Paraquaternionic CR-Submanifolds

  • Gabriel-Eduard Vîlcu


Paraquaternionic structures, at first known as quaternionic structures of second kind, are due to P. Libermann. Their study parallels that of quaternionic manifolds, yet relies on the algebra of paraquaternionic numbers. The counterpart in odd dimension of a paraquaternionic structure was introduced in 2006 by S. Ianuş, R. Mazzocco and G.E. Vîlcu and is referred to as a mixed 3-structure. It appears in a natural way on lightlike hypersurfaces in paraquaternionic manifolds. In this paper we review basic results concerning several types of submanifolds and semi-Riemannian submersions of manifolds endowed with paraquaternionic and mixed 3-structures.


Paraquaternionic structure Mixed 3-structure CR-submanifold Semi-Riemannian submersion Foliation 

2000 Mathematics Subject Classification




The author was supported by CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0118.


  1. 1.
    Al-Aqeel, A., Bejancu, A.: On normal semi-invariant submanifolds of paraquaternionic Kähler manifolds. Toyama Math. J. 30, 63–75 (2007)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alekseevsky, D.V., Cortes, V.: The twistor spaces of a para-quaternionic Kähler manifold. Osaka J. Math. 45(1), 215–251 (2008)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Alekseevsky, D., Kamishima, Y.: Quaternionic and para-quaternionic CR structure on \((4n+3)\)-dimensional manifolds. Central European J. Math. 2(5), 732–753 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alekseevsky, D.V., Cortes, V., Galaev, A.S., Leistner, T.: Cones over pseudo-Riemannian manifolds and their holonomy. J. Reine Angew. Math. 635, 23–69 (2009)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bejan, C.L., Druţă-Romaniuc, S.L.: Structures which are harmonic with respect to Walker metrics. Mediterr. J. Math. 12, 481–496 (2015)Google Scholar
  6. 6.
    Bejancu, A.: CR submanifolds of a Kaehler manifold I. Proc. Am. Math. Soc. 69, 135–142 (1978)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bejancu, A.: Hypersurfaces of manifolds with a Sasakian 3-structure. Demonstr. Math. 17, 197–209 (1984)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bejancu, A.: Normal semi-invariant submanifolds of paraquaternionic Kähler manifolds. Kuwait J. Sci. Eng. 33(2), 33–46 (2006)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bejancu, A., Farran, H.R.: Foliations and Geometric Structures. Springer, Berlin (2006)zbMATHGoogle Scholar
  10. 10.
    Besse, A.: Einstein Manifolds. Springer, New York (1987)CrossRefzbMATHGoogle Scholar
  11. 11.
    Blair, D.E.: Contact manifolds in Riemannian Geometry. Lectures Notes in Math, vol. 509 Springer, Berlin (1976)Google Scholar
  12. 12.
    Blair, D.E., Davidov, J., Muskarov, O.: Isotropic Kähler hyperbolic twistor spaces. J. Geom. Phys. 52(1), 74–88 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Blair, D.E., Davidov, J., Muskarov, O.: Hyperbolic twistor spaces. Rocky Mountain J. Math. 35, 1437–1465 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Blažić, N.: Para-quaternionic projective spaces and pseudo Riemannian geometry. Publ. Inst. Math. 60(74), 101–107 (1996)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Blažić, N., Vukmirović, S.: Solutions of Yang-Mills equations on generalized Hopf bundles. J. Geom. Phys. 41(1–2), 57–64 (2002)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Bonome, A., Castro, R., García-Río, E., Hervella, L., Vázquez-Lorenzo, R.: Pseudo-Riemannian manifolds with simple Jacobi operators. J. Math. Soc. Japan 54(4), 847–875 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Boyer, C., Galicki, K.: 3-Sasakian manifolds. Suppl. J. Differ. Geom. 6, 123–184 (1999)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Brozos-Vázquez, M., Gilkey, P., Nikčević, S., Vázquez-Lorenzo, R.: Geometric Realizations of para-Hermitian curvature models. Results Math. 56, 319–333 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Caldarella, A.: Paraquaternionic structures on smooth manifolds and related structures, Ph. D. thesis, University of Bari (2007)Google Scholar
  20. 20.
    Caldarella, A.: On paraquaternionic submersions between paraquaternionic Kähler manifolds. Acta Appl. Math. 112(1), 1–14 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Caldarella, A., Pastore, A.M.: Mixed 3-Sasakian structures and curvature. Ann. Polon. Math. 96, 107–125 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Calvaruso, G., Perrone, A.: Left-invariant hypercontact structures on three-dimensional Lie groups. Period Math Hung 69, 97–108 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Calvaruso, G., Perrone, D.: Metrics of Kaluza-Klein type on the anti-de Sitter space \(\mathbb{H}^3_1\). Math. Nachr. 287(8–9), 885–902 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Calvino-Louzao, E., García-Río, E., Gilkey, P., Vázquez-Lorenzo, R.: Higher-dimensional Osserman metrics with non-nilpotent Jacobi operators. Geom. Dedicata 156(1), 151–163 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Chen, B.-Y.: Pseudo-Riemannian Geometry, \(\delta \)-Invariants and Applications. World Scientific, Singapore (2011)CrossRefGoogle Scholar
  26. 26.
    Chiriac, N.C.: Normal anti-invariant submanifolds of paraquaternionic Kähler manifolds. Surv. Math. Appl. 1, 99–109 (2006)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Cruceanu, V., Fortuny, P., Gadea, P.M.: A survey on paracomplex geometry. Rocky Mt. J. Math. 26(1), 83–115 (1996)Google Scholar
  28. 28.
    Dancer, A.S., Jörgensen, H.R., Swann, A.F.: Metric geometries over the split quaternions. Rend. Sem. Mat. Univ. Pol. Torino 63(2), 119–139 (2005)MathSciNetzbMATHGoogle Scholar
  29. 29.
    David, L.: About the geometry of almost para-quaternionic manifolds. Differ. Geom. Appl. 27(5), 575–588 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Falcitelli, M., Ianuş, S., Pastore, A.M.: Riemannian Submersions and Related Topics. World Scientific, River Edge (2004)CrossRefzbMATHGoogle Scholar
  31. 31.
    Galicki, K., Lawson, B.: Quaternionic reduction and quaternionic orbifolds. Math. Ann. 282(1), 1–21 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    García-Río, E., Matsushita, Y., Vázquez-Lorenzo, R.: Paraquaternionic Kähler manifolds. Rocky Mt. J. Math. 31(1), 237–260 (2001)CrossRefzbMATHGoogle Scholar
  33. 33.
    Gray, A.: Curvature identities for Hermitian and almost-Hermitian manifolds. Tohoku Math. J. 28, 601–612 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Gray, A.: Einstein-like manifolds which are not Einstein. Geom. Dedicata 7, 259–280 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Ianuş, S.: Sulle strutture canoniche dello spazio fibrato tangente di una varieta riemanniana. Rend. Mat. 6, 75–96 (1973)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Ianuş, S., Vîlcu, G.E.: Some constructions of almost para-hyperhermitian structures on manifolds and tangent bundles. Int. J. Geom. Methods Mod. Phys. 5(6), 893–903 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Ianuş, S., Vîlcu, G.E.: Hypersurfaces of paraquaternionic space forms. J. Gen. Lie Theory Appl. 2, 175–179 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Ianuş, S., Vîlcu, G.E.: Semi-Riemannian hypersurfaces in manifolds with metric mixed 3-structures. Acta Math. Hung. 127(1–2), 154–177 (2010)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Ivanov, S., Zamkovoy, S.: Para-Hermitian and paraquaternionic manifolds. Differ. Geom. Appl. 23, 205–234 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ianuş, S., Mazzocco, R., Vîlcu, G.E.: Real lightlike hypersurfaces of paraquaternionic Kähler manifolds. Mediterr. J. Math. 3, 581–592 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Ianuş, S., Ionescu, A.M., Vîlcu, G.E.: Foliations on quaternion CR-submanifolds. Houston J. Math. 34(3), 739–751 (2008)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Ianuş, S., Visinescu, M., Vîlcu, G.E.: Conformal Killing-Yano tensors on manifolds with mixed 3-structures, SIGMA, Symmetry Integrability Geom. Methods Appl. 5, Paper 022, p. 12 (2009)Google Scholar
  43. 43.
    Ianuş, S., Visinescu, M., Vîlcu, G.E.: Hidden symmetries and Killing tensors on curved spaces. Phys. At. Nucl. 73(11), 1925–1930 (2010)CrossRefGoogle Scholar
  44. 44.
    Ianuş, S., Marchiafava, S., Vîlcu, G.E.: Paraquaternionic CR-submanifolds of paraquaternionic Kähler manifolds and semi-Riemannian submersions. Cent. Eur. J. Math. 8(4), 735–753 (2010)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Ivanov, S., Minchev, I., Zamkovoy, S.: Twistor and reflector spaces of almost para-quaternionic manifolds. In: Vicente Cortés (ed.) Handbook of Pseudo-Riemannian Geometry and Supersymmetry, pp. 477-496. European Mathematical Society (2010)Google Scholar
  46. 46.
    Ianuş, S., Ornea, L., Vîlcu, G.E.: Invariant and anti-invariant submanifolds in manifolds with metric mixed 3-structures. Mediterr. J. Math. 9(1), 105–128 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Kobayashi, S.: Submersions of CR submanifolds. Tôhoku Math. J. 39, 95–100 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 2. Interscience, New York (1963, 1969)Google Scholar
  49. 49.
    Krahe, M.: Para-pluriharmonic maps and twistor spaces. Ph.D. thesis, Universität Augsburg (2007)Google Scholar
  50. 50.
    Libermann, P.: Sur les structures presque quaternioniennes de deuxième espéce. C.R. Acad. Sci. Paris 234, 1030–1032 (1952)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Marchiafava, S.: Submanifolds of (para)-quaternionic Kähler manifolds. Note Mat. 28(S1), 295–316 (2008)MathSciNetGoogle Scholar
  52. 52.
    Marchiafava, S.: Twistorial maps between (para)quaternionic projective spaces. Bull. Math. Soc. Sci. Math. Roumanie 52, 321–332 (2009)MathSciNetzbMATHGoogle Scholar
  53. 53.
    O’Neill, B.: The fundamental equations of a submersion. Michigan Math. J. 39, 459–464 (1966)MathSciNetzbMATHGoogle Scholar
  54. 54.
    O’Neill, B.: Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, vol. 103. Academic Press, New York (1983)Google Scholar
  55. 55.
    Ornea, L.: \(CR\)-submanifolds. A class of examples. Rev. Roum. Math. Pures Appl. 51(1), 77–85 (2006)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Sasaki, S.: On differentiable manifolds with certain structures which are closely related to almost contact structure I. Tohoku Math. J. 12, 459–476 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Sato, I.: On a structure similar to the almost contact structure. Tensor, New Ser. 30, 219–224 (1976)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Song, Y.M., Kim, J.S., Tripathi, M.M.: On hypersurfaces of manifolds equipped with a hypercosymplectic 3-structure. Commun. Korean Math. Soc. 18, 297–308 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Vaccaro, M.: Kaehler and para-Kaehler submanifolds of a para-quaternionic Kaehler manifold. Ph. D. thesis, Università degli Studi di Roma II “Tor Vergata” (2007)Google Scholar
  60. 60.
    Vaccaro, M.: (Para-)Hermitian and (para-)Kähler submanifolds of a para-quaternionic Kähler manifold. Differ. Geom. Appl. 30(4), 347–364 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Vîlcu, G.E.: Submanifolds of an Almost Paraquaternionic Kähler Product Manifold. Int. Math. Forum 2(15), 735–746 (2007)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Vîlcu, G.E.: Normal semi-invariant submanifolds of paraquaternionic space forms and mixed 3-structures. BSG Proc. 15, 232–240 (2008)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Vîlcu, G.E.: Ruled CR-submanifolds of locally conformal Kähler manifolds. J. Geom. Phys. 62(6), 1366–1372 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Vîlcu, G.E.: Mixed paraquaternionic 3-submersions. Indag. Math. 24, 474–488 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Vîlcu, G.E.: Canonical foliations on paraquaternionic Cauchy-Riemann submanifolds. J. Math. Anal. Appl. 399(2), 551–558 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Vîlcu, G.E., Voicu, R.C.: Curvature properties of pseudo-sphere bundles over paraquaternionic manifolds. Int. J. Geom. Methods Mod. Phys. 9, 1250024, p. 23 (2012)Google Scholar
  67. 67.
    Vukmirović S.: Paraquaternionic reduction, math.DG/0304424Google Scholar
  68. 68.
    Zamkovoy, S.: Geometry of paraquaternionic Kähler manifolds with torsion. J. Geom. Phys. 57(1), 69–87 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Petroleum-Gas University of PloieştiPloieştiRomania
  2. 2.Faculty of Mathematics and Computer ScienceUniversity of Bucharest, Research Center in Geometry, Topology and AlgebraBucharestRomania

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