Ricci curvature for pointwise semi-slant warped products in non-Sasakian generalized Sasakian space forms and its applications

Year 2022, Volume: 51 Issue: 6, 1535 - 1549, 01.12.2022

Mohd Aquib , Siraj Uddin , M. Hasan Shahid

https://doi.org/10.15672/hujms.1034883

Cited By: 2

https://izlik.org/JA66XW25UJ

Abstract

We find Ricci curvature bounds for pointwise semi-slant warped products submanifolds in non-Sasakian generalized Sasakian space forms in this work, and analyze the equality case of the inequality. The derived inequality is also used to develop a number of applications.

Keywords

warped products , pointwise semi-slant submanifolds , generalized Sasakian space forms , Chen-Ricci inequality , harmonic function , Hessian function , Dirichlet energy

References

• [1] P. Alegre, D.E. Blair and A. Carriazo, Generalized Sasakian space forms, Israel J. Math. 141, 157–183, 2004.
• [2] P. Alegre and A. Carriazo, Structures on generalized Sasakian-space-forms, Differ. Geom. Appl. 26 (6), 656–666, 2008.
• [3] P. Alegre and A. Carriazo, Submanifolds of generalized Sasakian space forms, Taiwan. J. Math. 13, 923–941, 2009.
• [4] P. Alegre and A. Carriazo, Generalized Sasakian space forms and conformal changes of the metric, Results Math. 59, 485–493, 2011.
• [5] R. Al-Ghefari, F.R. Al-Solamy and M.H. Shahid, CR-submanifolds of generalized Sasakian space-forms, JP J. Geom. Topol. 6, 151–166, 2006.
• [6] M. Aquib, M.N. Boyom, M.H. Shahid and G. Vîlcu, The first fundamental equation and generalized wintgen-type inequalities for submanifolds in generalized space forms, Mathematics, 7, (2019) 1151. doi:10.3390/math7121151.
• [7] M. Atçeken , Contact CR-warped product submanifolds in Sasakian space forms, Hacet. J. Math. Stat. 44 (1), 23–32, 2015.
• [8] R.L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145, 1–49, 1967.
• [9] B.Y. Chen, On isometric minimal immersions from warped products into real space forms, Proc. Edinb. Math. Soc. 45, 579–587, 2002.
• [10] B.Y. Chen, Geometry of warped products as Riemannian submanifolds and related problem, Soochow J. Math. 28, 125–157, 2002.
• [11] B.Y. Chen, CR-warped products in complex projective spaces with compact holomorphic factor, Monatsh. Math. 141 (3), 177–186, 2004.
• [12] B.Y. Chen, Geometry of warped product submanifolds: a survey, J. Adv. Math. Stud. 6 (2), 1–43, 2013.
• [13] B.Y. Chen, Pseudo-Riemannian geometry, δ-invariants and applications, World Scientific, 2011.
• [14] A. Mustafa, A. De and S. Uddin, Characterization of warped product submanifolds in Kenmotsu manifolds, Balkan. J. Geom. Appl. 20, 86–97, 2015.
• [15] M. Falcitelli, Locally conformal C6-manifolds and generalized Sasakian-space-forms, Mediterr. J. Math. 7, 19–36, 2010.
• [16] S.K. Hui, S. Uddin, A. Alkhaldi and P. Mandal, Invariant submanifolds of generalized Sasakian-space-forms, Int. J. Geom. Methods Mod. Phys. 15 (9), 2018.
• [17] U.K. Kim, Conformally flat generalized Sasakian-space-forms and locally symmetric generalized Sasakian-space-forms, Note Mat. 26, 55–67, 2006.
• [18] F. Malek and V. Nejadakbary, A lower bound for the Ricci curvature of submanifolds in generalized Sasakian space forms, Adv. Geom. 13, 695–711, 2013.
• [19] M.I. Munteanu, Warped product contact CR-submanifolds of Sasakian space forms, Publ. Math. Debrecen, 66, 75–120, 2005.
• [20] C. Ozgur and S. Guvenc, On some classes of biharmonic Legendre curves in generalized Sasakian space forms, Collect. Math. 65, 203–218, 2014.
• [21] S. Uddin, B.Y. Chen and F.R. Al-Solamy, Warped product bi-slant immersions in Kaehler manifolds, Mediterr. J. Math. 14 (2), (2017) 11.
• [22] S. Uddin and F.R. Al-Solamy, Warped product pseudo-slant immersions in Sasakian manifolds, Publ.Math. Debrecen 91 ((3-4)), 331–348 2017.