Analysis and Mathematical Physics

, Volume 8, Issue 4, pp 493–520 | Cite as

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

  • Wolfram Bauer
  • Daisuke TaramaEmail author


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 \(G_{r,s}\).


Pseudo-Riemannian metric Hamilton’s equation Killing vector fields Pseudo-H-type nilmanifolds 

Mathematics Subject Classification

53C30 22E25 37K10 



We thank the referee for many useful hints that improved the presentation of the paper.


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institut für AnalysisLeibniz Universität HannoverHannoverGermany
  2. 2.Department of Mathematical SciencesRitsumeikan UniversityKusatsuJapan

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