Density of closed geodesics in a compact nilmanifold with Chevalley rational structure.

摘要

We continue the study of the distribution of closed geodesics on nilmanifolds Gamma N, constructed from a simply connected 2-step nilpotent Lie group N with a left invariant metric and a lattice Gamma in N. We consider a Lie group N with associated 2-step nilpotent Lie algebra N=U⊕g0 constructed from an irreducible representation of a compact semisimple Lie algebra g0 on a real finite dimensional vector space U.; K. B. Lee and K. Park have shown that if {lcub} N , ⟨ , ⟩{rcub} I satisfies a nonsingularity condition and a resonance condition, then the density of closed geodesics property will hold on Gamma N for all lattices Gamma. M. Mast has shown that the resonance condition is a necessary condition for the density of closed geodesics. L. DeMeyer proved that the nonsingularity condition of Lee and Park was not necessary in the case that g0= su2 .; We investigate the general case where the nonsingularity condition does not hold for a compact semisimple Lie algebra g0 and some g0 -module U. We determine sufficient conditions on the associated semisimple Lie algebra g0 for GammaN to have the density of closed geodesics property, where Gamma is a lattice arising from a Chevalley rational structure on N . We show that in almost all cases the density of closed geodesics property holds in GammaN. We list explicitly the exceptional cases where our method does not apply.