A study of generalized quasi Einstein spacetimes with applications in general relativity. (English) Zbl 1337.83010

Summary: The aim of this paper is to investigate some geometric and physical properties of the generalized quasi Einstein spacetime \(G(QE)_4\) under certain conditions. Firstly, we prove the existence of \(G(QE)_4\) by constructing a non trivial example. Then it is proved that the \(G(QE)_4\) spacetime with the conditions \(\mathcal B\cdot S=L_SQ(g,S)\), where \(\mathcal B\) denotes the Ricci tensor or the concircular curvature tensor is an \(N(\frac{a-b}{3})\)-quasi Einstein spacetime and in a \(G(QE)_4\) spacetime with \(C\cdot S=0\), where \(C\) is the conformal curvature tensor, \(a-b\) is an eigenvalue of the Ricci operator. Then, we deal with the Ricci recurrent \(G(QE)_4\) spacetime and prove that in this spacetime, the acceleration vector and the vorticity tensor vanish; but this spacetime has the non-vanishing expansion scalar and the shear tensor. Moreover, it is shown that every Ricci recurrent \(G(QE)_4\) is Weyl compatible, purely electric spacetime and its possible Petrov types are I or D.


83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
53Z05 Applications of differential geometry to physics
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
Full Text: DOI


[1] Besse, A.L.: Einstein manifolds, Ergeb. Math. Grenzgeb., 3. Folge, Bd. 10. Springer Verlag, Berlin (1987)
[2] Chaki, MC; Maity, RK, On-quasi Einstein manifolds, Pub. Math. Debrecen, 57, 297-306, (2000) · Zbl 0968.53030
[3] Chaki, MC, On generalized quasi-Einstein manifold, Publ. Math. Debrecen, 58, 638-691, (2001)
[4] De, U.C., Ghosh, G.C.: On quasi Einstein and special quasi Einstein manifolds. In: Proc. of the Int. Conf. of Mathematics and its Applications, pp. 178-191. Kuwait University (2004) · Zbl 1079.53064
[5] Tanno, S, Ricci curvatures of contact Riemannian manifolds, Tohoku Math. J, 40, 441-8, (1988) · Zbl 0655.53035
[6] Triphati, MM; Kim, JS, On N(k)-Einstein manifolds, Commun. Korean Math. Soc, 22, 411-417, (2007) · Zbl 1168.53321
[7] Özgür, C; Triphati, MM, On the concircular curvature tensor of an N(k)-Einstein manifolds, Math. Pannon., 18, 95-100, (2007) · Zbl 1164.53012
[8] Yano, K.: Concircular geometry. I-IV. Proc. Imp. Acad. 16, 195-200, 354-360, 442-448, 505-511 (1940) · Zbl 0024.08102
[9] Yano, K., Kon, M.: Structures on Manifolds. World Scientific Publishing, Singapore (1984) · Zbl 0557.53001
[10] Arslan, K; Çelik, Y; Deszcz, R; Ezentaş, R, On the equivalence of the Ricci-semisymmetry and semisymmetry, Colloquium Mathematicum, 76, 279-294, (1998) · Zbl 0901.53010
[11] Deszcz, R, On pseudosymmetric spaces, Bull. Belg. Math. Soc. Ser. A, 44, 1-34, (1992) · Zbl 0808.53012
[12] Deszcz, R; Hotlos, M; Şentürk, Z, On the equivalence of the Ricci-pseudosymmetry and pseudosymmetry, Colloquium Mathematicum, 79, 211-227, (1999) · Zbl 0926.53012
[13] Patterson, E.M.: Some theorems on Ricci-recurrent spaces. Journ. Lond. Math. Soc. (1952) · Zbl 0048.15604
[14] O’Neill, B.: Semi-Riemannian Geometry With Applications to Relativity, pp 336-341. Academic Press, New York (1983)
[15] Ferus, D.: A Remark on Codazzi Tensors on Constant Curvature Space, Lecture Notes Math, Global Differential Geometry and Global Analysis, vol. 838. Springer-Verlag, New York (1981) · Zbl 0437.53013
[16] Roter, W, On conformally symmetric Ricci-recurrent spaces, Colloquium Math., 31, 87-96, (1974) · Zbl 0292.53014
[17] Ellis, G.F.R.: In: Sachs R. K. (ed.) Relativistic Cosmology in General Relativity and Cosmology, pp. 104-179. Academic Press, London (1971)
[18] Mantica, CA; Molinari, LG, Extended derdzinski-shen theorem for curvature tensors, Colloq. Math., 128, 69-82, (2012) · Zbl 1269.53018
[19] Mantica, CA; Molinari, LG, Weyl compatible tensors, Int. J. Geom. Methods Mod. Phys., 11, 1450070, (2014) · Zbl 1307.53010
[20] Bertschinger, E; Hamilton, AJS, Lagrangian evolution of the Weyl tensor, Astroph. J., 435, 1-7, (1994)
[21] Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Hertl, E.: Exact Solutions of Einstein’s Field Equations, 2n edn. Cambridge University Press (2003) · Zbl 1057.83004
[22] Mantica, CA; Molinari, LG, Riemann compatible tensors, Colloq. Math., 128, 197-210, (2012) · Zbl 1276.53016
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.