×

zbMATH — the first resource for mathematics

A new class of almost Ricci solitons and their physical interpretation. (English) Zbl 1428.83010
Summary: We establish a link between a connection symmetry, called conformal collineation, and almost Ricci soliton (in particular Ricci soliton) in reducible Ricci symmetric semi-Riemannian manifolds. As a physical application, by investigating the kinematic and dynamic properties of almost Ricci soliton manifolds, we present a physical model of imperfect fluid spacetimes. This model gives a general relation between the physical quantities \((u, \mu, p, \alpha, \eta, \sigma_{i j})\) of the matter tensor of the field equations and does not provide any exact solution. Therefore, we propose further study on finding exact solutions of our viscous fluid physical model for which it is required that the fluid velocity vector \(u\) be tilted. We also suggest two open problems.
MSC:
83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)
35C08 Soliton solutions
53E20 Ricci flows
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Hamilton, R., Three-manifolds with positive Ricci curvature, Journal of Differential Geometry, 17, 2, 155-306 (1982) · Zbl 0504.53034
[2] Cao, H. D.; Chow, B.; Chu, S. C.; Yau, S. T., Collected Papers on Ricci Flow. Collected Papers on Ricci Flow, Series in Geometry and Topology, 37 (2003), Somerville, Mass, USA: International Press, Somerville, Mass, USA · Zbl 1108.53002
[3] Chow, B.; Knopf, D., The Ricci Flow: An Introduction (2004), Providence, RI, USA: AMS, Providence, RI, USA · Zbl 1086.53085
[4] Crasmareanu, M., Liouville and geodesic Ricci solitons, Comptes Rendus Mathematique, 1, 21-22, 1305-1308 (2009) · Zbl 1183.53036
[5] Onda, K., Lorentz Ricci solitons on 3-dimensional Lie groups, Geometriae Dedicata, 147, 313-322 (2010) · Zbl 1203.53044
[6] Pigola, S.; Rigoli, M.; Rimoldi, M.; Setti, A. G., Ricci almost solitons, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, 10, 4, 757-799 (2011) · Zbl 1239.53057
[7] Duggal, K. L.; Sharma, R., Symmetries of Spacetimes and Riemannian Manifolds, 487 (1999), Kluwer Academic · Zbl 0962.53003
[8] Tashiro, Y., On conformal collineations, Mathematical Journal of Okayama University, 10, 75-85 (1960) · Zbl 0103.14904
[9] Sharma, R.; Duggal, K. L., A characterization of affine conformal vector field, Comptes Rendus Mathématiques des l’Académie des Sciences. La Société Royale du Canada, 7, 201-205 (1985) · Zbl 0579.53020
[10] Eisenhart, L. P., Symmetric tensors of the second order whose first covariant derivatives are zero, Transactions of the American Mathematical Society, 25, 2, 297-306 (1923) · JFM 49.0539.01
[11] Patterson, E. M., On symmetric recurrent tensors of the second order, The Quarterly Journal of Mathematics, 2, 151-158 (1951) · Zbl 0045.43204
[12] Levy, H., Symmetric tensors of the second order whose covariant derivatives vanish, The Annals of Mathematics, 27, 2, 91-98 (1925) · JFM 51.0576.02
[13] Duggal, K. L., Affine conformal vector fields in semi-Riemannian manifolds, Acta Applicandae Mathematicae, 23, 3, 275-294 (1991) · Zbl 0734.53023
[14] Mason, D. P.; Maartens, R., Kinematics and dynamics of conformal collineations in relativity, Journal of Mathematical Physics, 28, 9, 2182-2186 (1987) · Zbl 0635.76132
[15] Barros, A.; Batista, R.; Ribeiro, E., Compact almost solitons with constant scalar curvature are gradient, Monatshefte für Mathematik, 174, 1, 29-39 (2014) · Zbl 1296.53092
[16] Barros, A.; Ribeiro, J., Some characterizations for compact almost Ricci solitons, Proceedings of the American Mathematical Society, 140, 3, 1033-1040 (2012) · Zbl 1245.53044
[17] Sharma, R., Almost Ricci solitons and \(K\)-contact geometry, Monatshefte für Mathematik, 175, 4, 621-628 (2014) · Zbl 1307.53038
[18] Wang, Y., Gradient Ricci almost solitons on two classes of almost Kenmotsu manifolds, Journal of the Korean Mathematical Society, 53, 5, 1101-1114 (2016) · Zbl 1358.53039
[19] Patterson, E. M., Some theorems on Ricci-recurrent spaces, Journal of the London Mathematical Society. Second Series, 27, 287-295 (1952) · Zbl 0048.15604
[20] Katzin, G. H.; Levine, J.; Davis, W. R., Curvature collineations in conformally at spaces, I, Tensor N.S., 21, 51-61 (1970) · Zbl 0216.43602
[21] Levine, J.; Katzin, G. H., Conformally at spaces admitting special quadratic first integrals, 1. Symmetric spaces, Tensor N. S., 19, 317-328 (1968) · Zbl 0159.51301
[22] Grycak, W., On affine collineations in conformally recurrent manifolds, The Tensor Society. Tensor. New Series, 35, 1, 45-50 (1981) · Zbl 0463.53007
[23] Adati, T.; Miyazawa, T., On Riemannian space with recurrent conformal curvature, Tensor, 18, 348-354 (1967) · Zbl 0152.39103
[24] Yano, K., Integral Formulas in Riemannian Geometry (1970), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0213.23801
[25] Maartens, R.; Mason, D. P.; Tsamparlis, M., Kinematic and dynamic properties of conformal Killing vectors in anisotropic fluids, Journal of Mathematical Physics, 27, 12, 2987-2994 (1986) · Zbl 0609.53031
[26] Greenberg, P. J., The general theory of space-like congruences with an application to vorticity in relativistic hydrodynamics, Journal of Mathematical Analysis and Applications, 30, 128-143 (1970) · Zbl 0195.23906
[27] Maartens, R.; Maharaj, S. D., Conformal Killing vectors in Robertson-Walker spacetimes, Classical and Quantum Gravity, 3, 5, 1005-1011 (1986) · Zbl 0596.53021
[28] Hall, G. S.; da Costa, J., Affine collineations in space-time, Journal of Mathematical Physics, 29, 11, 2465-2472 (1988) · Zbl 0661.53017
[29] Coley, A. A.; Tupper, B. O. J., Radiation-like imperfect fluid cosmologies, Astrophysical Journal, 288, 2, part 1, 418-421 (1985)
[30] Landau, L.; Lifshitz, E. M., Fluid Mechanics (1958), Reading, Mass, USA: Addison-Welsely, Reading, Mass, USA
[31] Brozos-Vázquez, M.; García-Río, E.; Gavino-Fernández, S., Locally conformally flat Lorentzian gradient Ricci solitons, Journal of Geometric Analysis, 23, 3, 1196-1212 (2013) · Zbl 1285.53059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.