Some Inequalities on Half Lightlike Submanifolds of a Lorentzian Manifold with Semi-Symmetric Metric Connection

Year 2021, , 1 - 14, 01.06.2021

Nergiz Poyraz , Burçin Doğan

https://doi.org/10.46572/naturengs.881981

Abstract

In this paper, we introduce some inequalities for screen homothetic half lightlike submanifolds of a real space form of constant sectional curvature , endowed with semi-symmetric metric connection. Using these inequalities, we derive some characterizations for such half lightlike submanifolds. Finally, Chen-Ricci inequalities are calculated. Morever, the equality cases are considered and we get some results.

Keywords

Chen inequalities;, Half lightlike submanifold;, Semi-symmetric metric connection

References

• [1] Kupeli, D. N., (1996). Singular semi-Riemannian Geometry, Kluwer Academic Publishers, Dordrecht.
• [2] Duggal, K. L., Bejancu, A., (1996). Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publishers, Dordrecht.
• [3] Duggal, K. L., Jin, D. H. (2007). Null curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scientific.
• [4] Duggal, K. L. and Sahin, B., (2010). Differential Geometry of Lightlike Submanifolds, Birkhäuser, Basel.
• [5] Friedmann, A and Schouten, J. A., (1924). Über die Geometrie der halbsymmetrischen Übertragungen, (German) Math. Z., Vol. I, 21, 211-223.
• [6] Hayden, H. A., (1932). Subspace of a space with torsion, Proceedings of the London Mathematical Society II Series, 34, 27-50.
• [7] Yano, K., (1970). On Semi-Symmetric Metric Connection, Rev. Roum. Math. Pures Et Appl., 15, 1579-1586.
• [8] Imai, T., (1972). Hypersurfaces of a Riemannian Manifold with Semi-Symmetric Metric Connection, Tensor, N.S., 23, 300-306.
• [9] Imai, T., (1972). Notes on Semi-Symmetric Metric Connection, Tensor, N.S., 24, 293-296.
• [10] Nakao, Z., (1976). Submanifolds of a Riemannian manifold with semi-symmetric metric connections, Proc. Amer. Math. Soc., 54, 261-266.
• [11] Duggal, K. L., and Sharma, R., (1986). Semi-Symmetric metric connection in a Semi-Riemannian Manifold, Indian J. Pure appl Math., 17, 1276-1283.
• [12] Konar, A. and Biswas, B., (2001). Lorentzian Manifold that Admits a type of Semi-Symmetric Metric Connection, Bull. Cal. Math., Soc., 93(5), 427-437.
• [13] Yaşar, E., Çöken, A. C., Yücesan, A., (2007). Lightlike Hypersurfaces of Semi-Riemannian Manifold with Semi-Symmetric Metric Connection, Kuweyt Journal of Science and Engineering, 34(2A), 11-24.
• [14] Akyol, M. A., Vanlı, A. T. And Fernandez, L. M., (2013). Curvature properties of a semi symmetric metric connection on S manifolds., Annales Polonici Mathematici, 107(1), 71-86. [15] Chen, B. Y., (1993). Some pinching and classification theorems for minimal submanifolds, Arch. math., (Basel), 60(6), 568-578.
• [16] Chen, B. Y., (1998). Strings of Riemannian invariants, inequalities, ideal immersions and their applications, The Third Pacific Rim Geometry Conference (Seoul, 1996), 7-60, Monogr. Geom. Topology, 25, Int. Press, Cambridge, MA,.
• [17] Chen, B. Y., (2000). Some new obstructions to minimal and Lagrangian isometric immersions, Japanese J. Math., 26, 105-127.
• [18] Chen, B. Y., (2008). 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.
• [19] Chen, B. Y., Dillen, F., Verstraelen, L. and Vrancken, V., (2000). Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces, Proc. Amer. Math. Soc., 128, 589-598.
• [20] Hong, S., Matsumoto K. and Tripathi, M. M. , (2005). Certain basic inequalities for submanifolds of locally conformal Kaehlerian space forms, SUT J. Math., 4(1), 75-94.
• [21] Kim, J. S., Choi, J., (2003). A basic inequality for submanifolds in a cosymplectic space form, Int. J. Math. Math. Sci., 9, 539-547.
• [22] Matsumoto, K., Mihai, I., Oiaga, A., (2001). Ricci curvature of submanifolds in complex space forms, Rev. Roumaine Math. Pures Appl., 46, 775-782.
• [23] Oiaga, A., Mihai, I., Chen, B. Y., (1999). Inequalities for slant submanifolds in complex space forms, Demonstratio Math., 32, 835-846.
• [24] Mihai, A. and Özgür, C., (2010). Chen inequalities for submanifolds of real space form with a semi-symmetric metric connection, Tawanese Journal of Mathematics, 14(4), 1465-1477.
• [25] Zhang, P., Zhang, L. and Song, W., (2014). 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.
• [26] Gülbahar, M., Kılıç, E. and Keleş, S., (2013). Chen-like inequalities on lightlike hypersurfaces of a Lorentzian manifold, J. Inequal. Appl., 266.
• [27] Gülbahar, M., Kılıç, E. and Keleş, S., (2013). Some inequalities on screen homothetic lightlike hypersurfaces of a Lorentzian manifold, Taiwanese Journal of Mathematics, 17(6), 2083-2100.
• [28] Poyraz, N. Ö., Doğan, B. and Yaşar, E., (2017). Chen Inequalities on Lightlike Hypersurface of a Lorentzian manifold with semi-symmetric metric connection, Int. Electronic Journal of Geometry, 10(1), 1-14.
• [29] Gülbahar, M., Kılıç, E., (2017). Some optimal inequalities for screen conformal half-lightlike submanifolds, Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, 33(2), 315-329.
• [30] Duggal, K. L. and Jin, D. H., (1999). Half lightlike submanifolds of codimension 2, Math. J. Toyama Univ, 22: 121-161.
• [31] Bejan, C. L. and Duggal, K. L., (2005). Global lightlike manifolds and harmonicity, Kodai Math. J., 28(1), 131-145.
• [32] Duggal, K. L. and Sahin, B., (2004). Screen conformal half-lightlike submanifolds, Int. J. Math. and Math. Sci., 68, 3737-3753.
• [33] Jin, D. H., (2011). 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.
• [34] Beem, J. K., Ehrlich, P. E. and Easley, K. L., (1996). Global Lorentzian geometry, Volume 202 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York.
• [35] De Smet, P. J., Dillen, F., Verstraelen, L. and Vrancken, V., (1999). A pointwise inequality in submanifold theory, Arch. Math. (Brno), 5(2), 115-128.
• [36] Tripathi, M. M., (2003). Certain Basic Inequalities for Submanifolds in ( ; ) Space, Recent Advances in Riemannian and Lorentzian Geometries, Baltimore, 187-202.