A Study on Chen-like Inequalities for Half Lightlike Submanifolds of a Lorentzian Manifold Endowed with Semi-Symmetric Metric Connection

Year 2020, Volume: 1 Issue: 2, 76 - 88, 31.12.2020

Nergiz Poyraz , Burçin Doğan

Abstract

In this paper, Chen-like inequalities of a half lightlike submanifolds of a real space form N(c) with constant sectional curvature c, equipped with semi-symmetric metric connection are established and some important characterization theorems for such submanifolds are proved using these inequalities.

Keywords

Chen inequality, Half-lightlike submanifold, Lorentzian manifold, Semi-symmetric metric connection

References

• J. K. Beem, P. E. Ehrlich, and K. L. Easley, Global Lorentzian geometry, Volume 202 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York, second edition, 1996.
• C. L. Bejan and K. L. Duggal, Global lightlike manifolds and harmonicity, Kodai Math. J., 28(1) 131-145, (2005).
• B. Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. math. (Basel), 60(6), 568-578, (1993), 7-60, Monogr. Geom. Topology, 25, Int. Press, Cambridge, MA, (1998).
• B. Y. Chen, Some new obstructions to minimal and Lagrangian isometric immersions, Japanese J. Math. 26, 105-127, (2000).
• B. Y. Chen, δ-invariants, Inequalities of Submanifolds and Their Applications, in Topics in Differential Geometry, Eds. A. Mihai, I. Mihai, R. Miron, Editura Academiei Romane, Bucuresti, 29-156,(2008).
• 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., 128,589-598,(2000).
• P. J. De Smet, F. Dillen, L. Verstraelen and V. Vrancken, A pointwise inequality in submanifold theory, Arch. Math. (Brno), 35(2),115-128,(1999).
• K. L. Duggal, A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publishers, Dordrecht 1996.
• K. L. Duggal and D.H.Jin, Half lightlike submanifolds of codimension 2, Math. J. Toyama Univ, 22:121-161,(1999).
• K. L. Duggal, D.H.Jin, Null curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scientific, 2007.
• K. L. Duggal and R. Sharma, Semi-Symmetric metric connection in a Semi-Riemannian Manifold, Indian J. Pure appl Math., 17 , 1276-1283,(1986).
• K. L. Duggal and B. Sahin, Screen conformal half-lightlike submanifolds, Int. J. Math. and Math. Sci., 68,3737-3753,(2004).
• K. L. Duggal and B. Sahin, Differential Geometry of Lightlike Submanifolds, Birkhäuser, Basel, 2010.
• A. Friedmann and J. A. Schouten, Über die Geometrie der halbsymmetrischen Übertragungen, (German) Math. Z., SoZ.I.21,211-223,(1924).
• M. Gülbahar, E. Kılıç, Some optimal inequalities for screen conformal half-lightlike submanifolds, Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, 33(2),315-329,(2017).
• M. Gülbahar, E. Kılıç and S. Keleş, Chen-like inequalities on lightlike hypersurfaces of a Lorentzian manifold, J. Inequal. Appl., 2013:266,(2013).
• M. Gülbahar, E. Kılıç and S. Keleş, Some inequalities on screen homothetic lightlike hypersurfaces of a Lorentzian manifold, Taiwanese Journal of Mathematics, 17(6),2083-2100,(2013).
• H. A. Hayden, Subspace of a space with torsion, Proceedings of the London Mathematical Society II Series, 34,27-50,(1932).
• S. Hong, K Matsumoto and M. M. Tripathi, Certain basic inequalities for submanifolds of locally conformal Kaehlerian space forms, SUT J. Math., 4(1),75-94,(2005).
• T. Imai, Hypersurfaces of a Riemannian Manifold with Semi-Symmetric Metric Connection, Tensor, N.S., SoZ.23,300-306,(1972).
• T. Imai, Notes on Semi-Symmetric Metric Connection, Tensor, N.S., SoZ.24,293-296,(1972).
• D. H. Jin, Half lightlike submanifolds with totally umbilical screen distributions, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math., 17(1):29-38,(2010).
• D. H. Jin, Geometry of half lightlike submanifolds of a semi-Riemannian space form with a semi-symmetric metric connection, J. Chungcheong Math. Soc., 24(4),769-780,(2011).
• J. S. Kim, J. Choi, A basic inequality for submanifolds in a cosymplectic space form, Int. J. Math. Math. Sci., 9,539-547,(2003).
• A. Konar and B. Biswas, Lorentzian Manifold that Admits a type of Semi-Symmetric Metric Connection, Bull. Cal. Math., Soc., 93(5),427-437,(2001).
• D. N. Kupeli, Singular semi-Riemannian Geometry, Kluwer Academic Publishers, Dordrecht 366,1996.
• K. Matsumoto, I. Mihai, A. Oiaga, Ricci curvature of submanifolds in complex space forms, Rev. Roumaine Math. Pures Appl., 46,775-782,(2001).
• A. Mihai A and C. Özgür, Chen inequalities for submanifolds of real space form with a semi-symmetric metric connection, Tawanese Journal of Mathematics, SoZ.14,Lo.4,pp.1465-1477,(2010).
• A. Oiaga, I. Mihai, B. Y. Chen, Inequalities for slant submanifolds in complex space forms, Demonstratio Math., 32,835-846,(1999).
• Z. Nakao, Submanifolds of a Riemannian manifold with semi-symmetric metric connections, Proc. Amer. Math. Soc., 54,261-266,(1976).
• N. Ö. Poyraz, B. Doğan and E. Yaşar, Chen Inequalities on Lightlike Hypersurface of a Lorentzian manifold with semi-symmetric metric connection, Int. Electronic Journal of Geometry, 10(1),1-14,(2017).
• M. M. Tripathi, Certain Basic Inequalities for Submanifolds in ( ; ) Space, Recent Advances in Riemannian and Lorentzian Geometries, Baltimore, pp.187-202,(2003).
• K. Yano, On Semi-Symmetric Metric Connection, Rev. Roum. Math. Pures Et Appl., 15,1579-1586,(1970).
• E. Yaşar, A. C. Çöken, A. Yücesan, Lightlike Hypersurfaces of Semi-Riemannian Manifold with Semi-Symmetric Metric Connection, Kuweyt Journal of Science and Engineering, 34(2A),11-24,(2007).
• P. Zhang, L. Zhang and W. Song, Chen's inequalities for submanifolds of a Riemannian manifold of quasi-constant curvature with a semi-symmetric metric connection, Taiwanese Journal of Mathematics, 18(6),1841-1862,(2014).