Journal of Geometry

, 109:18 | Cite as

The decomposition of almost paracontact metric manifolds in eleven classes revisited

  • Simeon Zamkovoy
  • Galia Nakova


This paper is a continuation of our previous work, where eleven basic classes of almost paracontact metric manifolds with respect to the covariant derivative of the structure tensor field were obtained. First we decompose one of the eleven classes into two classes and the basic classes of the considered manifolds become twelve. Also, we determine the classes of \(\alpha \)-para-Sasakian, \(\alpha \)-para-Kenmotsu, normal, paracontact metric, para-Sasakian, K-paracontact and quasi-para-Sasakian manifolds. Moreover, we study 3-dimensional almost paracontact metric manifolds and show that they belong to four basic classes from the considered classification. We define an almost paracontact metric structure on any 3-dimensional Lie group and give concrete examples of Lie groups belonging to each of the four basic classes, characterized by commutators on the corresponding Lie algebras.


Almost paracontact metric manifolds 3-Dimensional almost paracontact manifolds \(\alpha \)-Para-Sasakian manifolds \(\alpha \)-Para-Kenmotsu manifolds 


  1. 1.
    Alexiev, V., Ganchev, G.: On the classification of almost contact metric manifolds. In: Mathematics and Education in Mathematics, Proceedings of 15th Spring Conference, Sunny Beach, pp. 155–161 (1986)Google Scholar
  2. 2.
    Blair, D.E.: Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, vol. 509. Springer, Berlin (1976)Google Scholar
  3. 3.
    Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Birkhäuser, Boston (2002)Google Scholar
  4. 4.
    Ganchev, G., Borisov, A.: Note on the almost complex manifolds with a Norden metric. C. R. Acad. Bulg. Sci. 39, 31–34 (1986)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ganchev, G., Mihova, V., Gribachev, K.: Almost contact manifolds with B-metric. Math. Balk. 7(3–4), 261–276 (1993)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gray, A., Hervella, L.: The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann. Mat. Pura Appl. 123, 35–58 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kaneyuki, S., Willams, F.L.: Almost paracontact and parahodge structures on manifolds. Nagoya Math. J. 99, 173–187 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Manev, H.: On the structure tensors of almost contact B-metric manifolds. Filomat 29(3), 427–436 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Manev, H., Mekerov, D.: Lie groups as 3-dimensional almost contact B-metric manifolds. J. Geom. 106, 229–242 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Nakova, G., Zamkovoy, S.: Eleven classes of almost paracontact manifolds with semi-Riemannian metric of (n + 1; n). In: Adachi, T., Hashimoto, H., Hristov, M. (eds.) Recent Progress in Diffrential Geometry and its Related Fields, pp. 119–136. World Scientific Publ, Singapore (2012)Google Scholar
  11. 11.
    Naveira, A.M.: A classification of Riemannian almost product structures. Rend. Mat. Roma 3, 577–592 (1983)zbMATHGoogle Scholar
  12. 12.
    Welyczko, J.: On Legendre curves in 3-dimensional normal almost paracontact metric manifolds. Result. Math. 54, 377–387 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zamkovoy, S.: Canonical connections on paracontact manifolds. Ann. Glob. Anal. Geom. 36, 37–60 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsUniversity of Sofia “St. Kl. Ohridski”SofiaBulgaria
  2. 2.Department of Algebra and Geometry, Faculty of Mathematics and InformaticsUniversity of Veliko Tarnovo “St. Cyril and St. Methodius”Veliko TarnovoBulgaria

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