Skip to main content
SpringerLink
Log in

On weakly cyclic Z symmetric spacetimes

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

The object of the present paper is to study weakly cyclic Z symmetric spacetimes. At first we prove that a weakly cyclic Z symmetric spacetime is a quasi Einstein spacetime. Then we study \({{(WCZS)}_{4}}\) spacetimes satisfying the condition div \({C=0}\). Next we consider conformally flat \({{(WCZS)}_{4}}\) spacetimes. Finally, we characterise dust fluid and viscous fluid \({{(WCZS)}_{4}}\) spacetimes.

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

Access this article

Log in via an institution

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. L. Alías, A. Romero and M. Sánchez, Compact spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes, in: Geometry and Topology of Submanifolds VII. River Edge NJ, USA: World Scientific, Dillen F. editor (1995), pp. 67–70.

  2. Chen B-Y.: A simple characterization of generalized Robertson–Walker space-times. Gen. Rel. Grav. 46, 1833 (2014)

    Article  MATH  Google Scholar 

  3. J. K. Beem, P. E. Ehrlich and K. L. Easley, Global Lorentzian Geometry, 2nd Ed., Pure and Applied Mathematics, 202, Marcel Dekker (New York, 1996).

  4. Chaki M. C.: On pseudo Ricci symmetric manifolds. Bulg. J. Phys. 15, 525–531 (1988)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  6. Chaki M. C., Roy S.: Space-times with covariant-constant energy-momentum tensor. Int. J. Theo. Phys. 35, 1027–1032 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. M. C. Chaki and S. K. Saha, On pseudo-projective Ricci symmetric manifolds, Bulg. J. Phys., 21 (1994), 1–7.

  8. F. Brickell and R. S. Clark, Differentiable Manifolds, Van Nostrand Reinhold Comp. (London, 1978).

  9. De U. C., Ghosh G. C.: On weakly Ricci symmetric spacetime manifolds. Radovi Matematicki 13, 93–101 (2004)

    MathSciNet  MATH  Google Scholar 

  10. A. De, C. Özgür and U. C. De, On conformally flat almost pseudo-Ricci symmetric spacetimes, Int. J. Theor. Phys., 51 (2012), 2578–2887.

  11. De U. C., Mantica C. A., Suh Y. J.: On weakly cyclic Z symmetric manifolds. Acta. Math. Hungar. 146, 153–167 (2015)

    Article  MathSciNet  Google Scholar 

  12. L. P. Eisenhart, Riemannian Geometry, Princeton University Press (Princeton, N. J., 1949).

  13. Karchar H.: Infinitesimal characterization of Friedmann Universe. Arch. Math. Basel 38, 58–64 (1992)

    Article  Google Scholar 

  14. C. A. Mantica and L. G. Molinari, Weakly Z Symmetric manifolds, Acta Math. Hungar., 135(1–2) (2012), 80–96.

  15. C. A. Mantica, L. G. Molinari and U. C. De, A condition for a perfect-fluid space-time to be a generalized Robertson–Walker space-time, arXiv:1508.05883v1 [Math.DG], 24 (Aug. 2015).

  16. C. A. Mantica and Y. J. Suh, Pseudo Z symmetric Riemannian manifolds with harmonic curvature tensors, J. Math. Phys., 57(2), 022508 (2016), 6 pages.

  17. C. A. Mantica and Y. J. Suh, Recurrent Z forms on Riemannian and Kaehler manifolds, Int. J. Geom. Meth. Mod. Phys., 9 (2012), 1250059 (26 pages).

  18. C. A. Mantica and Y. J. Suh, Pseudo-Z symmetric space-times, J. Math Phys., 55, 042502 (2014), 12 pages.

  19. C. A. Mantica and Y. J. Suh, Pseudo- Z symmetric space-times with divergence-free Weyl tensor and pp-waves, Int. J. Geom. Meth. Mod Phys., 13(2) (2016), 1650015 (34 pages).

  20. J. Mike\({\check{s}}\) and L. Rachunek, Torse-forming vector fields in T-symmetric Riemannian spaces, in: Steps in Differential Geometry, Proc. of the Colloq. on Diff. Geometry (Debrecen, 2000), pp. 219–229.

  21. Novello M., Reboucas M. J.: The stability of a rotating universe. The Astrophysical Journal 225, 719–724 (1978)

    Article  Google Scholar 

  22. B. O’Neill, Semi-Riemannian Geometry with Applications to the Relativity, Academic Press (New York–London, 1983).

  23. Gȩbarowski A.: Nearly conformally symmetric warped product manifolds. Bulletin of the Institute of Mathematics Academia Sinica 20, 359–371 (1992)

    MathSciNet  MATH  Google Scholar 

  24. G. S. Hall, Symmetries and Curvature Structure in General Relativity, World Scientific (Singapore, 2004).

  25. Sánchez M.: On the geometry of generalized Robertson–Walker spacetimes: geodesics. Gen. Rel. Grav. 30, 915–932 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Sánchez, On the geometry of generalized Robertson–Walker spacetimes: curvature and Killing fields, Gen. Relativ. Grav., 31 (1999), 1–15.

  27. J. A. Schouten, Ricci-Calculus, An Introduction to Tensor Analysis and its Geometrical Applications, Springer-Verlag (Berlin–Göttingen–Heidelberg, 1954).

  28. Sharma R.: Proper conformal symmetries of spacetimes with divergence free Weyl tensor. J. Math. Phys. 34, 3582–3587 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. L. C. Shepley and A. H. Taub, Spacetimes containing perfect fluids and having a vanishing conformal divergence, Commun. Math. Phys., 5 (1967), 237–256.

  30. S. K. Srivastava, General Relativity and Cosmology, Prentice-Hall of India Private Limited (New Delhi, 2008).

  31. H. Sthepani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Hertl, Exact Solutions of Einstein’s Field Equations, Cambridge Monographs on Mathematical Physics, Cambridge University Press 2nd ed. (2003).

  32. Tamássy L., Binh T. Q.: On weakly symmetries of Einstein and Sasakian manifolds. Tensor, (N. S.) 53, 140–148 (1993)

    MathSciNet  MATH  Google Scholar 

  33. Yano K.: Concircular geometry I. Proc. Imp. Acad. Tokyo 16, 195–200 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yano K.: On the torseforming direction in Riemannian spaces. Proc. Imp. Acad. Tokyo 20, 340–345 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  35. F. O. Zengin, \({m}\)-Projectively flat spacetimes, Math. Reports, 14(64) (2012), 363–370.

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, University of Calcutta, 35, Ballygaunge Circular Road, Kolkata, 700019, West Bengal, India

    U. C. De

  2. Physics Department, Universitá degli Studi di Milano, Via Celoria 16, 20133, Milano, Italy

    C. A. Mantica

  3. I. I. S. Lagrange, Via L., Modignani 65, 20161, Milano, Italy

    C. A. Mantica

  4. Physics Department, Universitá degli Studi di Milano and I. N. F. N. Sez. Milano, Via Celoria 16, 20133, Milano, Italy

    L. G. Molinari

  5. Department of Mathematics, Kyungpook National University, Taegu, 702-701, Korea

    Y. J. Suh

Authors
  1. U. C. De
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. A. Mantica
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. G. Molinari
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Y. J. Suh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. A. Mantica.

Additional information

This work was supported by Grant Proj. No. NRF-2015-R1A2A1A-01002459 from National Research Foundation, and the fourth author by 2015 KNU (Boekhyeon).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

De, U.C., Mantica, C.A., Molinari, L.G. et al. On weakly cyclic Z symmetric spacetimes. Acta Math. Hungar. 149, 462–477 (2016). https://doi.org/10.1007/s10474-016-0612-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-016-0612-3

Key words and phrases

Mathematics Subject Classification

Search

Navigation