Geometric inequalities of $ \mathcal{PR} $-warped product submanifold in para-Kenmotsu manifold

Fatemah Mofarreh; S. K. Srivastava; Anuj Kumar; Akram Ali; Fatemah Mofarreh; S. K. Srivastava; Anuj Kumar; Akram Ali

doi:10.3934/math.20221069

AIMS Mathematics

2022, Volume 7, Issue 10: 19481-19509. doi: 10.3934/math.20221069

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Research article

Geometric inequalities of $ \mathcal{PR} $-warped product submanifold in para-Kenmotsu manifold

1.
Mathematical Science Department Faculty of Science Princess Nourah bint Abdulrahman University, Riyadh 11546, Saudi Arabia
2.
Department of Mathematics, Central University of Himachal Pradesh, Dharamshala 176215, Himachal Pradesh, India
3.
Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia

Received: 12 April 2022 Revised: 21 July 2022 Accepted: 31 July 2022 Published: 02 September 2022
MSC : 53C12, 53C25, 53B25, 53B30, 53D15

The main purpose of this paper is to study the properties of $ \mathcal{PR} $-semi-invariant submanifold of para-Kenmotsu manifold. We obtain the integrability conditions for the invariant distribution and anti-invariant distribution. We obtain some existence and non-existence results of $ \mathcal{PR} $-semi-invariant warped product submanifolds. We provide some necessary and sufficient conditions for $ \mathcal{PR} $-semi-invariant submanifold to be a $ \mathcal{PR} $-semi-invariant warped product submanifold in para-Kenmotsu manifold. We also derive some sharp inequalities for $ \mathcal{PR} $-semi-invariant warped product submanifold in para-Kenmotsu manifolds.
- para contact manifold,
- submanifold,
- warped product
Citation: Fatemah Mofarreh, S. K. Srivastava, Anuj Kumar, Akram Ali. Geometric inequalities of $ \mathcal{PR} $-warped product submanifold in para-Kenmotsu manifold[J]. AIMS Mathematics, 2022, 7(10): 19481-19509. doi: 10.3934/math.20221069

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Abstract

The main purpose of this paper is to study the properties of $ \mathcal{PR} $-semi-invariant submanifold of para-Kenmotsu manifold. We obtain the integrability conditions for the invariant distribution and anti-invariant distribution. We obtain some existence and non-existence results of $ \mathcal{PR} $-semi-invariant warped product submanifolds. We provide some necessary and sufficient conditions for $ \mathcal{PR} $-semi-invariant submanifold to be a $ \mathcal{PR} $-semi-invariant warped product submanifold in para-Kenmotsu manifold. We also derive some sharp inequalities for $ \mathcal{PR} $-semi-invariant warped product submanifold in para-Kenmotsu manifolds.

References

[1]	R. L. Bishop, B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1–49. https://doi.org/10.2307/1995057 doi: 10.2307/1995057
[2]	G. I. Kruchkovich, On motions in semi-reducible Riemann spaces, Usp. Mat. Nauk, 12 (1957), 149–156.
[3]	B. Y. Chen, Geometry of warped product $\mathcal{CR}$-submanifolds in Kaehler manifolds, Mh. Math., 133 (2001), 177–195. https://doi.org/10.1007/s006050170019 doi: 10.1007/s006050170019
[4]	B. Y. Chen, Geometry of warped product $\mathcal{CR}$-submanifolds in Kaehler manifolds, II, Mh. Math., 134 (2001), 103–119. https://doi.org/10.1007/s006050170002 doi: 10.1007/s006050170002
[5]	I. Hasegawa, I. Mihai, Contact $\mathcal{CR}$-warped product submanifolds in Sasakian manifolds, Geom. Dedicata, 102 (2003), 143–150. https://doi.org/10.1023/B:GEOM.0000006582.29685.22 doi: 10.1023/B:GEOM.0000006582.29685.22
[6]	V. A. Khan, K. A. Khan, A note on warped product submanifolds of Kenmotsu manifolds, Math. Slovaca, 61 (2011), 79–92. https://doi.org/10.2478/s12175-010-0061-3 doi: 10.2478/s12175-010-0061-3
[7]	S. Uddin, K. A. Khan, An inequality for contact $\mathcal{CR}$-warped product submanifolds of nearly cosymplectic manifolds, J. Inequal. Appl., 2012 (2012), 1–7. https://doi.org/10.1186/1029-242X-2012-304 doi: 10.1186/1029-242X-2012-304
[8]	F. R. Al-Solamy, M. A. Khan, Semi-invariant warped product submanifolds of almost contact manifolds, J. Inequal. Appl., 2012 (2012), 1–12. https://doi.org/10.1186/1029-242X-2012-127 doi: 10.1186/1029-242X-2012-127
[9]	I. Mihai, Contact $\mathcal{CR}$-warped product submanifolds in Sasakian space forms, Geom. Dedicata, 109 (2004), 165–173. https://doi.org/10.1007/s10711-004-5459-z doi: 10.1007/s10711-004-5459-z
[10]	S. Uddin, Geometry of warped product semi-slant submanifolds of Kenmotsu manifolds, Bull. Math. Sci., 8 (2018), 435–451. https://doi.org/10.1007/s13373-017-0106-9 doi: 10.1007/s13373-017-0106-9
[11]	B. Y. Chen, M. I. Munteanu, Geometry of PR-warped products in para-Keahler manifolds, Taiwanese J. Math., 16 (2012), 1293–1327.
[12]	S. K. Srivastava, A. Sharma, Geometry of ${\mathcal {P}\mathcal {R}}$-semi-invariant warped product submanifolds in paracosymplectic manifold, J. Geom., 108 (2017), 61–74. https://doi.org/10.1007/s00022-016-0325-3 doi: 10.1007/s00022-016-0325-3
[13]	S. K. Srivastava, A. Sharma, Pointwise pseudo-slant warped product submanifolds in a K$\ddot{a}$hler manifold, Mediterr. J. Math., 14 (2017), 20. https://doi.org/10.1007/s00009-016-0832-3 doi: 10.1007/s00009-016-0832-3
[14]	M. Dhiman, A. Kumar, S. K. Srivastava, $\mathcal {PR}$-semi slant warped product submanifold of paraKenmotsu manifolds, Results Math., 77 (2022), 1–30. https://doi.org/10.1007/s00025-022-01695-4 doi: 10.1007/s00025-022-01695-4
[15]	S. K. Srivastava, F. Mofarreh, A. Kumar, A. Ali, Characterizations of $\mathcal {PR} $-pseudo-slant warped product submanifold of para-Kenmotsu manifold with slant base, Symmetry, 14 (2022), 1001. https://doi.org/10.3390/sym14051001 doi: 10.3390/sym14051001
[16]	K. Srivastava, S. K. Srivastava, On a class of $\alpha$-para Kenmotsu manifolds, Mediterr. J. Math., 13 (2016), 391–399. https://doi.org/10.1007/s00009-014-0496-9 doi: 10.1007/s00009-014-0496-9
[17]	S. Zamkovoy, G. Nakova, The decomposition of almost paracontact metric manifolds in eleven classes revisited, J. Geom., 109 (2018), 1–23. https://doi.org/10.1007/s00022-018-0423-5 doi: 10.1007/s00022-018-0423-5
[18]	J. Nash, The imbedding problem for Riemannian manifolds, Ann. Math., 63 (1956), 20–63. https://doi.org/10.2307/1969989 doi: 10.2307/1969989
[19]	B. Y. Chen, Differential geometry of warped product manifolds and submanifolds, World Scientific, 2017.
[20]	S. Hiepko, Eine innere Kennzeichnung der verzerrten Produkte, Math. Ann., 241 (1979), 209–215. https://doi.org/10.1007/BF01421206 doi: 10.1007/BF01421206

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