# SCREEN ISOTROPIC LEAVES ON LIGHTLIKE HYPERSURFACES OF A LORENTZIAN MANIFOLD

• Gulbahar, Mehmet (Department of Mathematics Faculty Science and Art Siirt University)
• Published : 2017.03.31

#### Abstract

In the present paper, screen isotropic leaves on lightlike hypersurfaces of a Lorentzian manifold are introduced and studied which are inspired by the definition of isotropic immersions in the Riemannian context. Some examples of such leaves are mentioned. Furthermore, some relations involving curvature invariants are obtained.

#### References

1. T. Adachi and T. Sugiyama, A characterization of isotropic immersions by extrinsic shapes of smooth curves, Differential Geom. Appl. 26 (2008), no. 3, 307-312. https://doi.org/10.1016/j.difgeo.2007.11.022
2. S. S. Ahn, D. S. Kim, and Y. H. Kim, Submanifolds of Euclidean space with non-negative Ricci curvature, Kyungpook Math. J. 35 (1995), no. 2, 387-392.
3. C. Atindogbe and K. L. Duggal, Conformal screen on lightlike hypersurfaces, Int. J. Pure Appl. Math. 11 (2004), no. 4, 421-442.
4. J. K. Beem, P. E. Ehrlich, and K. L. Easley, Global Lorentzian Geometry, Marcel Dekker Inc., New York, Second Edition, 1996.
5. C. L. Bejan and K. L. Duggal, Global lightlike manifolds and harmonicity, Kodai Math. J. 28 (2005), no. 1, 131-145. https://doi.org/10.2996/kmj/1111588042
6. A. Bejancu and K. L. Duggal, Degenerated hypersurfaces of semi-Riemannian manifolds, Bul. Inst. Politehn. Iasi Sect. I 37(41) (1991), no. 1-4, 13-22.
7. O. Birembaux and L. Vrancken, Isotropic affine hypersurfaces of dimension 5, J. Math. Anal. Appl. 417 (2014), no. 2, 918-962.
8. N. Boumuki, Isotropic immersions with low codimension of space forms into space forms, Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci. 37 (2004), 1-4.
9. N. Boumuki, Isotropic immersions and parallel immersions of Cayley projective plane into a real space form, New Zealand J. Math. 36 (2007), 139-146.
10. K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publisher, 1996.
11. K. L. Duggal and D. H. Jin, Totally umbilical lightlike submanifolds, Kodai Math. J. 26 (2003), no. 1, 49-68. https://doi.org/10.2996/kmj/1050496648
12. K. L. Duggal and D. H. Jin, Null Curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scientific Publishing, Hackensack, NJ, USA, 2007.
13. K. L. Duggal and B. Sahin, Differential Geometry of Lightlike Submanifolds, Birkhauser Verlag AG., 2010.
14. N. Ejiri, Totally real isotropic submanifolds in a complex projective space, Unpublished.
15. M. Gulbahar, E. Kilic, and S. Keles, Chen-like inequalities on lightlike hypersurfaces of a Lorentzian manifold, J. Inequal. Appl. 2013 (2013), 266, 18 pp. https://doi.org/10.1186/1029-242X-2013-18
16. M. Gulbahar, E. Kilic, and S. Keles, Some inequalities on screen homothetic lightlike hypersurfaces of a Lorentzian manifold, Taiwanese J. Math. 17 (2013), no. 6, 2083-2100. https://doi.org/10.11650/tjm.17.2013.3185
17. T. Itoh and K. Ogiue, Isotropic immersions and Veronese manifolds, Trans. Amer. Math. Soc. 209 (1975), 109-117. https://doi.org/10.1090/S0002-9947-1975-0375172-8
18. D. H. Jin, Lightlike hypersurfaces with totally umbilical screen distributions, J. Chungcheong Math. Soc. 22 (2009), no. 3, 409-416.
19. Y. H. Kim, Isotropic submanifolds of real space forms, Kyungpook Math. J. 52 (2012), no. 3, 271-278. https://doi.org/10.5666/KMJ.2012.52.3.271
20. H. Li and X. Wang, Isotropic Lagrangian submanifolds in complex Euclidean space and complex hyperbolic space, Results Math. 56 (2009), no. 1-4, 387-403. https://doi.org/10.1007/s00025-009-0422-9
21. S. Maeda, Isotropic immersions with parallel second fundamental form, Canad. Math. Bull. 26 (1983) no. 3, 291-296. https://doi.org/10.4153/CMB-1983-047-9
22. S. Maeda, Isotropic immersions with parallel second fundamental form II, Yokohama Math. J. 31 (1983), no. 1-2, 131-138.
23. S. Maeda, Remarks on isotropic immersions, Yokohama Math. J. 34 (1986), no. 1-2, 83-90.
24. S. Maeda, A characterization of constant isotropic immersions by circles, Arch. Math. (Basel) 81 (2003), no. 1, 90-95. https://doi.org/10.1007/s00013-003-4677-1
25. S. Montiel and F. Urbano, Isotropic totally real submanifolds, Math. Z. 199 (1988), no. 1, 55-60. https://doi.org/10.1007/BF01160209
26. H. Naitoh, Isotropic submanifolds with parallel second fundamental form in $P^m(c)$, Osaka J. Math. 18 (1981), no. 2, 427-464.
27. H. Nakagawa and T. Itoh, On isotropic immersions of space forms into a sphere, Proc. of Japan-Unites Seminar on Minimal Submanifolds, including Geodesics, 1978.
28. B. O'Neill, Isotropic and Kahler immersions, Canad. J. Math. 17 (1965), 907-915. https://doi.org/10.4153/CJM-1965-086-7
29. L. Vrancken, Some remarks on isotropic submanifolds, Publ. Inst. Math. (Beograd) (N.S.) 51(65) (1992), 94-100.
30. X. Wang, H. Li, and L. Vrancken, Lagrangian submanifolds in 3-dimensional complex space forms with isotropic cubic tensor, Bull. Belg. Math. Soc. Simon Stevin 18 (2011), no. 3, 431-451.
31. X. Wang, H. Li, and L. Vrancken, Minimal Lagrangian isotropic immersions in indefinite complex space forms, J. Geom. Phys. 62 (2012), no. 4, 707-723. https://doi.org/10.1016/j.geomphys.2011.12.015