Research Article
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Year 2021, , 277 - 291, 29.10.2021
https://doi.org/10.36890/iejg.937419

Abstract

References

  • [1] Ahsan, Z., Siddiqui, S. A. : Concircular curvature tensor and fluid spacetimes. Int. J. Theor. Phys. 48, 3202–3212 (2009).
  • [2] Allison, D. E.: Energy conditions in standard static space-times. Gen. Rel. Grav., 20, 115–122 (1988).
  • [3] Barros, A., Batista, R., Ribeiro Jr. E.: Rigidity of gradient almost Ricci solitons. Illinois J. Math. 56(4), 1267–1279 (2012).
  • [4] Beem, J. K., Ehrlich, P. E., Easley, K. L.: Global Lorentzian Geometry. (2nd Ed.), Marcel Dekker. New York (1996).
  • [5] Bishop, R. L., O’Neill, B.:Manifolds of negative curvature. Trans. Amer. Math. Soc. 145, 1–49 (1969).
  • [6] Blair, D.E., Kim, J.-S., Tripathi, M.M.: On the concircular curvature tensor of a contact metric manifold. J. Korean Math. Soc. 42(5), 883–892 (2005).
  • [7] Catino, G.: Generalized quasi Einstein manifolds with harmonic Weyl tensor. Math. Z. 271, 751–756 (2012).
  • [8] Chaki, M.C.: On Generalized quasi-Einstein manifold. Publ. Math. Debrecen. 58, 638–691 (2001).
  • [9] Cheeger, J., Colding, T. H.: Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. of Math. 144(2), 189–237 (1996).
  • [10] Chen, B. Y. : A simple characterization of generalized Robertson-Walker space-times. Gen. Relativ. Gravit. 46, 18–33 (2014).
  • [11] De, U. C., Shenawy, S., Ünal, B.: Sequential warped products: curvature and conformal vector fields. Filomat. 33(13), 4071–4083 (2019).
  • [12] De, U. C., Shenawy, S., Ünal, B.: Concircular Curvature on warped product manifolds and applications. Bull. Malays. Math. Sci. Soc. 43, 3395– 3409 (2020).
  • [13] Dobarro, F, Ünal, B. : Special standard static spacetimes. Nonlinear Analysis: Theory, Methods and Applications. 59(5), 759–770 (2004).
  • [14] Güler, S., Altay Demirbag, S. : A Study of generalized quasi Einstein spacetimes with applications in general relativity. Int. J. Theor. Phys. 55, 548–562 (2016).
  • [15] Güler, S.: On a class of gradient almost Ricci solitons. Bull. Malays. Math. Sci. Soc. 43, 3635–3650 (2020).
  • [16] Karaca, F., Özgür, C.: On quasi-Einstein sequential warped product manifolds. Journal of Geom. Phys. 165, 104248 (2021).
  • [17] Mantica, C. A, Molinari, L. G., De, U. C.: A condition for a perfect fluid space-time to be a generalized Robertson-Walker space-time. J. Math. Phys. 57(2), 022508 (2016).
  • [18] Mantica, C. A, Suh, Y. J., De, U. C. : A note on generalized Robertson-Walker space-times. Int. J. Geom. Meth. Mod. Phys. 13, 1650079 (2016).
  • [19] O’Neill, B.: Semi Riemannian Geometry with Applications to Relativity. Pure and Applied Ser. Academic Press. New York (1983).
  • [20] Shenawy, S.: A note on sequential warped product manifolds. Preprint arxiv:1506.06056v1 (2015).
  • [21] Souso, M. L., Pina, R.: Gradient Ricci solitons with structure of warped product. Results Math. 17, 825–840 (2017).
  • [22] Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact solutions of Einstein’s field equations. Second Edition, Cambridge University Press. Cambridge (2003).
  • [23] Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc. 117, 251–275 (1965).
  • [24] Yano, K., Kon, M.: Structures on Manifolds. World Scientific Publishing. Singapore (1984).
  • [25] Yano, K.: Concircular geometry I. Concircular transformations. Proc. Imp. Acad. Tokyo. 16, 195–200 (1940).
  • [26] Yun, G., Co, J., Hwang, S. : Bach-flat h-almost gradient Ricci solitons. Pacific J. Math. 288(2), 475–488 (2017).

Sequential Warped Products and Their Applications

Year 2021, , 277 - 291, 29.10.2021
https://doi.org/10.36890/iejg.937419

Abstract

In this paper, we study the sequential warped product manifolds, which are the natural generalizations of singly warped products. Many spacetime models that characterize the universe and the solutions of Einstein's field equations are known to have this new structure. For this reason, first, we investigate the geometry of sequential warped product manifold under some conditions of concircular curvature tensor. We also study the conformal and gradient almost Ricci solitons on the sequential warped product. These conditions allow us to obtain some interesting expressions for the Riemann curvature and the Ricci tensors of its base and fiber from the geometrical and the physical point of view. Then, we give two important applications of this concept in the Lorentzian settings, which are sequential generalized Robertson-Walker spacetimes and sequential standard static spacetimes and obtain the form of the warping functions. Also, by considering generalized quasi Einsteinian conditions on these spacetimes, we find some specific formulas for the Ricci tensors of the bases and fibers. Finally, we terminate this work with some examples for this structure.

References

  • [1] Ahsan, Z., Siddiqui, S. A. : Concircular curvature tensor and fluid spacetimes. Int. J. Theor. Phys. 48, 3202–3212 (2009).
  • [2] Allison, D. E.: Energy conditions in standard static space-times. Gen. Rel. Grav., 20, 115–122 (1988).
  • [3] Barros, A., Batista, R., Ribeiro Jr. E.: Rigidity of gradient almost Ricci solitons. Illinois J. Math. 56(4), 1267–1279 (2012).
  • [4] Beem, J. K., Ehrlich, P. E., Easley, K. L.: Global Lorentzian Geometry. (2nd Ed.), Marcel Dekker. New York (1996).
  • [5] Bishop, R. L., O’Neill, B.:Manifolds of negative curvature. Trans. Amer. Math. Soc. 145, 1–49 (1969).
  • [6] Blair, D.E., Kim, J.-S., Tripathi, M.M.: On the concircular curvature tensor of a contact metric manifold. J. Korean Math. Soc. 42(5), 883–892 (2005).
  • [7] Catino, G.: Generalized quasi Einstein manifolds with harmonic Weyl tensor. Math. Z. 271, 751–756 (2012).
  • [8] Chaki, M.C.: On Generalized quasi-Einstein manifold. Publ. Math. Debrecen. 58, 638–691 (2001).
  • [9] Cheeger, J., Colding, T. H.: Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. of Math. 144(2), 189–237 (1996).
  • [10] Chen, B. Y. : A simple characterization of generalized Robertson-Walker space-times. Gen. Relativ. Gravit. 46, 18–33 (2014).
  • [11] De, U. C., Shenawy, S., Ünal, B.: Sequential warped products: curvature and conformal vector fields. Filomat. 33(13), 4071–4083 (2019).
  • [12] De, U. C., Shenawy, S., Ünal, B.: Concircular Curvature on warped product manifolds and applications. Bull. Malays. Math. Sci. Soc. 43, 3395– 3409 (2020).
  • [13] Dobarro, F, Ünal, B. : Special standard static spacetimes. Nonlinear Analysis: Theory, Methods and Applications. 59(5), 759–770 (2004).
  • [14] Güler, S., Altay Demirbag, S. : A Study of generalized quasi Einstein spacetimes with applications in general relativity. Int. J. Theor. Phys. 55, 548–562 (2016).
  • [15] Güler, S.: On a class of gradient almost Ricci solitons. Bull. Malays. Math. Sci. Soc. 43, 3635–3650 (2020).
  • [16] Karaca, F., Özgür, C.: On quasi-Einstein sequential warped product manifolds. Journal of Geom. Phys. 165, 104248 (2021).
  • [17] Mantica, C. A, Molinari, L. G., De, U. C.: A condition for a perfect fluid space-time to be a generalized Robertson-Walker space-time. J. Math. Phys. 57(2), 022508 (2016).
  • [18] Mantica, C. A, Suh, Y. J., De, U. C. : A note on generalized Robertson-Walker space-times. Int. J. Geom. Meth. Mod. Phys. 13, 1650079 (2016).
  • [19] O’Neill, B.: Semi Riemannian Geometry with Applications to Relativity. Pure and Applied Ser. Academic Press. New York (1983).
  • [20] Shenawy, S.: A note on sequential warped product manifolds. Preprint arxiv:1506.06056v1 (2015).
  • [21] Souso, M. L., Pina, R.: Gradient Ricci solitons with structure of warped product. Results Math. 17, 825–840 (2017).
  • [22] Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact solutions of Einstein’s field equations. Second Edition, Cambridge University Press. Cambridge (2003).
  • [23] Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc. 117, 251–275 (1965).
  • [24] Yano, K., Kon, M.: Structures on Manifolds. World Scientific Publishing. Singapore (1984).
  • [25] Yano, K.: Concircular geometry I. Concircular transformations. Proc. Imp. Acad. Tokyo. 16, 195–200 (1940).
  • [26] Yun, G., Co, J., Hwang, S. : Bach-flat h-almost gradient Ricci solitons. Pacific J. Math. 288(2), 475–488 (2017).
There are 26 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Sinem Güler 0000-0001-9994-2927

Publication Date October 29, 2021
Acceptance Date September 10, 2021
Published in Issue Year 2021

Cite

APA Güler, S. (2021). Sequential Warped Products and Their Applications. International Electronic Journal of Geometry, 14(2), 277-291. https://doi.org/10.36890/iejg.937419
AMA Güler S. Sequential Warped Products and Their Applications. Int. Electron. J. Geom. October 2021;14(2):277-291. doi:10.36890/iejg.937419
Chicago Güler, Sinem. “Sequential Warped Products and Their Applications”. International Electronic Journal of Geometry 14, no. 2 (October 2021): 277-91. https://doi.org/10.36890/iejg.937419.
EndNote Güler S (October 1, 2021) Sequential Warped Products and Their Applications. International Electronic Journal of Geometry 14 2 277–291.
IEEE S. Güler, “Sequential Warped Products and Their Applications”, Int. Electron. J. Geom., vol. 14, no. 2, pp. 277–291, 2021, doi: 10.36890/iejg.937419.
ISNAD Güler, Sinem. “Sequential Warped Products and Their Applications”. International Electronic Journal of Geometry 14/2 (October 2021), 277-291. https://doi.org/10.36890/iejg.937419.
JAMA Güler S. Sequential Warped Products and Their Applications. Int. Electron. J. Geom. 2021;14:277–291.
MLA Güler, Sinem. “Sequential Warped Products and Their Applications”. International Electronic Journal of Geometry, vol. 14, no. 2, 2021, pp. 277-91, doi:10.36890/iejg.937419.
Vancouver Güler S. Sequential Warped Products and Their Applications. Int. Electron. J. Geom. 2021;14(2):277-91.