  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).
