Wintgen Inequalities for Submanifolds of $\delta$-Lorentzian Trans-Sasakian Space Form

Year 2023, Volume: 11 Issue: 1, 52 - 60, 30.04.2023

Oğuzhan Bahadır

Abstract

The outline of this research article is that, $\delta$-Lorentzian trans Sasakian manifolds with a semi-symmetric-metric connection (briefly say $SSM$) have been investigated. Indeed, we obtain the expressions for Riemannian curvature tensor $\bar{R}$, Ricci curvature tensors $\bar{Ric}$ and scalar curvature $\bar{r}$ of the $\delta$-Lorentzian trans-Sasakian manifolds with a $SSM$ connection. Mainly, we discuss the generalized Wintgen inequalities for submanifolds in $\delta$-Lorentzian trans-Sasakian space form with a $SSM$ connection. Furthermore, we examine the generalized Wintgen inequality for submanifolds of $\delta$-Lorentzian trans-Sasakian space form.

Keywords

$\delta$-Lorentzian trans-Sasakian manifold, semi-symmetric metric connection, Wintgen inequalities

References

• [1] T. Takahashi, Sasakian manifolds with Pseudo -Riemannian metric, Tohoku Math.J. 21 (1969),271–290.
• [2] A. Bejancu and K. L. Duggal, Real hypersurfaces of indefinite Kaehler manifolds, Int. J. Math. Math. Sci. 16 (1993), no. 3, 545–556.
• [3] K. Matsumoto, On Lorentzian paracontact manifolds, Bull. Yamagata Univ. Nat. Science 2 (1989), 151–156.
• [4] H. Gill and K. K. Dube, Generalized CR- Submanifolds of a trans Lorentzian para Sasakian manifold, Proc. Nat. Acad. Sci. India Sec. A Phys. 2 (2006), 119–124.
• [5] S. S. Pujar and V. J. Khairnar, On Lorentzian trans-Sasakian manifold-I, Int.J.of Ultra Sciences of Physical Sciences 23(1) (2011),53–66.
• [6] M. D. Siddiqi, A. Haseeb and M. Ahmad, A Note On Generalized Ricci-Recurrent (ε,δ)- Trans-Sasakian Manifolds, Palestine J. Math. 4(1) (2015), 156–163
• [7] M. M. Tripathi, E. Kilic, S. Y. Perktas and S. Keles, Indefinite almost para-contact metric manifolds, Int. J. Math. and Math. Sci. (2010), art. id 846195, pp. 19.
• [8] S. M. Bhati, On weakly Ricci φ-symmetric δ-Lorentzian trans Sasakian manifolds, Bull. Math. Anal. Appl. 5(1) (2013), 36–43.
• [9] A. Haseeb, A. Ahamd and M. D. Siddiqi, On contact CR-submanifolds of a δ-Lorentzian trans-Sasakian manifold, Global J. Adv. Res. Class. Mod. Geom. 6(2) (2017), 73–82.
• [10] A. Friedmann and J. A. Schouten, Uber die Geometric der halbsymmetrischen Ubertragung, Math. Z. 21 (1924), 211–223.
• [11] E. Bartolotti, Sulla geometria della variata a connection affine. Ann. di Mat. 4(8) (1930), 53–101.
• [12] H. A. Hayden, Subspaces of space with torsion, Proc. London Math. Soc. 34 (1932), 27–50.
• [13] K. Yano, On semi-symmetric metric connections, Revue Roumaine De Math. Pures Appl. 15 (1970), 1579–1586.
• [14] I. E. Hirica and L. Nicolescu, Conformal connections on Lyra manifolds, ˘ Balkan J. Geom. Appl. 13 (2008), 43–49.
• [15] A. Haseeb, M. A. Khan and M. D. Siddiqi, Some results on an (ε)- Kenmotsu manifolds with a semi-symmetric semi- metric connection, Acta Mathematica Universitatis Comenianae 85(1) (2016), 9–20.
• [16] G. Pathak and U. C. De, On a semi-symmetric connection in a Kenmotsu manifold, Bull. Calcutta Math. Soc. 94 (2002), no. 4, 319–324.
• [17] A. Sharfuddin and S. I. Hussain, Semi-symmetric metric connections in almost contact manifolds, Tensor (N.S.) 30 (1976), 133–139.
• [18] M. D. Siddiqi, M. Ahmad and J. P. Ojha, CR-Submanifolds of a nearly Trans-Hyperbolic Sasakian manifold with a semi-symmetric-non-metric connection, African Diaspora Journal of Math., N.S., 17(1) (2012), 93–105.
• [19] M. M. Tripathi, On a semi-symmetric metric connection in a Kenmotsu manifold, J. Pure Math. 16 (1999), 67–71.
• [20] P. Wintgen, Sur linlit Chen-Wilmore, C. R. Acad. Sci. Paris Ser. A-B 288 (1979), A993–A995.
• [21] I. V. Guadalupe and L. Rodriguez, Normal curvature of surfaces in space forms, Pacific J. Math. 106 (1983), 95–103.
• [22] J. Ge and Z. Tang, A proof of the DDVV conjecture and its equality case, Pacific J. Math. 237 (2009), 87–95.
• [23] I. Mihai, On the generalized Wintgen inequality for lagrangian submanifolds in complex space form, Nonlinear Analysis, 95 (2014), 714–720.
• [24] S. Tanno, The automorphism groups of almost contact Riemannian manifolds, Tohoku Math.J. 21 (1969), 21–38.
• [25] A. Gray and L. M. Harvella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. 123(4) (1980), 35–58.
• [26] J. C. Marrero, The local structure of Trans-Sasakian manifolds, Annali di Mat. Pura ed Appl. 162 (1992), 77–86.
• [27] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture note in Mathematics 509, (Springer-Verlag, Berlin-New York, 1976).
• [28] J. A. Oubina, New classes of almost contact metric structures, Publ. Math. Debrecen 32 (1985), 187–193
• [29] Z. Lu, Normal scalar curvature conjecture and its applications, J. Fucnt. Anal. 261 (2011), 1284–1308