Geometry of Twisted Products and Applications on Static Perfect Fluid Spacetimes

Year 2023, Volume: 16 Issue: 2, 598 - 607, 29.10.2023

Sinem Güler , U.c. De , Bülent Ünal

https://doi.org/10.36890/iejg.1286525

Abstract

In this paper, first we study the harmonicity of the functions and forms on the twisted products, and then we determine its sectional curvature. We explore some characteristics of static perfect fluid and static vacuum spacetimes on twisted product manifolds by proving the existence and obstructions on Ricci curvature. Finally, we study the problem of the existence static perfect fluid spacetime associated with the twisted generalized Robertson-Walker and standard static spacetime metrics.

Keywords

Twisted product, gradient Ricci soliton, static perfect fluid, vacuum static space, generalized Robertson-Walker spacetime, standard static spacetime metric

References

• [1] Allison, D. E., Ünal, B.: Geodesic structure of standard static spacetimes. J. Geom. Phys. 46, 193–200 (2003).
• [2] Beem, J. K., Ehrlich, P. E., Easley, K. L.: Global Lorentzian Geometry. Marcel Dekker. Second Edition. NewYork (1996).
• [3] Bishop, R. L., O’Neill, B.: Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969) 1–49.
• [4] Blaga, A. M.: η-Ricci solitons on Lorentzian para-Sasakian manifolds. Filomat. 30 (2) 489–496 (2016).
• [5] Blaga, A. M., Özgür, C.: Almost η-Ricci and almost η-Yamabe solitons with torse-forming potential vector field. Quaestiones Mathematicae. 1–21 (2020).
• [6] Cao, H.-D., Chen, Q.: On locally conformally flat gradient steady Ricci solitons. Trans. Am. Math. Soc. 2377-2391 (2012).
• [7] Chen, B.-Y.: A simple characterization of generalized Robertson-Walker spacetimes. Gen. Relativ. Gravit. 46 18–33 (2014).
• [8] Chen, B.-Y.: Rectifying submanifolds of Riemannian manifolds and torqued vector fields, Kragujevac Journal of Mathematics 41(1), 93—103 (2017).
• [9] De, U. C., Shenawy, S., Ünal, B.: Concircular curvature on warped product manifolds and applications, Bull. Malays. Math. Sci. Soc. 43 3395–3409 (2020).
• [10] De, U. C., Chaubey, S. K., Shenawy, S.: Perfect fluid spacetimes and Yamabe solitons. J. Math. Phys. 62 032501 (2021).
• [11] Deshmukh, S., Turki, N. B., Vilcu, G.-E.: A note on static spaces. Results in Physics 27 104519 (2021).
• [12] Dobarro F., Ünal, B.: Special standard static spacetimes. Nonlinear Anal. Theory Methods Appl. 59(5) 759–770 (2004).
• [13] Dobarro, F., Ünal, B.: Implications of energy conditions on standard static spacetimes. Nonlinear Anal. 71(11) 5476–90 (2009).
• [14] Fernández-López, M., García-Río, E., Kupeli, D. N., Ünal, B.: A curvature condition for a twisted product to be a warped product. Manuscripta Math. 106 213–217 (2001).
• [15] Güler, S., Tastan, H. M. : Gradient solitons on twisted product manifolds and their applications in general relativity. Int. J. Geom. Meth. Mod. Phys. doi.org/10.1142/S0219887822501547 (2022).
• [16] Mantica, C. A, Suh, Y. J., De, U. C.: A note on generalized Robertson-Walker spacetimes. Int. J. Geom. Meth. Mod. Phys. 13 1650079 (2016).
• [17] Mantica, C. A., Molinari, L. G., De, U. C.: A condition for a perfect fluid spacetime to be a generalized Robertson-Walker spacetime, J. Math. Phys. 57 (2) 022508 (2016).
• [18] Mantica, C. A., Molinari, L. G.: Twisted Lorentzian manifolds: a characterization with torse-forming time-like unit vectors. Gen Relativ Gravit. 49 (51) (2017).
• [19] O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity, Academic Press Limited, London, (1983).
• [20] Ponge, R., Reckziegel, H.: Twisted Products in Pseudo-Riemannian Geometry. Geom. Dedicata. 48 15–25 (1993).
• [21] Qing, J., Yuan, W. : A note on static spaces and related problems. J. Geom. Phys. 74 18–27 (2013).
• [22] Sanchez, M.: On the geometry of generalized Robertson-Walker spacetimes: curvature and Killing fields. J. Geom. Phys. 31 1—15 (1999).