On gradient normalized Ricci-harmonic solitons in sequential warped products

Noura Alhouiti; Fatemah Mofarreh; Akram Ali; Fatemah Abdullah Alghamdi; Noura Alhouiti; Fatemah Mofarreh; Akram Ali; Fatemah Abdullah Alghamdi

doi:10.3934/math.20241129

AIMS Mathematics

2024, Volume 9, Issue 9: 23221-23233. doi: 10.3934/math.20241129

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On gradient normalized Ricci-harmonic solitons in sequential warped products

1.
Department of Basic Sciences, University College of Haqel, University of Tabuk, 71491 Tabuk, Saudi Arabia
2.
Mathematical Science Department, Faculty of Science, Princess Nourah bint Abdulrahman University. Riyadh 11546, Saudi Arabia
3.
Department of Mathematics, College of Science, King Khalid University, Abha, Saudi Arabia
4.
Financial Sciences Department, Applied College, Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia

Received: 22 March 2024 Revised: 30 April 2024 Accepted: 14 May 2024 Published: 31 July 2024
MSC : 53C21, 53C25, 53C50

Our investigation involved sequentially warped product manifolds that contained gradient-normalized Ricci-harmonic solitons. We presented the primary connections for a gradient-normalized Ricci-harmonic soliton on sequential warped product manifolds. In practical applications, our research investigated gradient-normalized Ricci-harmonic solitons for sequential generalized Robertson-Walker spacetimes and sequential standard static space-times. Our finding generalized all results proven in ^[26].
Citation: Noura Alhouiti, Fatemah Mofarreh, Akram Ali, Fatemah Abdullah Alghamdi. On gradient normalized Ricci-harmonic solitons in sequential warped products[J]. AIMS Mathematics, 2024, 9(9): 23221-23233. doi: 10.3934/math.20241129

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Abstract

Our investigation involved sequentially warped product manifolds that contained gradient-normalized Ricci-harmonic solitons. We presented the primary connections for a gradient-normalized Ricci-harmonic soliton on sequential warped product manifolds. In practical applications, our research investigated gradient-normalized Ricci-harmonic solitons for sequential generalized Robertson-Walker spacetimes and sequential standard static space-times. Our finding generalized all results proven in ^[26].

References

[1]	A. Abolarinwa, Evolution and monotonicity of the first eigenvalue of $p$-Laplacian under the Ricci-harmonic flow, J. Appl. Anal., 21 (2015), 147–160. https://doi.org/10.1515/jaa-2015-0013 doi: 10.1515/jaa-2015-0013
[2]	A. Abolarinwa, Gap theorems for compact almost Ricci-harmonic solitons, Int. J. Math., 30 (2019), 1950040. https://doi.org/10.1142/S0129167X1950040X doi: 10.1142/S0129167X1950040X
[3]	A. Abolarinwa, N. K. Oladejo, S. O. Salawu, On the entropy formulas and solitons for the Ricci-harmonic flow, Bull. Iran. Math. Soc., 45 (2019), 1177–1192. https://doi.org/10.1007/s41980-018-00192-1 doi: 10.1007/s41980-018-00192-1
[4]	A. Abolarinwa, Y. Chong, D. Zhang, On the spectrum of the $p$-biharmonic operator under the Ricci flow, Results Math., 75 (2020), 54. https://doi.org/10.1007/s00025-020-1182-9 doi: 10.1007/s00025-020-1182-9
[5]	A. Anselli, On the rigidity of harmonic-Ricci solitons, Adv. Geom., 22 (2022), 171–198. https://doi.org/10.1515/advgeom-2022-0003 doi: 10.1515/advgeom-2022-0003
[6]	S. Azami, Ricci-Bourguignon flow coupled with harmonic map flow, Int. J. Math., 30 (2019), 1950049. https://doi.org/10.1142/S0129167X19500496 doi: 10.1142/S0129167X19500496
[7]	S. Azami, V. Pirhadi, G. Fasihi-Ramandi, Complete shrinking Ricci-Bourguignon harmonic solitons, Int. J. Math., 33 (2022), 2250046. https://doi.org/10.1142/S0129167X2250046X doi: 10.1142/S0129167X2250046X
[8]	H. Al-Sodais, H. Alodan, S. Deshmukh, Hypersurfaces of Euclidean space as gradient Ricci solitons, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (NS), 61 (2015), 437–444.
[9]	A. Ali, F. Mofarreh, D. S. Patra, Geometry of almost Ricci solitons on paracontact metric manifolds, Quaest. Math., 45 (2022), 1167–1180. https://doi.org/10.2989/16073606.2021.1929539 doi: 10.2989/16073606.2021.1929539
[10]	A. Ali, N. Alshehri, F. Mofarreh, Y. Li, Geometry of gradient Einstein harmonic solitons in sequential warped products manifolds, Eur. Phys. J. Plus, 139 (2024), 339. https://doi.org/10.1140/epjp/s13360-024-05120-3 doi: 10.1140/epjp/s13360-024-05120-3
[11]	R. L. Bishop, B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1–49.
[12]	A. M. Blaga, H. M. Tastan, Gradient solitons on doubly warped product manifolds, Rep. Math. Phys., 89 (2022), 319–333. https://doi.org/10.1016/S0034-4877(22)00036-2 doi: 10.1016/S0034-4877(22)00036-2
[13]	U. C. De, M. Turan, A. Yildiz, A. De, Ricci solitons and gradient Ricci solitons on $3$-dimensional normal almost contact metric manifolds, Publ. Math. Debrecen, 80 (2012), 127–142.
[14]	U. C. De, K. Mandal, Ricci solitons and gradient Ricci solitons on N($ k $)-paracontact manifold, Zh. Mat. Fiz. Anal. Geom., 15 (2019), 307–320. https://doi.org/10.15407/mag15.03.307 doi: 10.15407/mag15.03.307
[15]	S. Dwivedi, Some results on Ricci-Bourguignon solitons and almost solitons, Can. Math. Bull., 64 (2021), 591–604. https://doi.org/10.4153/S0008439520000673 doi: 10.4153/S0008439520000673
[16]	S. Dwivedi, D. S. Patra, Some results on almost $$-Ricci-Bourguignon solitons, J. Geom. Phys., 178* (2022), 104519. https://doi.org/10.1016/j.geomphys.2022.104519 doi: 10.1016/j.geomphys.2022.104519
[17]	U. C. De, S. Shenawy, Generalized quasi-Einstein GRW space-times, Int. J. Geom. Methods M., 16 (2019), 1950124. https://doi.org/10.1142/S021988781950124X doi: 10.1142/S021988781950124X
[18]	U. C. De, S. Shenawy, B. Unal, Sequential warped products: Curvature and conformal vector fields, Filomat, 33 (2019), 4071–4083. https://doi.org/10.2298/FIL1913071D doi: 10.2298/FIL1913071D
[19]	F. Dobarro, B. Unal, Special standard static space–times, Nonlinear Anal.-Theor., 59 (2004), 759–770. https://doi.org/10.1016/j.na.2004.07.035 doi: 10.1016/j.na.2004.07.035
[20]	H. X. Guo, R. Philipowski, A. Thalmaier, On gradient solitons of the Ricci-harmonic flow, Acta Math. Sin. (Engl. Ser.), 31 (2015), 1798–1804. https://doi.org/10.1007/s10114-015-4446-7 doi: 10.1007/s10114-015-4446-7
[21]	S. Guler, Sequential warped products and their applications, Int. Electron. J. Geom., 14 (2021), 277–291. https://doi.org/10.36890/iejg.937419 doi: 10.36890/iejg.937419
[22]	D. Ganguly, S. Dey, A. Ali, A. Bhattacharyya, Conformal Ricci soliton and quasi-Yamabe soliton on generalized Sasakian space form, J. Geom. Phys., 169 (2021), 104339. https://doi.org/10.1016/j.geomphys.2021.104339 doi: 10.1016/j.geomphys.2021.104339
[23]	R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry, 17 (1982), 255–306. https://doi.org/10.4310/jdg/1214436922 doi: 10.4310/jdg/1214436922
[24]	R. S. Hamilton, The Ricci flow on surfaces, In: Mathematics and general relativity, Contemporary Mathematics, 1988,237–262. https://doi.org/10.1090/conm/071
[25]	F. Karaca, C. Ozgur, Gradient Ricci-harmonic solitons on doubly warped product manifolds, Filomat, 37 (2023), 5969–5977. https://doi.org/10.2298/FIL2318969K doi: 10.2298/FIL2318969K
[26]	F. Karaca, C. Ozgur, On sequential warped product manifolds admitting gradient Ricci-harmonic solitons, Phys. Scr., 98 (2023), 085213. https://doi.org/10.1088/1402-4896/ace1b4 doi: 10.1088/1402-4896/ace1b4
[27]	M. L. de Sousa, R. Pina, Gradient Ricci solitons with the structure of warped products, Results Math., 71 (2017), 825–840. https://doi.org/10.1007/s00025-016-0583-2 doi: 10.1007/s00025-016-0583-2
[28]	Y. Li, D. S. Patra, N. Alluhaibi, F. Mofarreh, A. Ali, Geometric classifications of $k$-almost Ricci solitons admitting paracontact metrices, Open Math., 21 (2023), 20220610. https://doi.org/10.1515/math-2022-0610 doi: 10.1515/math-2022-0610
[29]	R. Muller, Ricci flow coupled with harmonic map flow, 2012, arXiv: 0912.2907. https://doi.org/10.48550/arXiv.0912.2907
[30]	D. S. Patra, A. Ali, F. Mofarreh, Characterizations of Ricci-Bourguignon almost solitons on pseudo-Riemannian manifolds, Mediterr. J. Math., 19 (2022), 176. https://doi.org/10.1007/s00009-022-02085-4 doi: 10.1007/s00009-022-02085-4
[31]	X. Cao, A remark of Ricci-Bourguignon harmonic soliton, 2023, arXiv: 2309.16485. https://doi.org/10.48550/arXiv.2309.16485

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