Closed geodesics on compact nilmanifolds with Chevalley rational structure

We study the distribution of closed geodesics on nilmanifolds Γ N arising from a 2-step nilpotent Lie algebra constructed from an irreducible representation of a compact semisimple Lie algebra on a real finite dimensional vector space U. We determine sufficient conditions on the semisimple Lie algebra for Γ N to have the density of closed geodesics property where Γ is a lattice arising from a Chevalley rational structure on .