$ N(\kappa) $-paracontact metric manifolds admitting the Fischer-Marsden conjecture

Sudhakar Kumar Chaubey; Meraj Ali Khan; Amna Salim Rashid Al Kaabi; Sudhakar Kumar Chaubey; Meraj Ali Khan; Amna Salim Rashid Al Kaabi

doi:10.3934/math.2024111

AIMS Mathematics

2024, Volume 9, Issue 1: 2232-2243. doi: 10.3934/math.2024111

Previous Article Next Article

Research article Special Issues

$ N(\kappa) $-paracontact metric manifolds admitting the Fischer-Marsden conjecture

1.
Department of Information Technology, University of Technology and Applied Sciences, Shinas, P.O. Box 77, Postal Code 324, Oman
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia

Received: 18 September 2023 Revised: 04 December 2023 Accepted: 05 December 2023 Published: 21 December 2023
MSC : 53B30, 53C15, 53C25

We characterize $ N(\kappa) $-paracontact metric manifolds (NKPMM) $ M^{2n+1} $ satisfying the Fischer-Marsden conjecture. We demostrate that, if an $ M^{2n+1} $ satisfies the Fischer-Marsden equation, then either $ M^{2n+1} $ with $ \kappa > -1 $ is a non-Einstein manifold or $ M^{2n+1} $ is locally isometric to $ \mathbb{E}^{n+1} \times \mathbb{H}^{n}(-4) $ for $ n > 1 $. For the $ 3 $-dimensional case, we show that $ M^3 $ is an Einstein manifold.
- $ N(\kappa) $-paracontact metric manifolds,
- Fischer-Marsden conjecture,
- Einstein manifold
Citation: Sudhakar Kumar Chaubey, Meraj Ali Khan, Amna Salim Rashid Al Kaabi. $ N(\kappa) $-paracontact metric manifolds admitting the Fischer-Marsden conjecture[J]. AIMS Mathematics, 2024, 9(1): 2232-2243. doi: 10.3934/math.2024111

Related Papers:

Abstract

We characterize $ N(\kappa) $-paracontact metric manifolds (NKPMM) $ M^{2n+1} $ satisfying the Fischer-Marsden conjecture. We demostrate that, if an $ M^{2n+1} $ satisfies the Fischer-Marsden equation, then either $ M^{2n+1} $ with $ \kappa > -1 $ is a non-Einstein manifold or $ M^{2n+1} $ is locally isometric to $ \mathbb{E}^{n+1} \times \mathbb{H}^{n}(-4) $ for $ n > 1 $. For the $ 3 $-dimensional case, we show that $ M^3 $ is an Einstein manifold.

References

[1]	S. Kaneyuli, F. L. Williams, Almost paracontact and parahodge structures on manifolds, Nagoya Math. J., 99 (1985), 173–187. https://doi.org/10.1017/s0027763000021565 doi: 10.1017/s0027763000021565
[2]	S. Zamkovoy, Canonical connections on paracontact manifolds, Ann. Glob. Anal. Geom., 36 (2009), 37–60. https://doi.org/10.1007/s10455-008-9147-3 doi: 10.1007/s10455-008-9147-3
[3]	G. Calvaruso, V. Martin-Molina, Paracontact metric structures on the unit tangent sphere bundle, Ann. Mat. Pura Appl., 194 (2015), 1359–1380. https://doi.org/10.1007/s10231-014-0424-4 doi: 10.1007/s10231-014-0424-4
[4]	B. Cappelletti-Montano, Bi-paracontact structures and Legendre foliations, Kodai Math. J., 33 (2010), 473–512. https://doi.org/10.2996/kmj/1288962554 doi: 10.2996/kmj/1288962554
[5]	A. M. Blaga, $\eta$-Ricci solitons on Lorentzian para-Sasakian manifolds, Filomat, 30 (2016), 489–496. https://doi.org/10.2298/FIL1602489B doi: 10.2298/FIL1602489B
[6]	G. Calvaruso, Homogeneous paracontact metric three-manifolds, Illinois J. Math., 55 (2011), 697–718. https://doi.org/10.1215/ijm/1359762409 doi: 10.1215/ijm/1359762409
[7]	G. Calvaruso, A. Perrone, Ricci solitons in three-dimensional paracontact geometry, J. Geom. Phys., 98 (2015), 1–12. https://doi.org/10.1016/j.geomphys.2015.07.021 doi: 10.1016/j.geomphys.2015.07.021
[8]	I. K. Erken, Yamabe solitons on three-dimensional normal almost paracontact metric manifolds, Period. Math. Hungar., 80 (2020), 172–184. https://doi.org/10.1007/s10998-019-00303-3 doi: 10.1007/s10998-019-00303-3
[9]	Y. J. Suh, K. Mandal, Yamabe solitons on three-dimensional $N(k)$-paracontact metric manifolds, Bull. Iran. Math. Soc., 44 (2018), 183–191. http://doi.org/10.1007/s41980-018-0013-1 doi: 10.1007/s41980-018-0013-1
[10]	B. Cappelletti-Montano, I. Küpeli Erken, C. Murathan, Nullity conditions in paracontact geometry, Differ. Geom. Appl., 30 (2012), 665–693. https://doi.org/10.1016/j.difgeo.2012.09.006 doi: 10.1016/j.difgeo.2012.09.006
[11]	B. Cappelletti-Montano, A. Carriazo, V. Martin-Molina, Sasaki-Einstein and paraSasaki-Einstein metrics from $(\kappa, \mu)$-stuctures, J. Geom. Phys., 73 (2013), 20–36. https://doi.org/10.1016/j.geomphys.2013.05.001 doi: 10.1016/j.geomphys.2013.05.001
[12]	B. Cappelletti-Montano, L. Di Terlizzi, Geometric structures associated to a contact metric $(\kappa, \mu)$-space, Pacific J. Math., 246 (2010), 257–292. https://doi.org/10.2140/pjm.2010.246.257 doi: 10.2140/pjm.2010.246.257
[13]	S. Kaneyuki, M. Kozai, Paracomplex structures and affine symmetric spaces, Tokyo J. Math., 8 (1985), 81–98. https://doi.org/10.3836/tjm/1270151571 doi: 10.3836/tjm/1270151571
[14]	D. G. Prakasha, K. K. Mirji, On $\phi$-symmetric $N(k)$-paracontact metric manifolds, J. Math., 2015 (2015), 728298. https://doi.org/10.1155/2015/728298 doi: 10.1155/2015/728298
[15]	V. Martin-Molina, Paracontact metric manifolds without a contact metric counterpart, Taiwanese J. Math., 19 (2015), 175–191. https://doi.org/10.11650/tjm.19.2015.4447 doi: 10.11650/tjm.19.2015.4447
[16]	V. Martin-Molina, Local classification and examples of an important class of paracontact metric manifolds, Filomat, 29 (2015), 507–515. https://doi.org/10.2298/FIL1503507M doi: 10.2298/FIL1503507M
[17]	J. P. Bourguignon, Une stratification de l'espace des structures riemanniennes, Compos. Math., 30 (1975), 1–41.
[18]	A. E. Fischer, J. E. Marsden, Manifolds of Riemannian metrics with prescribed scalar curvature, Bull. Amer. Math. Soc., 80 (1974), 479–484. https://doi.org/10.1090/S0002-9904-1974-13457-9 doi: 10.1090/S0002-9904-1974-13457-9
[19]	J. Corvino, Scalar curvature deformations and a gluing construction for the Einstein constraint equations, Commun. Math. Phys., 214 (2000), 137–189. https://doi.org/10.1007/PL00005533 doi: 10.1007/PL00005533
[20]	I. Al-Dayal, S. Deshmukh, M. D. Siddiqi, A characterization of GRW spacetimes, Mathematics, 9 (2021), 2209. https://doi.org/10.3390/math9182209 doi: 10.3390/math9182209
[21]	O. Kobayashi, A differential equation arising from scalar curvature function, J. Math. Soc. Japan, 34 (1982), 665–675. https://doi.org/10.2969/jmsj/03440665 doi: 10.2969/jmsj/03440665
[22]	P. Cernea, D. Guan, Killing fields generated by multiple solutions to the Fischer-Marsden equation, Int. J. Math., 26 (2015), 93–111. https://doi.org/10.1142/S0129167X15400066 doi: 10.1142/S0129167X15400066
[23]	D. S. Patra, A. Ghosh, The Fischer-Marsden conjecture and contact geometry, Period. Math. Hungar., 76 (2018), 207–216. https://doi.org/10.1007/s10998-017-0220-1 doi: 10.1007/s10998-017-0220-1
[24]	D. G. Prakasha, P. Veeresha, Venkatesha, The Fischer-Marsden conjecture on non-Kenmotsu $(\kappa, \mu)$-almost Kenmotsu manifolds, J. Geom., 110 (2019), 1. https://doi.org/10.1007/s00022-018-0457-8 doi: 10.1007/s00022-018-0457-8
[25]	S. K. Chaubey, U. C. De, Y. J. Suh, Kenmotsu manifolds satisfying the Fischer-Marsden equation, J. Korean Math. Soc., 58 (2021), 597–607. https://doi.org/10.4134/JKMS.j190602 doi: 10.4134/JKMS.j190602
[26]	S. K. Chaubey, Y. J. Suh, Ricci-Bourguignon solitons and Fischer-Marsden conjecture on generalized Sasakian-space-forms with $\beta$-Kenmotsu structure, J. Korean Math. Soc., 60 (2023), 341–358. https://doi.org/10.4134/JKMS.j220057 doi: 10.4134/JKMS.j220057
[27]	S. K. Chaubey, G. E. Vîlcu, Gradient Ricci solitons and Fischer-Marsden equation on cosymplectic manifolds, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 116 (2022), 186. https://doi.org/10.1007/s13398-022-01325-2 doi: 10.1007/s13398-022-01325-2
[28]	S. Deshmukh, H. Al-Sodais, G. E. Vîlcu, A note on some remarkable differential equations on a Riemannian manifold, J. Math. Anal. Appl., 519 (2023), 126778. https://doi.org/10.1016/j.jmaa.2022.126778 doi: 10.1016/j.jmaa.2022.126778
[29]	D. H. Hwang, Y. J. Suh, Fischer-Marsden conjecture for hypersurfaces of the complex hyperbolic space, J. Geom. Phys., 186 (2023), 104768. https://doi.org/10.1016/j.geomphys.2023.104768 doi: 10.1016/j.geomphys.2023.104768
[30]	Y. J. Suh, Fischer-Marsden conjecture on real hypersurfaces in the complex quadric, J. Math. Pures Appl., 177 (2023), 129–153. http://doi.org/10.1016/j.matpur.2023.06.010 doi: 10.1016/j.matpur.2023.06.010
[31]	Y. J. Suh, Fischer-Marsden conjecture on real hypersurfaces in the complex hyperbolic two-plane Grassmannians, Anal. Math. Phys., 12 (2022), 126. http://doi.org/10.1007/s13324-022-00738-x doi: 10.1007/s13324-022-00738-x
[32]	V. Venkatesha, D. M. Naik, M. Devaraja, H. A. Kumara, Real hypersurfaces of complex space forms satisfying Fischer-Marsden equation, Ann. Univ. Ferrara, 67 (2021), 203–216. https://doi.org/10.1007/s11565-021-00361-x doi: 10.1007/s11565-021-00361-x
[33]	U. C. De, S. Deshmukh, K. Mandal, On three-dimensional $N(k)$-paracontact metric manifolds and Ricci solitons, Bull. Iran. Math. Soc., 43 (2017), 1571–1583.
[34]	S. Zamkovoy, V. Tzanov, Non-existence of flat paracontact metric structure in dimension greater than or equal to five, arXiv: 0910.5838v1, 2011. https://doi.org/10.48550/arXiv.0910.5838

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)