Taiwanese Journal of Mathematics

SOME INEQUALITIES ON SCREEN HOMOTHETIC LIGHTLIKE HYPERSURFACES OF A LORENTZIAN MANIFOLD

Mehmet Gülbahar, Erol Kılıç, and Sadık Keleş

Full-text: Open access

Abstract

In this paper, we establish some inequalities involving $k$-Ricci curvature, $k$-scalar curvature, the screen scalar curvature on a screen homothetic lightlike hypersurface of a Lorentzian manifold. We compute Chen-Ricci inequality and Chen inequality on a screen homothetic lightlike hypersurface of a Lorentzian manifold. We give an optimal inequality involving the $\delta(n_{1},\ldots ,n_{k})$-invariant and some characterizations (totally umbilicity, totally geodesicity, minimality, etc.) for lightlike hypersurfaces.

Article information

Source
Taiwanese J. Math., Volume 17, Number 6 (2013), 2083-2100.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706286

Digital Object Identifier
doi:10.11650/tjm.17.2013.3185

Mathematical Reviews number (MathSciNet)
MR3141875

Zentralblatt MATH identifier
1286.53063

Subjects
Primary: 53C40: Global submanifolds [See also 53B25] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53C50: Lorentz manifolds, manifolds with indefinite metrics

Keywords
curvature lightlike hypersurface Lorentzian manifold

Citation

Gülbahar, Mehmet; Kılıç, Erol; Keleş, Sadık. SOME INEQUALITIES ON SCREEN HOMOTHETIC LIGHTLIKE HYPERSURFACES OF A LORENTZIAN MANIFOLD. Taiwanese J. Math. 17 (2013), no. 6, 2083--2100. doi:10.11650/tjm.17.2013.3185. https://projecteuclid.org/euclid.twjm/1499706286


Export citation

References

  • C. Atindogbe and K. L. Duggal, Conformal screen on lightlike hypersurfaces, Int. J. Pure Appl. Math., 11(4) (2004), 421-442.
  • J. K. Beem, P. E. Ehrlich and K. L. Easley, Global Lorentzian Geometry, 2nd Edition, Marcel Dekker, Inc., New York, 1996.
  • C. L. Bejan and K. L. Duggal, Global lightlike manifolds and harmonicity, Kodai Math. J., 28(1) (2005), 131-145.
  • B.-Y. Chen, A Riemannian invariant for submanifolds in space forms and its applications, Geometry and Topology of Submanifolds VI, (Leuven, 1993/Brussels, 1993), (NJ: World Scientific Publishing, River Edge), 1994, pp. 58-81, no. 6, 568-578.
  • B. Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math. $($Basel$)$, 60(6) (1993), 568-578.
  • B.-Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Mathematic Journal, 41 (1999), 33-41.
  • B.-Y. Chen, Some new obstructions to minimal and Lagrangian isometric immersions, Japan J. Math., 26 (2000), 105-127.
  • B.-Y. Chen, Ricci curvature of real hypersurfaces in complex hyperbolic space, Arch. Math. $($Brno$)$, 38(1) (2002), 73-80.
  • B.-Y. Chen, F. Dillen, L. Verstraelen and V. Vrancken, Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces, Proc. Amer. Math. Soc., 128 (2000), 589-598.
  • P. J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken, A pointwise inequality in submanifold theory, Arch. Math. $($Brno$)$, 35(2) (1999), 115-128.
  • K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publisher, 1996.
  • K. L. Duggal and B. Sahin, Differential Geometry of Lightlike Submanifolds, Birkhäuser Verlag AG., 2010.
  • K. L. Duggal, On scalar curvature in lightlike geometry, Journal of Geometry and Physics, 57(2) (2007), 473-481.
  • M. Gülbahar, E. K\ptmr \il\ptmr \iç and S. Keleş, Chen-like inequalities on lightlike hypersurfaces of a Lorentzian manifold, J. of Ineq. and Appl., (2013), Doi: 10.1186/1029-242x-2013-266.
  • S. Hong, K. Matsumoto and M. M. Tripathi, Certain basic inequalities for submanifolds of locally conformal Kaehlerian space forms, SUT J. Math., 4(1) (2005), 75-94.
  • S. Hong and M. M. Tripathi, On Ricci curvature of submanifolds, Int. J. Pure Appl. Math. Sci., 2(2) (2005), 227-245.
  • A. Mihai and B. Y. Chen, Inequalities for slant submanifolds in generalized complex space forms, Rad. Mat., 12(2) (2004), 215-231.
  • C. Özgür and M. M. Tripathi, On submanifolds satisfying Chen's equality in a real space form, Arab. J. Sci. Eng. Sect. A Sci., 33(2) (2008), 321-330.
  • M. M. Tripathi, Certain Basic Inequalities for Submanifolds in $(\kappa,\mu)$ Space, Recent advances in Riemannian and Lorentzian geometries Baltimore, MD, 2003, pp. 187-202.
  • M. M. Tripathi, Chen-Ricci inequality for submanifolds of contact metric manifolds, J. Adv. Math. Stud., 1(1-2) (2008), 111-134.
  • M. M. Tripathi, Improved Chen-Ricci inequality for curvature like tensors and its applications, Diff. Geom. Appl., 29(5) (2011), 685-692.