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ON LORENTZIAN QUASI-EINSTEIN MANIFOLDS

  • Received : 2008.11.11
  • Published : 2011.07.01

Abstract

The notion of quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. The object of the present paper is to study Lorentzian quasi-Einstein manifolds. Some basic geometric properties of such a manifold are obtained. The applications of Lorentzian quasi-Einstein manifolds to the general relativity and cosmology are investigated. Theories of gravitational collapse and models of Supernova explosions [5] are based on a relativistic fluid model for the star. In the theories of galaxy formation, relativistic fluid models have been used in order to describe the evolution of perturbations of the baryon and radiation components of the cosmic medium [32]. Theories of the structure and stability of neutron stars assume that the medium can be treated as a relativistic perfectly conducting magneto fluid. Theories of relativistic stars (which would be models for supermassive stars) are also based on relativistic fluid models. The problem of accretion onto a neutron star or a black hole is usually set in the framework of relativistic fluid models. Among others it is shown that a quasi-Einstein spacetime represents perfect fluid spacetime model in cosmology and consequently such a spacetime determines the final phase in the evolution of the universe. Finally the existence of such manifolds is ensured by several examples constructed from various well known geometric structures.

Keywords

References

  1. P. Amendt and H. Weitzner, Relativistically covariant warm charged uid beam modeling, The Physics of Fluids 28 (1985), 949-957. https://doi.org/10.1063/1.865066
  2. D. E. Blair, T. Koufogiorgos, and R. Sharma, A classifcation of 3-dimensional contact metric manifolds with Q$\phi$ = $\phi$Q, Kodai Math. J. 13 (1990), no. 3, 391-401. https://doi.org/10.2996/kmj/1138039284
  3. J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry, Marcel Dekker, 1981.
  4. B. Y. Chen and K. Yano, Hypersurfaces of a conformally at space, Tensor (N.S.) 26 (1972), 318-322.
  5. R. A. Chevalier, Hydrodynamic models of supernova explosions, Fundamentals of Cosmic Physics 7 (1981), 1-58.
  6. U. C. De, K. Matsumoto, and A. A. Shaikh, On Lorentzian para-Sasakian manifolds, Rendiconti del Seminario Mat. de Messina, al n. 3 (1999), 149-158.
  7. F. Defever, R. Deszcz, M. Hotlos, M. Kucharski, and Z. Senturk, Generalisations of Robertson-Walker spaces, Annales Univ. Sci. Budapest. Eotvos Sect. Math. 43 (2000), 13-24.
  8. J. Deprez, W. Roter, and L. Verstraelen, Conditions on the projective curvature tensor of conformally at Riemannian manifolds, Kyungpook Math. J. 29 (1989), no. 2, 153- 165.
  9. R. Deszcz, On pseudosymmetric spaces, Bull. Soc. Math. Belg. Ser. A 44 (1992), no. 1, 1-34.
  10. R. Deszcz, F. Dillen, L. Verstraelen, and L. Vrancken, Quasi-Einstein totally real sub- manifolds of the nearly Kahler 66-sphere, Tohoku Math. J. (2) 51 (1999), no. 4, 461-478. https://doi.org/10.2748/tmj/1178224715
  11. R. Deszcz, M. Glogowska, M. Hotlos, and Z. Senturk, On certain quasi-Einstein semi- symmetric hypersurfaces, Annales Univ. Sci. Budapest. Eotvos Sect. Math. 41 (1998), 151-164.
  12. R. Deszcz, M. Hotlos, and Z. Senturk, Quasi-Einstein hypersurfaces in semi-Riemann- ian space forms, Colloq. Math. 81 (2001), no. 1, 81-97.
  13. R. Deszcz, M. Hotlos, and Z. Senturk, Quasi-Einstein hypersurfaces in semi-Riemann- ian space forms, Colloq. Math. 81 (2001), no. 1, 81-97.
  14. R. Deszcz, M. Hotlos, and Z. Senturk, On curvature properties of quasi-Einstein hypersurfaces in semi-Euclidean spaces, Soochow J. Math. 27 (2001), no. 4, 375-389.
  15. R. Deszcz, P. Verheyen, and L. Verstraelen, On some generalized Einstein metric con- ditions,Publ. Inst. Math. (Beograd) (N.S.) 60(74) (1996), 108-120.
  16. R. Deszcz, L. Verstraelen, and S. Yaprak, Warped products realizing a certain condition of pseudosymmetry type imposed on the Weyl curvature tensor, Chinese J. Math. 22 (1994), no. 2, 139-157.
  17. R. Deszcz, L. Verstraelen, and S. Yaprak, Pseudosymmetric hypersurfaces in 4-dimensional spaces of constant curvature, Bull. Inst. Math. Acad. Sinica 22 (1994), no. 2, 167-179.
  18. R. Deszcz and M. Hotlos, On some pseudosymmetry type curvature condition, Tsukuba J. Math. 27 (2003), no. 1, 13-30. https://doi.org/10.21099/tkbjm/1496164557
  19. R. Deszcz and M. Hotlos, On hypersurfaces with type number two in space forms, Ann. Univ. Sci. Bu- dapest. Eotvos Sect. Math. 46 (2003), 19-34.
  20. R. Deszcz and M. Glogowska, Examples of nonsemisymmetric Ricci-semisymmetric hypersurfaces, Colloq. Math. 94 (2002), no. 1, 87-101. https://doi.org/10.4064/cm94-1-7
  21. D. Ferus, A Remark on Codazzi Tensors on Constant Curvature Space, Lecture Notes Math. 838, Global Differential Geometry and Global Analysis, Springer-Verlag, New York, 1981.
  22. M. Glogowska, Semi-Riemannian manifolds whose weyl tensor is a Kulkarni-Nomizu square, Publ. Inst. Math. (Beograd), Tome 72 (2002), no. 86, 95-106. https://doi.org/10.2298/PIM0272095G
  23. M. Glogowska, On quasi-Einstein Cartan type hypersurfaces, J. Geom. Phys. 58 (2008), no. 5, 599-614. https://doi.org/10.1016/j.geomphys.2007.12.012
  24. F. Gouli-Andreou and E. Moutaf, Two classes of pseudosymmetric contact metric 3- manifolds, Pacifc J. Math. 239 (2009), no. 1, 17-37. https://doi.org/10.2140/pjm.2009.239.17
  25. T. Ikawa and M. Erdogan, Sasakian manifolds with Lorentzian metric, Kyungpook Math. J. 35 (1996), no. 3, 517-526.
  26. H. Karchar, Infnitesimal characterization of Friedmann Universe, Arch. Math. Basel 38 (1992), 58-64.
  27. K. Matsumoto, On Lorentzian paracontact manifolds, Bull. Yamagata Univ. Natur. Sci. 12 (1989), no. 2, 151-156.
  28. K. Matsumoto and I. Mihai, On a certain transformation in a Lorentzian para-Sasakian manifold, Tensor N. S. 47 (1988), no. 2, 189-197.
  29. I. Mihai and R. Rosca, On Lorentzian P-Sasakian manifolds, Classical analysis (Kaz- imierz Dolny, 1991), 155-169, World Sci. Publ., River Edge, NJ, 1992.
  30. I. Mihai, A. A. Shaikh, and U. C. De, On Lorentzian para-Sasakian manifolds, Korean J. Math. Sciences 6 (1999), 1-13. https://doi.org/10.1186/2251-7456-6-1
  31. M. Novello and M. J. Reboucas, The stability of a rotating universe, The Astrophysics J. 225 (1978), 719-724. https://doi.org/10.1086/156533
  32. B. O'Neill, Semi Riemannian Geometry with Applications to Relativity, Academic Press, Inc., 1983.
  33. P. J. E. Peebles, The Large-Scale Structure of the Universe, Princeton Univ. Press, 1980.
  34. J. A. Schouten, Ricci Calculus, (2nd Ed.), Springer-Verlag, Berlin, 1954.
  35. A. A. Shaikh, On Lorentzian almost paracontact manifolds with a structure of the con- circular type, Kyungpook Math. J. 43 (2003), no. 2, 305-314.
  36. A. A. Shaikh, T. Basu, and K. K. Baishya, On the existence of locally $\phi$-recurrent LP-Sasakian manifold, Bull. Allahabad Math. Soc. 24 (2009), no. 2, 281-295.
  37. A. A. Shaikh and K. K. Baishya, On $\phi$-symmetric LP-Sasakian manifolds, Yokohama Math. J. 52 (2006), no. 2, 97-112.
  38. A. A. Shaikh and K. K. Baishya, Some results on LP-Sasakian manifolds, Bull. Math. Soc. Sc. Math. Roumanie Tome, 49(97) (2006), no. 2, 193-205.
  39. A. A. Shaikh and K. K. Baishya, On concircular structure spacetimes, J. Math. Stat. 1 (2005), no. 2, 129-132. https://doi.org/10.3844/jmssp.2005.129.132
  40. A. A. Shaikh and K. K. Baishya, On concircular structure spacetimes II, Amer. J. Appl. Sci. 3 (2006), 1790-1794. https://doi.org/10.3844/ajassp.2006.1790.1794
  41. M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish and Perish, Vol. I, 1970.
  42. T. Takahashi, Sasakian $\phi$-symmetric spaces, Tohoku Math. J. 29 (1977), no. 1, 91-113. https://doi.org/10.2748/tmj/1178240699

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