Generalized warped product submanifolds of Lorentzian concircular structure manifolds

Tanumoy Pal; Ibrahim Al-Dayel; Meraj Ali Khan; Biswabismita Bag; Shyamal Kumar Hui; Foued Aloui; Tanumoy Pal; Ibrahim Al-Dayel; Meraj Ali Khan; Biswabismita Bag; Shyamal Kumar Hui; Foued Aloui

doi:10.3934/math.2024877

AIMS Mathematics

2024, Volume 9, Issue 7: 17997-18012. doi: 10.3934/math.2024877

Previous Article Next Article

Research article

Generalized warped product submanifolds of Lorentzian concircular structure manifolds

1.
A. M. J. High School, Mankhamar, Bankura-722144, West Bengal, India
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box-65892, Riyadh 11566, Saudi Arabia
3.
Department of Mathematics, The University of Burdwan, Golapbag, Burdwan-713104, West Bengal, India

Received: 15 March 2024 Revised: 14 May 2024 Accepted: 15 May 2024 Published: 27 May 2024
MSC : 53C15, 53C25, 53C40

We began by considering invariant, anti-invariant, proper slant, and pointwise slant submanifolds of a Lorentzian concircular structure manifold. Subsequently, we explored two distinct categories of warped product submanifolds. The first category encompassed the fiber submanifold as an anti-invariant submanifold, while the second category included the fiber submanifold as a pointwise slant submanifold. We established several fundamental results concerning these submanifold classes. Additionally, we demonstrated the existence of such submanifold classes through specific examples. Moreover, we derived inequalities for the squared norm of the second fundamental form.
- (LCS)_n-manifold,
- skew CR-submanifold,
- warped product
Citation: Tanumoy Pal, Ibrahim Al-Dayel, Meraj Ali Khan, Biswabismita Bag, Shyamal Kumar Hui, Foued Aloui. Generalized warped product submanifolds of Lorentzian concircular structure manifolds[J]. AIMS Mathematics, 2024, 9(7): 17997-18012. doi: 10.3934/math.2024877

Related Papers:

Abstract

We began by considering invariant, anti-invariant, proper slant, and pointwise slant submanifolds of a Lorentzian concircular structure manifold. Subsequently, we explored two distinct categories of warped product submanifolds. The first category encompassed the fiber submanifold as an anti-invariant submanifold, while the second category included the fiber submanifold as a pointwise slant submanifold. We established several fundamental results concerning these submanifold classes. Additionally, we demonstrated the existence of such submanifold classes through specific examples. Moreover, we derived inequalities for the squared norm of the second fundamental form.

References

[1]	M. Atceken, S. K. Hui, Slant and pseudo-slant submanifolds in LCS-manifolds, Czech. Math. J., 63 (2013), 177–190. https://doi.org/10.1007/s10587-013-0012-6 doi: 10.1007/s10587-013-0012-6
[2]	R. L. Bishop, B. O'Neill, Manifolds of negative curvature, T. Am. Math. Soc., 145 (1969), 1–49.
[3]	B. Y. Chen, Slant immersions, B. Aust. Math. Soc., 41 (1990), 135–147. https://doi.org/10.1017/S0004972700017925
[4]	B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifold, Mh. Math., 133 (2001), 177–195. https://doi.org/10.1007/s006050170019 doi: 10.1007/s006050170019
[5]	B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds II, Mh. Math., 134 (2001), 103–119. https://doi.org/10.1007/s006050170002 doi: 10.1007/s006050170002
[6]	B. Y. Chen, Differential geometry of warped product manifolds and submanifolds, Hackensack: World Scientific, 2017.
[7]	F. Etayo, On quasi-slant submanifolds of an almost Hermitian manifold, Publ. Math. Debrecen, 53 (1998), 217–223.
[8]	S. K. Hui, T. Pal, L. I. Piscoran, Characterization of warped product submanifolds of Lorentzian concircular structure manifolds, arXiv Preprint, 2018. https://doi.org/10.48550/arXiv.1803.02526
[9]	A. Lotta, Slant submanifolds in contact geometry, Bull. Math. Soc. Sci. Math. Roumanie, 39 (1996), 183–198.
[10]	C. A. Mantica, L. G. Molinari, A note on concircular structure structure space-times, arXiv Preprint, 2018.
[11]	K. Matsumoto, On Lorentzian almost paracontact manifolds, Bull. Yamagata Univ. Nat. Sci., 12 (1989), 151–156.
[12]	I. K. Mihai, R. H. Qjha, On Lorentzian para-Sasakian manifolds, Singapore: World Scientific, 2001.
[13]	I. Mihai, S. Uddin, A. Mihai, Warped product pointwise semi-slant submanifolds of Sasakian manifolds, Kragujev. J. Math., 45 (2021), 721–738.
[14]	A. Mustafa, S. Uddin, V. Khan, B. Wong, Contact CR-warped product submanifolds of nearly trans Sasakian manifolds, Taiwanese J. Math., 17 (2013), 1473–1486. https://doi.org/10.11650/tjm.17.2013.2601 doi: 10.11650/tjm.17.2013.2601
[15]	M. F. Naghi, I. Mihai, S. Uddin, F. R. Al-Solamy, Warped product skew CR-submanifolds of Kenmotsu manifolds and their applications, Filomat, 32 (2018), 3505–3528. https://doi.org/10.2298/FIL1810505N doi: 10.2298/FIL1810505N
[16]	B. O'Neill, Semi Riemannian Geometry with applications to relativity, New York: Academic Press, 1983.
[17]	K. S. Park, Pointwise slant and pointwise semi slant submanifolds in almost contact metric manifold, arXiv Preprint, 2014.
[18]	A. A. Shaikh, On Lorentzian almost paracontact manifolds with a structure of the concircular type, Kyungpook Math. J., 43 (2003), 305–314.
[19]	A. A. Shaikh, K. K. Baishya, On concircular structure spacetimes, J. Math. Stat., 1 (2005), 129–132.
[20]	A. A. Shaikh, K. K. Baishya, On concircular structure spacetimes II, American J. Appl. Sci., 3 (2006), 1790–1794.
[21]	S. K. Srivastava, A. Sharma, Pointwise semi-slant warped product submanifold in a Lorentzian paracosympletic manifold, arXiv Preprint, 2016.
[22]	S. Uddin, Warped product CR-submanifolds of LP-cosymplectic manifolds, Filomat, 24 (2010), 87–95. https://doi.org/10.2298/FIL1001087U doi: 10.2298/FIL1001087U
[23]	S. Uddin, L. S. Alqahtani, A. H. Alkhaldi, F. Y. Mofarreh, CR-slant warped product submanifolds in nearly Kaehler manifolds, Int. J. Geom. Method M., 17 (2020), 2050003. https://doi.org/10.1142/S0219887820500036 doi: 10.1142/S0219887820500036
[24]	S. Uddin, V. A. Khan, K. A. Khan, Warped product submanifolds of a Kenmotsu manifold, Turk. J. Math., 36 (2012), 319–330. https://doi.org/10.3906/mat-0912-42 doi: 10.3906/mat-0912-42
[25]	S. Uddin, M. Z. Ullah, A geometric obstruction for CR-slant warped products in a nearly cosympletic manifolds, Mathematics, 8 (2020), 1622. https://doi.org/10.3390/math8091622 doi: 10.3390/math8091622

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)