Central European Journal of Mathematics

, Volume 8, Issue 4, pp 735–753 | Cite as

Paraquaternionic CR-submanifolds of paraquaternionic Kähler manifolds and semi-Riemannian submersions

  • Stere Ianuş
  • Stefano Marchiafava
  • Gabriel Eduard Vîlcu
Research Article

Abstract

In this paper we introduce paraquaternionic CR-submanifolds of almost paraquaternionic hermitian manifolds and state some basic results on their differential geometry. We also study a class of semi-Riemannian submersions from paraquaternionic CR-submanifolds of paraquaternionic Kähler manifolds.

Keywords

Paraquaternionic Kähler manifold Foliation CR-submanifold Semi-Riemannian submersion 

MSC

53C15 

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References

  1. [1]
    Al-Aqeel A., Bejancu A., On normal semi-invariant submanifolds of paraquaternionic Kähler manifolds, Toyama Math. J., 2007, 30, 63–75MATHMathSciNetGoogle Scholar
  2. [2]
    Alekseevsky D., Cortés V., The twistor spaces of a para-quaternionic Kähler manifold, Osaka J. Math., 2008, 45(1), 215–251MATHMathSciNetGoogle Scholar
  3. [3]
    Alekseevsky D., Kamishima Y., Quaternionic and para-quaternionic CR structure on (4n+3)-dimensional manifolds, Cent. Eur. J. Math., 2004, 2(5), 732–753MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Alekseevsky D.V., Marchiafava S., Almost complex submanifolds of quaternionic manifolds, In: Proceedings of the colloquium on differential geometry, Debrecen (Hungary), 25–30 July 2000, Inst. Math. Inform. Debrecen, 2001, 23–38Google Scholar
  5. [5]
    Barros M., Chen B.-Y., Urbano F., Quaternion CR-submanifolds of quaternion manifolds, Kodai Math. J., 1981, 4(3), 399–417MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Bejancu A., CR submanifolds of a Kaehler manifold I, Proc. Am. Math. Soc., 1978, 69(1), 135–142MATHMathSciNetGoogle Scholar
  7. [7]
    Bejancu A., Geometry of CR-submanifolds, Mathematics and its Applications (East European Series), 23, D. Reidel Publishing Co., Dordrecht, 1986Google Scholar
  8. [8]
    Bejancu A., Normal semi-invariant submanifolds of paraquaternionic Kähler manifolds, Kuwait J. Sci. Engrg., 2006, 33(2), 33–45MathSciNetGoogle Scholar
  9. [9]
    Bejancu A., Farran H.R., Foliations and Geometric Structures, Mathematics and Its Applications, 580, Springer, Dordrecht, 2006Google Scholar
  10. [10]
    Blažić N., Paraquaternionic projective spaces and pseudo-Riemannian geometry, Publ. Inst. Math. (Beograd), 1996, 60(74), 101–107MathSciNetGoogle Scholar
  11. [11]
    Blažić N., Vukmirović S., Solutions of Yang-Mills equations on generalized Hopf bundles, J. Geom. Phys., 2002, 41(1–2), 57–64MATHMathSciNetGoogle Scholar
  12. [12]
    Chen B.-Y., Cohomology of CR-submanifolds, Ann. Fac. Sci. Toulouse Math., 1981, 3(2), 167–172MATHMathSciNetGoogle Scholar
  13. [13]
    Dancer A.S., Jørgensen H.R., Swann A.F., Metric geometries over the split quaternions, Rend. Sem. Mat. Univ. Politec. Torino, 2005, 63(2), 119–139MATHMathSciNetGoogle Scholar
  14. [14]
    Falcitelli M., Ianuş S., Pastore A.M., Riemannian Submersions and Related Topics, World Scientific, River Edge, 2004MATHCrossRefGoogle Scholar
  15. [15]
    Galicki K., Lawson H.B., Quaternionic reduction and quaternionic orbifolds, Math. Ann., 1988, 282(1), 1–21MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Garcia-Rio E., Matsushita Y., Vásquez-Lorenzo R., Paraquaternionic Kähler manifolds, Rocky Mountain J. Math., 2001, 31(1), 237–260MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Ianuş S., Ionescu A.M., Vîlcu G.E., Foliations on quaternion CR-submanifolds, Houston J. Math., 2008, 34(3), 739–751MATHMathSciNetGoogle Scholar
  18. [18]
    Ianuş S., Mazzocco R., Vîlcu G.E., Real lightlike hypersurfaces of paraquaternionic Kähler manifolds, Mediterr. J. Math., 2006, 3(3–4), 581–592MATHMathSciNetGoogle Scholar
  19. [19]
    Ianuş S., Vîlcu G.E., Some constructions of almost para-hyperhermitian structures on manifolds and tangent bundles, Int. J. Geom. Methods Mod. Phys., 2008, 5(6), 893–903MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Ionescu A.M., Vîlcu G.E., A note on paraquaternionic manifolds, Missouri J. Math. Sci., 2007, 19(3), 213–218MATHGoogle Scholar
  21. [21]
    Ivanov S., Zamkovoy S., Parahermitian and paraquaternionic manifolds, Differential Geom. Appl., 2005, 23(2), 205–234MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    Kobayashi S., Submersions of CR submanifolds, Tôhoku Math. J., 1987, 39(1), 95–100MATHGoogle Scholar
  23. [23]
    Libermann P., Sur le problème d’équivalence de certaines structures infinitésimales, Ann. Mat. Pura Appl., 1954, 36, 27–120MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    Marchiafava S., Submanifolds of (para)-quaternionic Kähler manifolds, Note Mat., 2008, 28(suppl.1), 295–316MathSciNetGoogle Scholar
  25. [25]
    O’Neill B., Semi-Riemannian Geometry. With Applications to Relativity, Pure and Applied Mathematics, 103, Academic Press, New York, 1983Google Scholar
  26. [26]
    Ornea L., CR-submanifolds. A class of examples, Rev. Roumaine Math. Pures Appl., 2006, 51(1), 77–85MATHMathSciNetGoogle Scholar
  27. [27]
    Vaccaro M., Kaehler and para-Kaehler submanifolds of a para-quaternionic Kaehler manifold, Ph.D. thesis, Università degli Studi di Roma II’ Tor Vergata’, Rome, Italy, 2007Google Scholar
  28. [28]
    Vîlcu G.E., Submanifolds of an Almost Paraquaternionic Kähler Product Manifold, Int. Math. Forum, 2007, 2(15), 735–746MATHMathSciNetGoogle Scholar
  29. [29]
    Vukmirovic S., Paraquaternionic reduction, preprint available at http://arxiv.org/abs/math/0304424Google Scholar
  30. [30]
    Wu H., On the de Rham decomposition theorem, Illinois J. Math., 1964, 8, 291–311MATHMathSciNetGoogle Scholar
  31. [31]
    Yano K., Kon M., CR Submanifolds of Kaehlerian and Sasakian Manifolds, Progress in Mathematics, 30, Birkhäuser, Boston, 1983Google Scholar
  32. [32]
    Zamkovoy S., Geometry of paraquaternionic Kähler manifolds with torsion, J. Geom. Phys., 2006, 57(1), 69–87MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© © Versita Warsaw and Springer-Verlag Wien 2010

Authors and Affiliations

  • Stere Ianuş
    • 1
  • Stefano Marchiafava
    • 2
  • Gabriel Eduard Vîlcu
    • 3
  1. 1.Department of MathematicsUniversity of BucharestBucharestRomania
  2. 2.Dipartimento di MatematicaUniversità di Roma “La Sapienza”RomeItaly
  3. 3.Department of Mathematics and Computer SciencePetroleum-Gas University of PloieştiPloieştiRomania

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