Quaternionic and para-quaternionic CR structure on (4n+3)-dimensional manifolds

Dmitri Alekseevsky; Yoshinobu Kamishima

Open Mathematics (2004)

  • Volume: 2, Issue: 5, page 732-753
  • ISSN: 2391-5455

Abstract

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We define notion of a quaternionic and para-quaternionic CR structure on a (4n+3)-dimensional manifold M as a triple (ω1,ω2,ω3) of 1-forms such that the corresponding 2-forms satisfy some algebraic relations. We associate with such a structure an Einstein metric on M and establish relations between quaternionic CR structures, contact pseudo-metric 3-structures and pseudo-Sasakian 3-structures. Homogeneous examples of (para)-quaternionic CR manifolds are given and a reduction construction of non homogeneous (para)-quaternionic CR manifolds is described.

How to cite

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Dmitri Alekseevsky, and Yoshinobu Kamishima. "Quaternionic and para-quaternionic CR structure on (4n+3)-dimensional manifolds." Open Mathematics 2.5 (2004): 732-753. <http://eudml.org/doc/268728>.

@article{DmitriAlekseevsky2004,
abstract = {We define notion of a quaternionic and para-quaternionic CR structure on a (4n+3)-dimensional manifold M as a triple (ω1,ω2,ω3) of 1-forms such that the corresponding 2-forms satisfy some algebraic relations. We associate with such a structure an Einstein metric on M and establish relations between quaternionic CR structures, contact pseudo-metric 3-structures and pseudo-Sasakian 3-structures. Homogeneous examples of (para)-quaternionic CR manifolds are given and a reduction construction of non homogeneous (para)-quaternionic CR manifolds is described.},
author = {Dmitri Alekseevsky, Yoshinobu Kamishima},
journal = {Open Mathematics},
keywords = {53C55; 57S25},
language = {eng},
number = {5},
pages = {732-753},
title = {Quaternionic and para-quaternionic CR structure on (4n+3)-dimensional manifolds},
url = {http://eudml.org/doc/268728},
volume = {2},
year = {2004},
}

TY - JOUR
AU - Dmitri Alekseevsky
AU - Yoshinobu Kamishima
TI - Quaternionic and para-quaternionic CR structure on (4n+3)-dimensional manifolds
JO - Open Mathematics
PY - 2004
VL - 2
IS - 5
SP - 732
EP - 753
AB - We define notion of a quaternionic and para-quaternionic CR structure on a (4n+3)-dimensional manifold M as a triple (ω1,ω2,ω3) of 1-forms such that the corresponding 2-forms satisfy some algebraic relations. We associate with such a structure an Einstein metric on M and establish relations between quaternionic CR structures, contact pseudo-metric 3-structures and pseudo-Sasakian 3-structures. Homogeneous examples of (para)-quaternionic CR manifolds are given and a reduction construction of non homogeneous (para)-quaternionic CR manifolds is described.
LA - eng
KW - 53C55; 57S25
UR - http://eudml.org/doc/268728
ER -

References

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  1. [1] D.V. Alekseevsky, V. Cortés: “Classification of pseudo-Riemannian symmetric spaces of quaternionic Kähler type”, Preprint Institut Eli Cartan, Vol. 11, (2004). Zbl1077.53040
  2. [2] D.A. Alekseevsky and Y. Kamishima: “Locally pseudo-conformal quaternionic CR structure”, preprint. Zbl1223.53054
  3. [3] A. Besse: Einstein manifolds, Springer Verlag, 1987. 
  4. [4] O. Biquard: Quaternionic structures in mathematics and physics, World Sci. Publishing, Rome, 1999, River Edge, NJ, 2001, pp. 23–30. Zbl0993.53017
  5. [5] D.E. Blair: “Riemannian geometry of contact and symplectic manifolds Contact manifolds”, Birkhäuser, Progress in Math., Vol. 203, (2002). Zbl1011.53001
  6. [6] C.P. Boyer, K. Galicki and B.M. Mann: “The geometry and topology of 3-Sasakian manifolds”, Jour. reine ange. Math., Vol. 455, (1994), pp. 183–220. Zbl0889.53029
  7. [7] N.J. Hitchin: “The self-duality equations on a Riemannian surface”, Proc. London Math. Soc., Vol. 55, (1987), pp. 59–126. Zbl0634.53045
  8. [8] S. Ishihara: “Quaternion Kählerian manifolds”, J. Diff. Geom., Vol. 9, (1974), pp. 483–500. Zbl0297.53014
  9. [9] S. Ishihara: “Quaternion Kählerian manifolds and fibred Riemannian spaces with Sasakian 3-structure”, Kōdai Math. Sem. Rep., Vol. 25, (1973), pp. 321–329. Zbl0267.53023
  10. [10] S. Ishihara and M. Konishi: “Real contact and complex contact structure”, Sea. Bull. Math., Vol. 3, (1979), pp. 151–161. Zbl0429.53026
  11. [11] W. Jelonek: “Positive and negative 3-K-contact structures”, Proc. of A.M.S., Vol. 129, (2000), pp. 247–256. http://dx.doi.org/10.1090/S0002-9939-00-05527-1 Zbl0977.53074
  12. [12] R. Kulkarni: “Proper actions and pseudo-Riemannian space forms”, Advances in Math., Vol. 40, (1981), pp. 10–51. http://dx.doi.org/10.1016/0001-8708(81)90031-1 
  13. [13] T. Kashiwada: “On a contact metric structure”, Math. Z., Vol. 238, (2001), pp. 829–832. http://dx.doi.org/10.1007/s002090100279 Zbl1004.53058
  14. [14] M. Konishi: “On manifolds with Sasakian 3-structure over quaternionic Kaehler manifolds”, |emphKodai Math. Jour., Vol. 29, (1975), pp. 194–200. Zbl0308.53035
  15. [15] S. Kobayashi and K. Nomizu: Foundations of differential geometry I, II, Interscience John Wiley & Sons, New York, 1969. Zbl0175.48504
  16. [16] A. Swann: “Aspects symplectiques de la géométrie quaternionique”, C. R. Acad. Sci. Paris, Seria I, Vol. 308, (1989), pp. 225–228. Zbl0661.53023
  17. [17] S. Tanno: “Killing vector fields on contact Riemannian manifolds and fibering related to the Hopf fibrations”, |emphTôhoku Math. Jour., Vol. 23, (1971), pp. 313–333. 
  18. [18] S. Tanno: “Remarks on a triple of K-contact structures”, Tôhoku Math. Jour., Vol. 48, (1996), pp. 519–531. Zbl0881.53040
  19. [19] J. Wolf: Spaces of constant curvature, McGraw-Hill, Inc., 1967. Zbl0162.53304

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