameter subgroup of a group G of isometries of (M,g). The metric g is then called a G-GO metric in M. For an arbitrary compact homogeneous manifold M=G/H, we simplify the general problem of determining the G-GO metrics in M. In particular, if the isotropy representation of H induces equivalent irreducible submodules in the tangent space of M, we obtain algebraic conditions, under which, any G-GO metric in M admits a reduced form. As an application we determine the U(n)-GO metrics in the complex Stiefel manifolds V_k(C^n).