Cs. Gordon et al., SYMPLECTIC RIGIDITY OF GEODESIC-FLOWS ON 2-STEP NILMANIFOLDS, Annales Scientifiques de l'Ecole Normale Superieure, 30(4), 1997, pp. 417-427
We show that if two 2-step Riemannian nilmanifolds have symplectically
conjugate geodesic hows, then they must be isometric. By 2-step Riema
nnian nilmanifold, we mean a Riemannian manifold of the form (Gamma\N,
g), where N is a 2-step nilpotent Lie group, Gamma is a cocompact disc
rete subgroup of N, and g is a metric whose pullback to N is left inva
riant.