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Journal of Geometry

, 110:43 | Cite as

Lie groups as 3-dimensional almost paracontact almost paracomplex Riemannian manifolds

  • Mancho ManevEmail author
  • Veselina Tavkova
Article

Abstract

Almost paracontact almost paracomplex Riemannian manifolds of the lowest dimension 3 are considered. Such structures are constructed on a family of Lie groups and the obtained manifolds are studied. Curvature properties of these manifolds are investigated. An example is commented as support of the obtained results.

Keywords

Almost paracontact structure almost paracomplex structure Riemannian metric Lie group Lie algebra curvature properties 

Mathematics Subject Classification

53C15 53C25 

Notes

Acknowledgements

The authors were supported by Projects MU19-FMI-020 and FP19-FMI-002 of the Scientific Research Fund, University of Plovdiv Paisii Hilendarski, Bulgaria.

References

  1. 1.
    Abbena, E., Garbiero, S.: Almost Hermitian homogeneous manifolds and Lie groups. Nihonkai Math. J. 4, 1–15 (1993)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Barberis, M.L.: Hypercomplex structures on four-dimensional Lie groups. Proc. Am. Math. Soc. 128(4), 1043–1054 (1997)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics, vol. 203. Birkhäuser, Boston (2002)CrossRefGoogle Scholar
  4. 4.
    Boeckx, E., Bueken, P., Vanhecke, L.: \(\varphi \)-symmetric contact metric spaces. Glasg. Math. J. 41, 409–416 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dotti, I.G., Fino, A.: HyperKähler torsion structures invariant by nilpotent Lie groups. Class. Quantum Gravity 19(3), 551–562 (2002)CrossRefGoogle Scholar
  6. 6.
    Fernández-Culma, E.A, Godoy, Y.: Anti-Kählerian geometry on Lie groups. Math. Phys. Anal. Geom, 21 (1), Art.no. 8 (2018)Google Scholar
  7. 7.
    Gribachev, K., Manev, M.: Almost hypercomplex pseudo-Hermitian manifolds and a 4-dimensional Lie group with such structure. J. Geom. 88(1–2), 41–52 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gribachev, K., Manev, M., Mekerov, D.: A Lie group as a 4-dimensional quasi-Kähler manifold with Norden metric. JP J. Geom. Topol. 6(1), 55–68 (2006)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Gribacheva, D., Mekerov, D.: Canonical connection on a class of Riemannian almost product manifolds. J. Geom. 102(1–2), 53–71 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ivanova, M., Manev, H.: Five-dimensional Lie groups which are almost contact B-metric manifolds with three natural connections. In: Adachi, T., Hashimoto, H., Hristov, M.J. (eds.) Current Developments in Differential Geometry and its Related Fields, pp. 115–128. World Scientific Publishing, Singapore (2016)Google Scholar
  11. 11.
    Manev, H.: On the structure tensors of almost contact B-metric manifolds. Filomat 29(3), 427–436 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Manev, M.: Hypercomplex structures with Hermitian–Norden metrics on four-dimensional Lie algebras. J. Geom. 105(1), 21–31 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Manev, H., Mekerov, D.: Lie groups as 3-dimensional almost contact B-metric manifolds. J. Geom. 106, 229–242 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Manev, M., Staikova, M.: On almost paracontact Riemannian manifolds of type \((n, n)\). J. Geom. 72(1–2), 108–114 (2001)Google Scholar
  15. 15.
    Manev, M., Tavkova, V.: On the almost paracontact almost paracomplex Riemannian manifolds. Facta Univ. Ser. Math. Inform. 33, 637–657 (2018)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Martín-Molina, V.: Paracontact metric manifolds without a contact metric counterpart. Taiwan. J. Math. 19(1), 175–191 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Naveira, A.M.: A classification of Riemannian almost product manifolds. Rend. Math. 3, 577–592 (1983)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Okumura, M.: Some remarks on space with a certain contact structure. Tohoku Math. J. 14(2), 135–145 (1962)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Olszak, Z.: On almost cosymplectic manifolds. Kodai Math. J. 4(2), 239–250 (1981)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Parchetalab, M.: On a class of paracontact Riemannian manifold. Int. J. Nonlinear Anal. Appl. 7(1), 195–205 (2016)zbMATHGoogle Scholar
  21. 21.
    Satō, I.: On a structure similar to the almost contact structure. Tensor (N.S.) 30, 219–224 (1976)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Shtarbeva, D.K.: Lie groups as four-dimensional Riemannian product manifolds. In: Dimiev, S., Sekigawa, K. (eds.) Topics in Contemporary Differential Geometry, Complex Analysis and Mathematical Physics, pp. 290–298. World Scientific Publishing, Hackensack (2007)CrossRefGoogle Scholar
  23. 23.
    Sinha, B., Sharma, R.: On para-A-Einstein manifolds. Publ. Inst. Math. (Beograd) (N.S.) 34(48), 211–215 (1983)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Teofilova, M.: Lie groups as four-dimensional conformal Kähler manifolds with Norden metric. In: Dimiev, S., Sekigawa, K. (eds.) Topics in Contemporary Differential Geometry, Complex Analysis and Mathematical Physics, pp. 319–326. World Scientific Publishing, Hackensack (2007)CrossRefGoogle Scholar
  25. 25.
    Zamkovoy, S., Nakova, G.: The decomposition of almost paracontact metric manifolds in eleven classes revisited. J. Geom. 109, 18 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Algebra and Geometry, Faculty of Mathematics and InformaticsUniversity of Plovdiv Paisii HilendarskiPlovdivBulgaria
  2. 2.Department of Medical Informatics, Biostatistics and E-Learning, Faculty of Public HealthMedical University of PlovdivPlovdivBulgaria

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