Strongly minimal complex lightlike hypersurfaces

Year 2022, , 958 - 969, 01.08.2022

Mehmet Gülbahar , Erol Kılıç

https://doi.org/10.15672/hujms.631290

Cited By: 1

Abstract

In this paper, complex lightlike hypersurfaces of an indefinite Kähler manifold are studied. An optimal inequality characterized to strongly minimality for coisotropic lightlike submanifolds is proved. Strongly minimal Monge-type hypersurfaces in $\mathcal{C}_{1}^{4}$ are examined and some examples of these hypersurfaces are given.

Keywords

minimal submanifold, lightlike submanifold, indefinite Kähler manifold

References

• [1] F.R. Al-Solamy and S. Uddin, An inequality for warped product submanifolds of a locally product Riemannian manifold, Hacet. J. Math. Stat. 48 (2), 351-358, 2019.
• [2] P. Alegre, B.Y. Chen and M.I. Munteanu, Riemannian submersions, $\delta$ -invariants, and optimal inequality, Ann. Global Anal. Geom. 42 (3), 317-331, 2012.
• [3] C. Atindogbe and K.L. Duggal, Conformal screen on lightlike hypersurfaces, Int. J. Pure Appl. Math. 11, 421442, 2004.
• [4] C.L. Bejan and K.L. Duggal, Global lightlike manifolds and harmonicity, Kodai Math. J. 28, 131-145, 2005.
• [5] B.Y. Chen, A Riemannian invariant for submanifolds in space forms and its applications, World Sci. Publ., River Edge, NJ, 6, 58-81, 1994.
• [6] B.Y. Chen, A series of Kählerian invariants and their applications to Kählerian geometry, Beitr. Algebra Geom. 42, 165-178, 2001.
• [7] B.Y. Chen, Pseudo-Riemannian geometry, $\delta$-invariants and applications, World Scientific 2011.
• [8] S.S. Chern, Minimal submanifolds in a Riemannian manifold, University of Kansas, Department of Mathematics 19, 1968.
• [9] R. Deszcz, M. Gogowska, M. Petrović-Torgašev and L. Verstraelen, On the Roter type of Chen ideal submanifolds, Results Math. 59, 401-413, 2011.
• [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, 49-68, 2003.
• [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, Screen Cauchy Riemann lightlike submanifolds, Acta Math. Hungar. 106 (1), 137-165, 2005.
• [14] K.L. Duggal and B. Sahin, Generalized Cauchy Riemann lightlike submanifolds, Acta Math. Hungar. 112 (1-2), 107-130, 2006.
• [15] K.L. Duggal and B. Sahin, Differential geometry of lightlike submanifolds, Birkhäuser Verlag AG. 2010.
• [16] M. Gülbahar, E. Kılıç and S. Keleş, Chen-like inequalities on lightlike hypersurfaces of a Lorentzian manifold, J. Inequal. Appl. 266, 2013.
• [17] D.N. Kupeli, Singular Semi-Riemannian Geometry, Kluwer Academic 1996.
• [18] C. Özgür and C. Murathan, Chen inequalities for submanifolds of a cosymplectic space form with a semi-symmetric metric connection, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), 58 (2), 395-408, 2012.
• [19] B. Sahin, Chen’s first inequality for Riemannian maps, Ann. Polon. Math. 117, 249- 258, 2016.
• [20] Z. Şentürk and L. Verstraelen, Chen ideal Kaehler hypersurfaces, Taiwan. J. Math. 12 (7), 1597-1608, 2008.
• [21] V. Slesar, B. Sahin and G.E. Vilcu, Inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms, J. nequal. Appl. 123, 2014.
• [22] B.D. Suceavă, On strongly minimal Kähler surfaces in ${\mathbf {C}^ 3} $ and the equality ${scal (p)= 4\inf sec (\pi^ r)}$, Results Math. 68 (1-2), 45-69, 2015.
• [23] T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18, 380-385, 1966.
• [24] F. Tricerri and L. Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Am. Math. Soc. 267, 365-398, 1981.
• [25] K. Yano and M. Kon, Structures on manifolds, Series in Pure Mathematics 3, 1984.
• [26] P. Zhang and L. Zhang, Inequalities for Casorati curvatures of submanifolds in real space forms, Adv. Geom. 16 (3), 329-335, 2016.