Skip to main content
Log in

Ricci soliton and geometrical structure in a perfect fluid spacetime with torse-forming vector field

  • Published:
Afrika Matematika Aims and scope Submit manuscript

Abstract

In this paper geometrical aspects of perfect fluid spacetime with torse-forming vector field \(\xi \) are described and Ricci soliton in perfect fluid spacetime with torse-forming vector field \(\xi \) are determined. Conditions for the Ricci soliton to be expanding, steady or shrinking are also given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahsan, Z., Siddiqui, S.A.: Concircular curvature tensor and fluid spacetimes. Int. J. Theor. Phys. 48, 3202–3212 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blaga, A.M.: Solitons and geometrical structures in a perfect fluid spacetime. arXiv:1705.04094 [math.DG] (2017)

  3. Bejan, C.L., Crasmareanu, M.: Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry. Ann. Glob. Anal. Geom. 46, 117–127 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Calvaruso, G., Zaeim, A.: A complete classification of Ricci and Yamabe solitons of non-reductive homogeneous 4-spaces. J. Geom. Phys. 80, 15–25 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Calvaruso, G., Perrone, A.: Ricci solitons in three-dimensional paracontact geometry. arXiv:1407.3458v1 (2014)

  6. Chaki, M.C., Ray, S.: Spacetimes with covariant constant energy momentum tensor. Int. J. Theor. Phys. 35(5), 1027–1032 (1996)

    Article  MATH  Google Scholar 

  7. Chaki, M.C., Maity, R.K.: On quasi Einstein manifolds. Publ. Math. Debr. 57, 297–306 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Chen, B., Yano, K.: Hypersurfaces of a conformally flat space. Tensor NS 26, 315–321 (1972)

    MathSciNet  MATH  Google Scholar 

  9. De, U.C., Velimirović, L.: Spacetimes with semisymmetric energy momentum tensor. Int. J. Theor. Phys. 54, 1779–1783 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. De, U.C., Ghosh, G.C.: On quasi-Einstein and special quasi-Einstein manifolds. In: Proceedings of the International Conference of Mathematics and its Applications, Kuwait University, April 5–7, 178–191 (2004)

  11. De, U.C., Ghosh, G.C.: On quasi-Einstein manifolds. Period. Math. Hung. 48(12), 223–231 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Deszcz, R., Hotlos, M., Senturk, Z.: On curvature properties of quasi-Einstein hypersurfaces in semi-Euclidean spaces. Soochow J. Math. 27, 375–389 (2001)

    MathSciNet  MATH  Google Scholar 

  13. Duggal, K.L., Sharma, R.: Symmetries of Spacetime and Riemannian Manifold, vol. 487. Springer, Berlin (1999)

    Book  Google Scholar 

  14. Güler, S., Demirbağ, S.A.: A study of generalized quasi Einstein spacetime with application in general relativity. Int. J. Theor. Phys. 55, 548–562 (2016)

    Article  MATH  Google Scholar 

  15. Hamilton, R.S.: The Ricci flow on surfaces. Contemp. Math. 71, 237–261 (1988)

    Article  MathSciNet  Google Scholar 

  16. Maartens, R., Mason, D.P., Tsamparlis, M.: Kinematic and dynamic properties of conformal Killing vectors in anisotropic fluids. J. Math. Phys. 27, 2987 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mallick, S., De, U.C.: Spacetimes with pseudosymmetric energy momentum tensor. Commun. Phys. 26(2), 121–128 (2016)

    Article  Google Scholar 

  18. Manjonjo, A.M., Maharaj, S.D., Moopanar, S.: Static models with conformal symmetry. Class. Quantum Gravity 35, 045015 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Moopanar, S., Maharaj, S.D.: Conformal Symmetries of Spherical Spacetimes. Int. J. Theor. Phys. 49, 1878–1885 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mason, D.P., Maartens, R.: Kinematics and dynamics of conformal collineations in relativity. J. Math. Phys. 28, 2182 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  21. O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983)

    MATH  Google Scholar 

  22. Ray-Guha, S.: On perfect fluid pseudo Ricci symmetric spacetime. Tensor NS 67, 101–107 (2006)

    MathSciNet  MATH  Google Scholar 

  23. Sharma, R., Ghosh, A.: Sasakian 3-manifold as a Ricci soliton represents the Heisenberg group. Int. J. Geom. Methods Mod. Phys. 8(1), 149–154 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Stephani, H.: General Relativity: an Introduction to the Theory of Gravitational Field. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  25. Tupper, B.O.J., Keane, A.J., Carot, J.: A classification of spherically symmetric spacetimes. Class. Quantum Gravity 29, 145016 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Venkatesha, Devaraja, M.N.: Certain results on \(K\)-paracontact and para Sasakian manifolds. J. Geom. 108, 939–952 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970)

    MATH  Google Scholar 

  28. Yano, K.: On torse forming direction in a Riemannian space. Proc. Imp. Acad. Tokyo 20, 340–345 (1994)

    Article  MathSciNet  Google Scholar 

  29. Yano, K., Kon, M.: Structure on Manifold, vol. 3. Series in pure mathematics. World Scientific publishing Co. Pte. Ltd., Singapore (1984)

    MATH  Google Scholar 

Download references

Acknowledgements

The author’s is very much grateful to the reviewer’s for some valuable comments and many editorial corrections.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Venkatesha.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Venkatesha, Kumara, H.A. Ricci soliton and geometrical structure in a perfect fluid spacetime with torse-forming vector field. Afr. Mat. 30, 725–736 (2019). https://doi.org/10.1007/s13370-019-00679-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13370-019-00679-y

Keywords

Mathematics Subject Classification

Navigation