On the complete integrability of the geodesic flow of pseudo-H-type Lie groups

Abstract

Pseudo-H-type groups $$G_{r,s}$$ form a class of step-two nilpotent Lie groups with a natural pseudo-Riemannian metric. In this paper the question of complete integrability in the sense of Liouville is studied for the corresponding (pseudo-)Riemannian geodesic flow. Via the isometry group of $$G_{r,s}$$ families of first integrals are constructed. A modification of these functions gives a set of $$\dim G_{r,s}$$ functionally independent smooth first integrals in involution. The existence of a lattice L in $$G_{r,s}$$ is guaranteed by recent work of K. Furutani and I. Markina. The complete integrability of the pseudo-Riemannian geodesic flow of the compact nilmanifold $$L \backslash G_{r,s}$$ is proved under additional assumptions on the group $$G_{r,s}$$ .