Optimization on Submanifolds of $\delta$-Lorentzian trans-Sasakian Manifolds with Casorati Curvatures


  • Aliya Naaz Siddiqui Department of Mathematics, Maharishi Markandeshwar Deemed to be University, Mullana, 133207, Ambala-Haryana, India
  • Mohd Danish Siddiqi Department of Mathematics, College of Science, Jazan University, Jazan, Kingdom of Saudi Arabia
  • Mohammad Hasan Shahid Department of Mathematics, Jamia Millia Islamia, New Delhi- 110025, India




δ-Lorentzian trans-Sasakian manifold, semi-symmetric metric connec- tion, Casorati curvatures.


The present research article is concerned about a couple of optimal inequalities for submanifolds of $\delta$-Lorentzian trans-Sasakian manifolds endowed with semi-symmetric metric connection (briefly says $SSM$). Some examples of $\delta$-Lorentzian trans-Sasakiam manifolds are also discussed here. This paper ends with some open problems.


Author Biographies

Aliya Naaz Siddiqui, Department of Mathematics, Maharishi Markandeshwar Deemed to be University, Mullana, 133207, Ambala-Haryana, India



Mohammad Hasan Shahid, Department of Mathematics, Jamia Millia Islamia, New Delhi- 110025, India




S. M. Bhati, On weakly Ricci φ-symmetric δ-Lorentzian trans-Sasakian manifolds, Bull. Math. Anal. Appl., 5 (2013), 36-43.

E. Bartolotti, Sulla geometria della variata a connection affine, Ann. di Mat., 4 (1930), 53-101.

A. Bejancu and K. L. Duggal, Real hypersurfaces of indefinite Kaehler manifolds, Int. J. Math. Math. Sci., 16 (1993), 545-556.

D. E. Blair, Contact manifolds in Riemannian geometry, Lecture note in Mathematics, Springer-Verlag Berlin-New York, 509 (1976).

B.-Y. Chen, Recent developments in δ-Casorati curvature invariants, Turk. J. Math., 45 (2021), 1-46.

S. Decu, S. Haesen and L. Verstraelen, Optimal inequalities involving Casorati curvatures, Bull. Transilv. Univ. Brasov Ser. B., 14 (2007), 85-93.

A. Friedmann and J. A. Schouten, Uber die Geometric der halbsymmetrischen Ubertragung, Math. Z., 21 (1924), 211-223.

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.

A. Gray and L. M. Harvella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl., 123 (1980), 35-58.

H. A. Hayden, Subspaces of space with torsion, Proc. London Math. Soc., 34 (1932), 27-50.

I. E. Hirică and L. Nicolescu, Conformal connections on Lyra manifolds, Balkan J. Geom. Appl., 13 (2008), 43-49.

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 (2017), 73-82.

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, (2016), 9-20.

J. B. Jun, U. C. De and G. Pathak, On Kenmotsu manifolds, J. Korean Math. Soc., 42 (2005), 435-445.

C. W. Lee, D. W. Yoon and J. W. Lee, Optimal inequalities for the Casorati curvatures of submanifolds of real space forms endowed with semi-symmetric metric connections, J. Inequal. Appl., 2014, 327 (2014).

J. C. Marrero, The local structure of trans-Sasakian manifolds, Annali di Mat. Pura ed Appl., 162 (1992), 77-86.

K. Matsumoto, On Lorentzian paracontact manifolds, Bull. Yamagata Univ. Nat. Science, 2 (1989), 151-156.

T. Oprea, Optimization methods on Riemannian submanifolds, An. Univ. Bucur., Mat., 54 (2005), 127-136.

T. Oprea, Chen’s inequality in the Lagrangian case, Colloq. Math., 108 (2007), 163-169.

T. Oprea, Ricci curvature of Lagrangian submanifolds in complex space forms, Math. Inequal. Appl., 13 (2010), 851-858.

J. A. Oubina, New classes of almost contact metric structures, Publ. Math. Debrecen, 32 (1985), 187-193.

G. Pathak and U. C. De, On a semi-symmetric connection in a Kenmotsu manifold, Bull. Calcutta Math. Soc., 94 (2002), 319-324.

S. S. Pujar and V. J. Khairnar, On Lorentzian trans-Sasakian manifold-I, Int. J. Ultra Sciences of Physical Sciences, 23 (2011), 53-66.

A. Sharfuddin and S. I. Hussain, Semi-symmetric metric connections in almost contact manifolds, Tensor (N.S.), 30 (1976), 133-139.

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 (2012), 93-105.

M. D. Siddiqi, A. Haseeb and M. Ahmad, A Note On Generalized Ricci-Recurrent (ϵ, δ)-trans-Sasakian Manifolds, Palestine J. Math., 4 (2015), 156-163

A. N. Siddiqui, Upper bound inequalities for δ-Casorati curvatures of submanifolds in generalized Sasakian space forms admitting a semi-symmetric metric connection, Inter. Elec. J. Geom., 11 (2018), 57-67.

S. Tanno, The automorphism groups of almost contact Riemannian manifolds, Tohoku Math. J., 21 (1969), 21-38.

K. Yano, On semi-symmetric metric connections, Revue Roumaine De Math. Pures Appl., 15 (1970), 1579-1586.

P. Zhang and L. Zhang, Remarks on inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms, J. Inequal. App., 2014, 2014:452.




How to Cite

Siddiqui, A. N. ., Siddiqi, M. D., & Shahid, M. H. (2021). Optimization on Submanifolds of $\delta$-Lorentzian trans-Sasakian Manifolds with Casorati Curvatures. Tamkang Journal of Mathematics, 53. https://doi.org/10.5556/j.tkjm.53.2022.4075