Homogeneous geodesics in the flag manifold SO(2l+1)/U(l-m)×SO(2m+1)

A geodesic curve in a Riemannian homogeneous manifold (M=G/K,g) is called a homogeneous geodesic if it is an orbit of an one-parameter subgroup of the Lie group G. We investigate G-invariant metrics such that all geodesics are homogeneous for the flag manifold M=SO(2l+1)/U(l-m)×SO(2m+1). By reformulating the problem into a matrix form we show that SO(2ℓ+1)/U(ℓ-m)×SO(2m+1) has homogeneous geodesics with respect to any SO(2ℓ+1)-invariant metric if and only if m=0. In all other cases this space admits at least one non-homogeneous geodesic. We also give examples of finding homogeneous geodesics in the above flag manifold for special values of l and m.