On almost cosymplectic (−1, μ, 0)-spaces

Piotr Dacko; Zbigniew Olszak

Open Mathematics (2005)

  • Volume: 3, Issue: 2, page 318-330
  • ISSN: 2391-5455

Abstract

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In our previous paper, almost cosymplectic (κ, μ, ν)-spaces were defined as the almost cosymplectic manifolds whose structure tensor fields satisfy a certain special curvature condition. Amongst other results, it was proved there that any almost cosymplectic (κ, μ, ν)-space can be 𝒟 -homothetically deformed to an almost cosymplectic −1, μ′, 0)-space. In the present paper, a complete local description of almost cosymplectic (−1, μ, 0)-speces is established: “models” of such spaces are constructed, and it is noted that a given almost cosymplectic (−1, μ 0)-space is locally isomorphic to a corresponding model. In the case when μ is constant, the models can be constructed on the whole of ℝ2n+1 and it is shown that they are left invariant with respect to Lie group actions.

How to cite

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Piotr Dacko, and Zbigniew Olszak. "On almost cosymplectic (−1, μ, 0)-spaces." Open Mathematics 3.2 (2005): 318-330. <http://eudml.org/doc/268912>.

@article{PiotrDacko2005,
abstract = {In our previous paper, almost cosymplectic (κ, μ, ν)-spaces were defined as the almost cosymplectic manifolds whose structure tensor fields satisfy a certain special curvature condition. Amongst other results, it was proved there that any almost cosymplectic (κ, μ, ν)-space can be \[\mathcal \{D\}\] -homothetically deformed to an almost cosymplectic −1, μ′, 0)-space. In the present paper, a complete local description of almost cosymplectic (−1, μ, 0)-speces is established: “models” of such spaces are constructed, and it is noted that a given almost cosymplectic (−1, μ 0)-space is locally isomorphic to a corresponding model. In the case when μ is constant, the models can be constructed on the whole of ℝ2n+1 and it is shown that they are left invariant with respect to Lie group actions.},
author = {Piotr Dacko, Zbigniew Olszak},
journal = {Open Mathematics},
keywords = {53C25; 53D15},
language = {eng},
number = {2},
pages = {318-330},
title = {On almost cosymplectic (−1, μ, 0)-spaces},
url = {http://eudml.org/doc/268912},
volume = {3},
year = {2005},
}

TY - JOUR
AU - Piotr Dacko
AU - Zbigniew Olszak
TI - On almost cosymplectic (−1, μ, 0)-spaces
JO - Open Mathematics
PY - 2005
VL - 3
IS - 2
SP - 318
EP - 330
AB - In our previous paper, almost cosymplectic (κ, μ, ν)-spaces were defined as the almost cosymplectic manifolds whose structure tensor fields satisfy a certain special curvature condition. Amongst other results, it was proved there that any almost cosymplectic (κ, μ, ν)-space can be \[\mathcal {D}\] -homothetically deformed to an almost cosymplectic −1, μ′, 0)-space. In the present paper, a complete local description of almost cosymplectic (−1, μ, 0)-speces is established: “models” of such spaces are constructed, and it is noted that a given almost cosymplectic (−1, μ 0)-space is locally isomorphic to a corresponding model. In the case when μ is constant, the models can be constructed on the whole of ℝ2n+1 and it is shown that they are left invariant with respect to Lie group actions.
LA - eng
KW - 53C25; 53D15
UR - http://eudml.org/doc/268912
ER -

References

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  1. [1] D.E. Blair: Riemannian geometry of contact and symplectic manifolds, Progress in Math., Vol. 203, Birkhäuser, Boston, 2001. 
  2. [2] D.E. Blair, T. Koufogiorgos and B.J. Papantoniou: “Contact metric manifolds satisfying a nullity condition”, Israel. J. Math., Vol. 91, (1995), pp. 189–214. Zbl0837.53038
  3. [3] E. Boeckx: “A full classification of contact metric (κ, μ)-spaces”, Ill. J. Math., Vol. 44, (2000), pp. 212–219. Zbl0969.53019
  4. [4] D. Chinea, M. de León and J.C. Marrero: “Stability of invariant foliations on almost contact manifolds”, Publ. Math. Debrecen, Vol. 43, (1993), pp. 41–52. Zbl0798.53032
  5. [5] D. Chinea and C. Gonzáles: An example of an almost cosymplectic homogeneous manifold, Lec. Notes Math., Vol. 1209, Springer, Berlin, 1986, pp. 133–142. 
  6. [6] L.A. Cordero, M. Fernández and M. De León: “Examples of compact almost contact manifolds admitting neither Sasakian nor cosymplectic structures”, Atti Sem. Mat. Fis. Univ. Modena, Vol. 34, (1985–86) pp. 43–54. Zbl0616.53032
  7. [7] P. Dacko: “On almost cosymplectic manifolds with the structure vector field ξ belonging to the k-nullity distribution”, Balkan. J. Geom. Appl., Vol. 5(2), (2000), pp. 47–60. Zbl0981.53075
  8. [8] P. Dacko and Z. Olszak: “On conformally flat almost cosymplectic manifolds with Kählerian leaves”, Rend. Sem. Mat. Univ. Pol. Torino, Vol. 56, (1998), pp. 89–103. Zbl0981.53074
  9. [9] P. Dacko and Z. Olszak: “On almost cosymplectic (κ, μ, ν)-spaces”, in print. Zbl1114.53028
  10. [10] H. Endo: “On some properties of almost cosymplectic manifolds”, An. §tiint. Univ. “Al. I. Cuza” Ia§i, Mat., Vol. 42, (1996), pp. 79–94. 
  11. [11] H. Endo: “On some invariant submanifolds in certain almost cosymplectic manifolds”, An. §tiint. Univ. “Al. I. Cuza” Ia§i, Mat., Vol. 43, (1997), pp. 383–395. 
  12. [12] H. Endo: “Non-existence of almost cosymplectic manifolds satisfying a certain condition”, Tensor N. S., Vol. 63, (2002), pp. 272–284. Zbl1119.53316
  13. [13] S.I. Goldberg and K. Yano: “Integrability of almost cosymplectic structure”, Pacific J. Math., Vol. 31, (1969), pp. 373–382. Zbl0185.25104
  14. [14] Z. Olszak: “On almost cosymplectic manifolds”, Kodai Math. J., Vol. 4, (1981), pp. 239–250. http://dx.doi.org/10.2996/kmj/1138036371 Zbl0451.53035
  15. [15] Z. Olszak: “Curvature properties of quasi-Sasakian manifolds”, Tensor N.S., Vol. 38, (1982), pp. 19–28. Zbl0507.53031
  16. [16] Z. Olszak: “Almost cosymplectic manifolds with Kählerian leaves”, Tensor N.S., Vol. 46, (1987), pp. 117–124. Zbl0631.53027
  17. [17] S. Tanno: “Ricci curvatures of contact Riemannian manifolds”, Tôhoku Math. J., Vol. 40, (1988), pp. 441–448. Zbl0655.53035

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