Bayram ŞAHİN, Sema KAZAN

Pseudosymmetric lightlike hypersurfaces

We study lightlike hypersurfaces of a semi-Riemannian manifold satisfying pseudosymmetry conditions. We give sufficient conditions for a lightlike hypersurface to be pseudosymmetric and show that there is a close relationship of the pseudosymmetry condition of a lightlike hypersurface and its integrable screen distribution. We obtain that a pseudosymmetric lightlike hypersurface is a semisymmetric lightlike hypersurface or totally geodesic under certain conditions. Moreover, we give an example of pseudosymmetric lightlike hypersurfaces and investigate pseudoparallel lightlike hypersurfaces. Furthermore, we introduce Ricci-pseudosymmetric lightlike hypersurfaces, obtain characterizations, and give an example for such hypersurfaces.

Anahtar Kelimeler:

Semisymmetric lightlike hypersurface, Ricci-semisymmetric lightlike hypersurface, pseudosymmetric lightlike hypersurface, pseudoparallel lightlike hypersurface

Pseudosymmetric lightlike hypersurfaces

Keywords:

Semisymmetric lightlike hypersurface, Ricci-semisymmetric lightlike hypersurface, pseudosymmetric lightlike hypersurface, pseudoparallel lightlike hypersurface,

PDF

___

Arslan K, C¸ elik Y, Deszcz R, Ezenta¸s R. On the equivalence of Ricci-semisymmetry and semisymmetry. Colloq Math 1998; 76: 279–294.
Arslan K, Deszcz R, Ezenta¸s R, Hotlo´s M, Murathan C. On generalized Robertson-Walker spacetimes satisfying some curvature condition. Turk J Math 2014; 38: 353–373.
Asperti AC, Lobos GA, Mercuri F. Pseudo-parallel submanifolds of a space form. Adv Geom 2002; 2: 57–71.
Boeckx E, Kowalski O, Vanhecke L. Riemannian Manifolds of Conullity Two. Singapore: World Scientific, 1996.
Cartan E. Sur une classse remarqable d’espaces de Rieamnnian. Bull Soc Math France 1926; 54: 214–264 (in French).
Defever F. Ricci-semisymmetric hypersurfaces. Balk J Geom Appl 2000; 5: 81–91.
Deprez J. Semiparallel surfaces in Euclidean space. J Geom 1985; 25: 192–200.
Deszcz R. On Ricci-pseudosymmetric warped products. Demonstratio Math 1989; 22: 1053–1065.
Deszcz R. On pseudosymmetric spaces. Bull Soc Math Belg S´er. A 1992; 44: 1–34.
Deszcz R. On certain classes of hypersurfaces in spaces of constant curvature. In: Dillen F, Komrakov B, Simon U, Verstraelen L, Van De Woestyne I, editors. Geometry and Topology of Submanifolds. Vol. 8. Singapore: World Scientific, 1996, pp. 101–110.
Deszcz R, Glogowska M, Hotlo´s M, Sawicz K. A survey on generalized Einstein metric conditions. In: Plaue M, Rendall AD, Scherfner M, editors. Advances in Lorentzian Geometry: Proceedings of the Lorentzian Geometry Conference in Berlin, AMS/IP Studies in Advanced Mathematics. Providence, RI, USA: American Mathematical Society, 2011, pp. 27–46.
Deszcz R, Grycak W. On some class of warped product manifolds. Bull Inst Math Acad Sinica 1987; 15: 311–322.
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 of Sciences, 2008, pp. 249–308.
Deszcz R, Hotlo´s M. Remarks on Riemannian manifolds satisfying certain curvature condition imposed on the Ricci tensor. Pr Nauk Pol Szczec 1988; 11: 23–34.
Deszcz R, Hotlo´s M. On some pseudosymmetry type curvature condition. Tsukuba J Math 2003; 27: 13–30.
Deszcz R, Verheyen P, Verstraelen L. On some generalized Einstein metric conditions. Publ Inst Math (Beograd) (N.S.) 1996; 60: 108–120.
Duggal KL, Bejancu A. Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications. Dordrecht, the Netherlands: Kluwer, 1996.
Duggal KL, S¸ahin B. Differential Geometry of Lightlike Submanifolds. Basel, Switzerland: Birkh¨auser, 2010.
Lungiambudila O, Massamba F, Tossa J. Symmetry properties of lightlike hypersurfaces in indefinite Sasakian manifolds. SUT J Math 2010; 46: 177–204.
Massamba F. Semi-parallel lightlike hypersurfaces of indefinite Sasakian manifolds. International Journal of Con- temporary Mathematical Sciences 2008; 3: 629–634.
Massamba F. On weakly Ricci symmetric lightlike hypersurfaces. STU J Math 2008; 44: 181–201.
Massamba F. On semi-parallel lightlike hypersurfaces of indefinite Kenmotsu manifolds. J Geom 2009; 95: 73–89.
Massamba F. Symmetries of null geometry in indefinite Kenmotsu manifolds. Mediterr J Math 2013; 10: 1079–1099.
Murathan C, Arslan K, Ezentas. R. Ricci generalized pseudo-parallel immersions. In: Bures J, Kowalski O, Krupka D, Slovak J, editors. Differential Geometry and Its Applications. Prague, Czech Republic: Matfyzpress, 2005, pp. 99–108.
¨Ozg¨ur, C. Pseudo Simetrik Manifoldlar. PhD, Uluda˘g University, Bursa, Turkey, 2001 (in Turkish). S
¸ahin B. Lightlike hypersurfaces of semi-Euclidean spaces satisfying curvature conditions of semisymmetry type. Turk J Math 2007; 31: 139–162.
Takagi R. An example of Riemannian manifold satisfying R(X, Y )· R = 0 but not ∇R = 0. Tˆohoku Math J 1972; 24: 105–108.
Upadhyay A, Gupta RS, Sharfuddin A. Semi-symmetric and Ricci semi-symmetric lightlike hypersurfaces of an indefinite generalized Sasakian space form. International Electronic Journal of Geometry 2012; 5: 140–150.