Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 021, 4 pages      arXiv:1003.0189      https://doi.org/10.3842/SIGMA.2010.021

On the Existence of a Codimension 1 Completely Integrable Totally Geodesic Distribution on a Pseudo-Riemannian Heisenberg Group

Wafaa Batat a and Salima Rahmani b, c
a) Ecole Normale Supérieure de L'Enseignement Technologique d'Oran, Département de Mathématiques et Informatique, B.P. 1523 El M'Naouar Oran, Algeria
b) Laboratoire de Mathématiques-Informatique et Applications, Université de Haute Alsace, 68093 Mulhouse Cedex, France
c) Ecole Doctorale de Systèmes Dynamiques et Géométrie, Département de Mathématiques, Faculté des Sciences, Université des Sciences et de la Technologie d'Oran, B.P. 1505 Oran El M'Naouer, Algeria

Received December 23, 2009, in final form February 23, 2010; Published online February 28, 2010

Abstract
In this note we prove that the Heisenberg group with a left-invariant pseudo-Riemannian metric admits a completely integrable totally geodesic distribution of codimension 1. This is on the contrary to the Riemannian case, as it was proved by T. Hangan.

Key words: Heisenberg group; pseudo-Riemannian metrics; geodesics; codimension 1 distributions; completely integrable distributions.

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References

  1. Cordero L.A., Parker P.E., Pseudoriemannian 2-step nilpotent Lie groups, math.DG/9905188.
  2. Eberlein P., Geometry of 2-step nilpotent groups with a left invariant metric, Ann. Sci. École Norm. Sup. (4) 27 (1994), 611-660.
  3. Guediri M., Sur la complétude des pseudo-métriques invariantes a gauche sur les groupes de Lie nilpotents, Rend. Sem. Mat. Univ. Politec. Torino 52 (1994), 371-376.
  4. Guediri M., Lorentz geometry of 2-step nilpotent Lie groups, Geom. Dedicata 100 (2003), 11-51.
  5. Hangan T., Au sujet des flots riemanniens sur le groupe nilpotent de Heisenberg, Rend. Circ. Mat. Palermo (2) 35 (1986), 291-305.
  6. Hangan T., Sur les distributions totalement géodésiques du groupe nilpotent riemannien H2p+1, Rend. Sem. Fac. Sci. Univ. Cagliari 55 (1985), 31-47.
  7. Jang C., Parker P.E., Examples of conjugate loci of pseudoRiemannian 2-step nilpotent Lie groups with nondegenerate center, in Recent Advances in Riemannian and Lorentzian Geometries (Baltimore, MD, 2003), Editors K.L. Duggal and R. Sharma, Contemp. Math., Vol. 337, Amer. Math. Soc., Providence, RI, 2003, 91-108.
  8. Rahmani S., Métriques de Lorentz sur les groupes de Lie unimodulaires, de dimension 3, J. Geom. Phys. 9 (1992), 295-302.
  9. Rahmani N., Rahmani S., Lorentzian geometry of the Heisenberg group, Geom. Dedicata 118 (2006), 133-140.

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