Journal of Geometry

, 109:18 | Cite as

The decomposition of almost paracontact metric manifolds in eleven classes revisited

Article
  • 9 Downloads

Abstract

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.

Keywords

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

References

  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)MathSciNetMATHGoogle Scholar
  5. 5.
    Ganchev, G., Mihova, V., Gribachev, K.: Almost contact manifolds with B-metric. Math. Balk. 7(3–4), 261–276 (1993)MathSciNetMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kaneyuki, S., Willams, F.L.: Almost paracontact and parahodge structures on manifolds. Nagoya Math. J. 99, 173–187 (1985)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Manev, H.: On the structure tensors of almost contact B-metric manifolds. Filomat 29(3), 427–436 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Manev, H., Mekerov, D.: Lie groups as 3-dimensional almost contact B-metric manifolds. J. Geom. 106, 229–242 (2015)MathSciNetCrossRefMATHGoogle 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)MATHGoogle Scholar
  12. 12.
    Welyczko, J.: On Legendre curves in 3-dimensional normal almost paracontact metric manifolds. Result. Math. 54, 377–387 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Zamkovoy, S.: Canonical connections on paracontact manifolds. Ann. Glob. Anal. Geom. 36, 37–60 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© 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

Personalised recommendations