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).
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.
Keywords
Sequential warped product concircular curvature tensor generalized Robertson-Walker spacetime standard static spacetime conformal soliton generalized quasi Einstein manifold
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).
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Article |
| Authors | |
| Publication Date | October 29, 2021 |
| Acceptance Date | September 10, 2021 |
| Published in Issue | Year 2021 Volume: 14 Issue: 2 |
