Abstract
In this paper geometrical aspects of perfect fluid spacetime with torse-forming vector field \(\xi \) are described and Ricci soliton in perfect fluid spacetime with torse-forming vector field \(\xi \) are determined. Conditions for the Ricci soliton to be expanding, steady or shrinking are also given.
Similar content being viewed by others
References
Ahsan, Z., Siddiqui, S.A.: Concircular curvature tensor and fluid spacetimes. Int. J. Theor. Phys. 48, 3202–3212 (2009)
Blaga, A.M.: Solitons and geometrical structures in a perfect fluid spacetime. arXiv:1705.04094 [math.DG] (2017)
Bejan, C.L., Crasmareanu, M.: Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry. Ann. Glob. Anal. Geom. 46, 117–127 (2014)
Calvaruso, G., Zaeim, A.: A complete classification of Ricci and Yamabe solitons of non-reductive homogeneous 4-spaces. J. Geom. Phys. 80, 15–25 (2014)
Calvaruso, G., Perrone, A.: Ricci solitons in three-dimensional paracontact geometry. arXiv:1407.3458v1 (2014)
Chaki, M.C., Ray, S.: Spacetimes with covariant constant energy momentum tensor. Int. J. Theor. Phys. 35(5), 1027–1032 (1996)
Chaki, M.C., Maity, R.K.: On quasi Einstein manifolds. Publ. Math. Debr. 57, 297–306 (2000)
Chen, B., Yano, K.: Hypersurfaces of a conformally flat space. Tensor NS 26, 315–321 (1972)
De, U.C., Velimirović, L.: Spacetimes with semisymmetric energy momentum tensor. Int. J. Theor. Phys. 54, 1779–1783 (2015)
De, U.C., Ghosh, G.C.: On quasi-Einstein and special quasi-Einstein manifolds. In: Proceedings of the International Conference of Mathematics and its Applications, Kuwait University, April 5–7, 178–191 (2004)
De, U.C., Ghosh, G.C.: On quasi-Einstein manifolds. Period. Math. Hung. 48(12), 223–231 (2004)
Deszcz, R., Hotlos, M., Senturk, Z.: On curvature properties of quasi-Einstein hypersurfaces in semi-Euclidean spaces. Soochow J. Math. 27, 375–389 (2001)
Duggal, K.L., Sharma, R.: Symmetries of Spacetime and Riemannian Manifold, vol. 487. Springer, Berlin (1999)
Güler, S., Demirbağ, S.A.: A study of generalized quasi Einstein spacetime with application in general relativity. Int. J. Theor. Phys. 55, 548–562 (2016)
Hamilton, R.S.: The Ricci flow on surfaces. Contemp. Math. 71, 237–261 (1988)
Maartens, R., Mason, D.P., Tsamparlis, M.: Kinematic and dynamic properties of conformal Killing vectors in anisotropic fluids. J. Math. Phys. 27, 2987 (1986)
Mallick, S., De, U.C.: Spacetimes with pseudosymmetric energy momentum tensor. Commun. Phys. 26(2), 121–128 (2016)
Manjonjo, A.M., Maharaj, S.D., Moopanar, S.: Static models with conformal symmetry. Class. Quantum Gravity 35, 045015 (2018)
Moopanar, S., Maharaj, S.D.: Conformal Symmetries of Spherical Spacetimes. Int. J. Theor. Phys. 49, 1878–1885 (2010)
Mason, D.P., Maartens, R.: Kinematics and dynamics of conformal collineations in relativity. J. Math. Phys. 28, 2182 (1987)
O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983)
Ray-Guha, S.: On perfect fluid pseudo Ricci symmetric spacetime. Tensor NS 67, 101–107 (2006)
Sharma, R., Ghosh, A.: Sasakian 3-manifold as a Ricci soliton represents the Heisenberg group. Int. J. Geom. Methods Mod. Phys. 8(1), 149–154 (2011)
Stephani, H.: General Relativity: an Introduction to the Theory of Gravitational Field. Cambridge University Press, Cambridge (1982)
Tupper, B.O.J., Keane, A.J., Carot, J.: A classification of spherically symmetric spacetimes. Class. Quantum Gravity 29, 145016 (2012)
Venkatesha, Devaraja, M.N.: Certain results on \(K\)-paracontact and para Sasakian manifolds. J. Geom. 108, 939–952 (2017)
Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970)
Yano, K.: On torse forming direction in a Riemannian space. Proc. Imp. Acad. Tokyo 20, 340–345 (1994)
Yano, K., Kon, M.: Structure on Manifold, vol. 3. Series in pure mathematics. World Scientific publishing Co. Pte. Ltd., Singapore (1984)
Acknowledgements
The author’s is very much grateful to the reviewer’s for some valuable comments and many editorial corrections.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Venkatesha, Kumara, H.A. Ricci soliton and geometrical structure in a perfect fluid spacetime with torse-forming vector field. Afr. Mat. 30, 725–736 (2019). https://doi.org/10.1007/s13370-019-00679-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13370-019-00679-y