Kadri ARSLAN, Ryszard DESZCZ, Ridvan EZENTAS, Marian HOTLOS, Cengizhan MURATHAN

On generalized Robertson--Walker spacetimes satisfying some curvature condition

We give necessary and sufficient conditions for warped product manifolds (M,g), of dimension \geqslant 4, with 1-dimensional base, and in particular, for generalized Robertson--Walker spacetimes, to satisfy some generalized Einstein metric condition. Namely, the difference tensor R . C - C . R, formed from the curvature tensor R and the Weyl conformal curvature tensor C, is expressed by the Tachibana tensor Q(S,R) formed from the Ricci tensor S and R. We also construct suitable examples of such manifolds. They are quasi-Einstein, i.e. at every point of M rank (S - a g) \leqslant 1, for some a \in R, or non-quasi-Einstein.

Anahtar Kelimeler:

Warped product, generalized Robertson--Walker spacetime, Einstein manifold, quasi-Einstein manifold, essentially conformally symmetric manifold, Tachibana tensor, generalized Einstein metric condition, pseudosymmetry type curvature condition, Ricci-pseud

On generalized Robertson--Walker spacetimes satisfying some curvature condition

Keywords:

Warped product, generalized Robertson--Walker spacetime, Einstein manifold, quasi-Einstein manifold, essentially conformally symmetric manifold, Tachibana tensor, generalized Einstein metric condition, pseudosymmetry type curvature condition, Ricci-pseud,

PDF

___

Al´ıas L, Romero A, S´ anchez M. Compact spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes. In: Dillen F, editor. Geometry and Topology of Submanifolds VII. River Edge, NJ, USA: World Scientic, 1995, pp. 67–70.
Al´ıas L, Romero A, S´ anchez M. Spacelike hypersurfaces of constant mean curvature and Calabi-Einstein type problems, Tˆ ohoku Math J 1997; 49: 337–345.
Beem JK, Ehrlich PE. Global Lorentzian Geometry. New York, NY, USA: Marcel Dekker, 1981.
Beem JK, Ehrlich PE, Powell TG. Warped product manifolds in relativity. In: Rassias TM, editor. Selected Studies: Physics-Astrophysics, Mathematics, History of Sciences, A Volume Dedicated to the Memory of Albert Einstein. Amsterdam, the Netherlands: North-Holland, 1982, pp. 41–56.
Belkhelfa M, Deszcz R, Glogowska M, Hotlo´ s M, Kowalczyk D, Verstraelen L. On some type of curvature conditions. Banach Center Publ Inst Math Polish Acad Sci 2002; 57: 179–194.
Besse AL. Einstein Manifolds. Berlin, Germany; Springer-Verlag, 1987.
Blanco OF, S´ anchez M, Senovilla JMM. Structure of second-order symmetric Lorentzian manifolds. J Eur Math Soc 2013; 15: 595–634.
Chojnacka-Dulas J, Deszcz R, Glogowska M, Prvanovi´ c M. On warped product manifolds satisfying some curvature conditions. J Geom Phys 2013; 74: 328–341.
Defever F, Deszcz R. On semi-Riemannian manifolds satisfying the condition R · R = Q(S, R). In: Verstraelen L, editor. Geometry and Topology of Submanifolds, III. Teaneck, NJ, USA: World Scientific, 1991, pp. 108–130.
Defever F., Deszcz R., Hotlo´ s M., Kucharski M. and S.ent¨urk Z.: Generalisations of Robertson-Walker spaces. Ann. Univ. Sci. Budapest. E¨ otv¨ os Sect. Math. 43, 2000, 13-24.
Defever F, Deszcz R, Prvanovi´ c M. On warped product manifolds satisfying some curvature condition of pseudosymmetry type. Bull Greek Math Soc 1994; 36: 43–67.
Defever F, Deszcz R, Verstraelen L, Vrancken L. On pseudosymmetric space-times. J Math Phys 1994; 35: 5908– 59
Derdzi´ nski A, Roter W. Some theorems on conformally symmetric manifolds. Tensor (NS) 1978; 32: 11–23.
Derdzi´ nski A, Roter W. Some properties of conformally symmetric manifolds which are not Ricci-recurrent . Tensor (NS) 1980; 34: 11–20.
Derdzinski A, Roter W. Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds. Tohoku Math J 2007; 59: 565–602.
Derdzinski A, Roter W. On compact manifolds admitting indefinite metrics with parallel Weyl tensor. J Geom Phys 2008; 58: 1137–1147.
Derdzinski A, Roter W. The local structure of conformally symmetric manifolds. Bull Belg Math Soc Simon Stevin 2009; 16: 117–128.
Derdzinski A, Roter W. Compact pseudo-Riemannian manifolds with parallel Weyl tensor. Ann Glob Anal Geom 2010; 37: 73–90.
Deszcz R. On pseudosymmetric spaces. Bull Soc Belg Math Ser A 1992; 44: 1–34.
Deszcz R. On pseudosymmetric warped product manifolds. In: Dillen F, editor. Geometry and Topology of Submanifolds, V. River Edge, NJ, USA: World Scientific, 1993, pp. 132–146.
Deszcz R, Glogowska M. Some nonsemisymmetric Ricci-semisymmetric warped product hypersurfaces. Publ Inst Math (Beograd) (NS) 2002; 72: 81–93.
Deszcz R, Glogowska M, Hotlo´ s M, Sawicz K. A survey on generalized Einstein metric conditions. In: Yau ST, editor. Advances in Lorentzian Geometry: Proceedings of the Lorentzian Geometry Conference in Berlin, AMS/IP Studies in Advanced Mathematics, 2011; 49, 27–46.
Deszcz R, Glogowska M, Hotlo´ s M, S ¸ent¨ urk Z. On certain quasi-Einstein semisymmetric hypersurfaces. Ann Univ Sci Budapest E¨ otv¨ os Sect Math 1998; 41: 151–164.
Deszcz R, Glogowska M, Hotlo´ s M, Verstraelen L. On some generalized Einstein metric conditions on hypersurfaces in semi-Riemannian space forms. Colloq Math 2003; 96: 149–166.
Deszcz R, Glogowska M, Hotlo´ s M, Zafindratafa G. Hypersurfaces in spaces of constant curvature satisfying some curvature conditions of pseudosymmetric type. In press. Deszcz R, Glogowska M, Plaue M, Sawicz K, Scherfner M. On hypersurfaces in space forms satisfying particular curvature conditions of Tachibana type. Kragujevac J Math 2011; 35: 223–247.
Deszcz R, Haesen S, Verstraelen L. Classification of space-times satisfying some pseudo-symmetry type conditions. Soochow J Math 2004; 30: 339–349.
Deszcz R, Haesen S, Verstraelen L. On natural symmetries. In: Mihai A, Mihai I, Miron R, editors. Topics in Differential Geometry. Bucharest, Romania: Romanian Academy, 2008, pp. 249–308.
Deszcz R, Hotlo´ s M. On a certain subclass of pseudosymmetric manifolds. Publ Math Debrecen 1998; 53: 29–48. Deszcz R, Hotlo´ s M. On hypersurfaces with type number two in spaces of constant curvature, Ann Univ Sci Budapest E¨ otv¨ os Sect Math 2003; 46: 19–34.
Deszcz R, Hotlo´ s M. On some pseudosymmetry type curvature condition. Tsukuba J Math 2003; 27: 13–30.
Deszcz R, Hotlo´ s M, S ¸ent¨ urk Z. On some family of generalized Einstein metric conditions. Demonstratio Math 2001; 34: 943–954.
Deszcz R, Hotlo´ s M, S ¸ent¨ urk Z. On curvature properties of certain quasi-Einstein hypersurfaces. Int J Math 2012; 23: 1250073.
Deszcz R, Kowalczyk D. On some class of pseudosymmetric warped products. Colloq Math 2003; 97: 7–22.
Deszcz R, Kucharski M. On curvature properties of certain generalized Robertson-Walker spacetimes. Tsukuba J Math 1999; 23: 113–130.
Deszcz R, Scherfner M. On a particular class of warped products with fibres locally isometric to generalized Cartan hypersurfaces. Colloq Math 2007; 109: 13–29.
Deszcz R, Verstraelen L, Vrancken L. On the symmetry of warped product spacetimes. Gen Relativ Grav 1991; 23: 671–681.
Ehrlich PE, Jung YT, Kim SB. Constant scalar curvatures on warped product manifolds. Tsukuba J Math 1996; 20: 239–256.
Eriksson I, Senovilla JMM. Note on (conformally) semi-symmetric spacetimes. Class Quantum Grav 2010; 27: 02700
Glogowska M. On a curvature characterization of Ricci-pseudosymmetric hypersurfaces. Acta Math Scientia 2004; 24 B: 361–375.
Glogowska M. On quasi-Einstein Cartan type hypersurfaces. J Geom Phys 2008; 58: 599–614.
Haesen S, Verstraelen L. Classification of the pseudosymmetric space-times. J Math Phys 2004 45: 2343–2346.
Haesen S, Verstraelen L. Properties of a scalar curvature invariant depending on two planes. Manuscripta Math 2007; 122: 59–72.
Haesen S, Verstraelen L. Natural intrinsic geometrical symmetries. Symmetry Integrability Geom Meth Appl 2009; 5: 0
Hotlo´ s M. On a certain curvature condition of pseudosymmetry type. To appear. Jahanara B, Haesen S, S ¸ent¨ urk Z, Verstraelen L. On the parallel transport of the Ricci curvatures. J Geom Phys 2007; 57: 1771–1777.
Kowalczyk D. On some subclass of semisymmetric manifolds. Soochow J Math 2001; 27: 445–461.
Kowalczyk D. On the Reissner-Nordstr¨ om-de Sitter type spacetimes. Tsukuba J Math 2006; 30: 363–381.
Kruchkovich GI. On some class of Riemannian spaces. Trudy Sem. po Vekt. i Tenz. Analizu 1961; 11: 103–120 (in Russian).
Murathan C, Arslan K, Deszcz R, Ezenta¸s R, ¨ Ozg¨ ur C. On a certain class of hypersurfaces of semi-Euclidean spaces. Publ Math Debrecen 2001; 58: 587–604.
O’Neill B. Semi-Riemannian Geometry with Applications to Relativity. New York, NY, USA: Academic Press, 1983. ¨ Ozg¨ ur C. Hypersurfaces satisfying some curvature conditions in the semi-Euclidean space. Chaos Solitons Fractals 2009; 39: 2457–2464.
S´ anchez M. On the geometry of generalized Robertson-Walker spacetimes: geodesics. Gen Relativ Grav 1998; 30: 915–932.
S´ anchez M. On the geometry of generalized Robertson-Walker spacetimes: curvature and Killing fields. J Geom Phys 1999; 31: 1–15.
Senovilla JMM. Second-order symmetric Lorentzian manifolds: I. Characterization and general results. Class Quantum Grav 2008; 25: 245011.
Shaikh AA, Kim YH, Hui SK. On Lorentzian quasi-Einstein manifolds. J Korean Math Soc 2011; 48: 669–689.
Stephani H, Kramer D, MacCallum M, Hoenselaers C, Herlt E. Exact Solutions of the Einstein’s Field Equations. Cambridge Monographs on Mathematical Physics. 2nd ed. Cambridge, UK: Cambridge University Press, 2003.
Szab´ o ZI. Structure theorems on Riemannian spaces satisfying R(X, Y ) · R = 0. I. The local version. J Differential Geom 1982; 17: 531–582.
Szab´ o ZI. Structure theorems on Riemannian spaces satisfying R(X, Y ) · R = 0. II. The global version. Geom Dedicata 1985; 19: 65–108.