Mediterranean Journal of Mathematics

, Volume 10, Issue 3, pp 1551–1571 | Cite as

Almost Cosymplectic and Almost Kenmotsu (κ, μ, ν)-Spaces



We study the Riemann curvature tensor of (κ, μ, ν)-spaces when they have almost cosymplectic and almost Kenmotsu structures, giving its writing explicitly. This leads to the definition and study of a natural generalisation of the contact metric (κ, μ, ν)-spaces. We present examples or obstruction results of these spaces in all possible cases.

Mathematics Subject Classification (2010)

Primary 53C15 Secondary 53C25 


Generalized (κ, μ)-spaces (κ, μ, ν)-space generalized Sasakian space form almost cosymplectic almost Kenmotsu 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alegre P., Blair D. E., Carriazo A.: Generalized Sasakian-space-forms, Israel J. Math. 141, 157–183 (2004)MATHGoogle Scholar
  2. 2.
    Alegre P., Carriazo A.: Structures on generalized Sasakian-space-forms, Differential Geom. Appl. 26((6), 656–666 (2008)MathSciNetMATHGoogle Scholar
  3. 3.
    Alegre P., Carriazo A.: Submanifolds of generalized Sasakian space forms, Taiwanese J. Math. 13((3), 923–941 (2009)MathSciNetMATHGoogle Scholar
  4. 4.
    K. Arslan, A. Carriazo, V. Martín-Molina and C. Murathan, The curvature tensor of (κ, μ, ν)-contact metric manifolds, arXiv:1109.625v1Google Scholar
  5. 5.
    D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Second edition. Birkh¨auser, Boston, 2010.Google Scholar
  6. 6.
    Blair D. E., Koufogiorgos T., Papantoniou B. J.: Contact metric manifolds satisfying a nullity condition, Israel J. Math. 91, 189–214 (1995)Google Scholar
  7. 7.
    A. Carriazo and V. Martín-Molina, Generalized (κ, μ)-space forms and Dahomothetic deformations, Balkan J. Geom. Appl. 6 (1) (2011), 37–47Google Scholar
  8. 8.
    A. Carriazo, V. Martín-Molina and M. M. Tripathi, Generalized (κ, μ)-space forms, Mediterr. J. Math. DOI  10.1007/s00009-012-0196-2
  9. 9.
    P. Dacko, On almost cosymplectic manifolds with the structure vector field ξ belonging to the κ-nullity distribution, Balkan J. Geom. Appl. 5 (no. 2) (2000), 47–60.Google Scholar
  10. 10.
    Dacko P., Olszak Z.: On almost cosymplectic (κ, μ, ν)-spaces, Banach Center Publ. 69, 211–220 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    P. Dacko and Z. Olszak, On almost cosymplectic (−1, μ, 0)-spaces, Cent. Eur. J. Math. 3 (no. 2) (2005), 318–330.Google Scholar
  12. 12.
    H. Endo, On Ricci curvatures of almost cosymplectic manifolds, An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.) 40 (1994), 75–83.Google Scholar
  13. 13.
    H. Endo, On some properties of almost cosymplectic manifolds, An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.) 42 (1996), 79–94.Google Scholar
  14. 14.
    H. Endo, On some invariant submanifolds in certain almost cosymplectic manifolds, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 43 (1997), 383–395.Google Scholar
  15. 15.
    H. Endo, Non-existence of almost cosymplectic manifolds satisfying a certain condition, Tensor (N.S.) 63 (2002), 272–284.Google Scholar
  16. 16.
    Dileo G., Pastore A. M.: Almost Kenmotsu Manifolds and local symmetry, Bull. Belg. Math. Soc. Simon Stevin 14, 343–354 (2007)MATHGoogle Scholar
  17. 17.
    Dileo G., Pastore A. M.: Almost Kenmotsu Manifolds and Nullity Distributions, J. Geom. 93, 46–61 (2009)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    T. W. Kim and H. K. Pak, Canonical foliations of certain classes of almost contact metric structures, Acta Math. Sin. (Engl. Ser.) 21 (2005), 841–846.Google Scholar
  19. 19.
    T. Koufogiorgos, Contact Riemannian manifolds with constant φ-sectional curvature, Tokyo J. Math. 20 (no. 1) (1997), 55–67.Google Scholar
  20. 20.
    T. Koufogiorgos, M. Markellos and V. J. Papantoniou, The harmonicity of the Reeb vector fields on contact metric 3-manifolds, Pacific J. Math. 234 (no. 2) (2008), 325–344.Google Scholar
  21. 21.
    T. Koufogiorgos and C. Tsichlias, On the existence of a new class of contact metric manifolds, Canad. Math. Bull. 43 (no. 4) (2000), 400–447.Google Scholar
  22. 22.
    Olszak Z.: On almost cosymplectic manifolds, Kodai Math. J. 4, 239–250 (1981)MathSciNetMATHGoogle Scholar
  23. 23.
    Z. Olszak and R. Rosca, Normal locally conformal almost cosymplectic manifolds, Publ. Math. Debrecen 39 (no. 3-4) (1991), 315–323.Google Scholar
  24. 24.
    H. Öztürk, N. Aktan and C. Murathan, Almost α-cosymplectic (κ, μ, ν)-spaces, arXiv:1007.0527v1.Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of Geometry and Topology, Faculty of MathematicsUniversity of SevillaSevillaSpain
  2. 2.Centro Universitario de la DefensaAcademia General MilitarZaragozaSpain

Personalised recommendations