{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:31:25Z","timestamp":1760149885044,"version":"build-2065373602"},"reference-count":56,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,9,28]],"date-time":"2023-09-28T00:00:00Z","timestamp":1695859200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this manifestation, we explain the geometrisation of \u03b7-Ricci\u2013Yamabe soliton and gradient \u03b7-Ricci\u2013Yamabe soliton on Riemannian submersions with the canonical variation. Also, we prove any fiber of the same submersion with the canonical variation (in short CV) is an \u03b7-Ricci\u2013Yamabe soliton, which is called the solitonic fiber. Also, under the same setting, we inspect the \u03b7-Ricci\u2013Yamabe soliton in Riemannian submersions with a \u03c6(Q)-vector field. Moreover, we provide an example of Riemannian submersions, which illustrates our findings. Finally, we explore some applications of Riemannian submersion along with cohomology, Betti number, and Pontryagin classes in number theory.<\/jats:p>","DOI":"10.3390\/sym15101841","type":"journal-article","created":{"date-parts":[[2023,9,29]],"date-time":"2023-09-29T02:47:30Z","timestamp":1695955650000},"page":"1841","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Significance of Solitonic Fibers in Riemannian Submersions and Some Number Theoretic Applications"],"prefix":"10.3390","volume":"15","author":[{"given":"Ali H.","family":"Hakami","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, P.O. Box 277, Jazan 4512, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1713-6831","authenticated-orcid":false,"given":"Mohd Danish","family":"Siddiqi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, P.O. Box 277, Jazan 4512, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Siddiqi, M.D., Mofarreh, F., and SChaubey, S.K. (2023). Solitonic Aspect of Relativistic Magneto-Fluid Spacetime with Some Specific Vector Fields. Mathematics, 11.","DOI":"10.3390\/math11071596"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Siddiqi, M.D., Mofarreh, F., Siddiqui, A.N., and Siddiqui, S.A. (2023). Geometrical Structure in a Relativistic Thermodynamical Fluid Spacetime. Axioms, 12.","DOI":"10.3390\/axioms12020138"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2485804","DOI":"10.1155\/2021\/2485804","article-title":"Imperfect Fluid Generalized Robertson Walker Spacetime Admitting Ricci\u2013Yamabe Metric","volume":"2021","author":"Alkhaldi","year":"2021","journal-title":"Adv. Math. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1090\/conm\/071\/954419","article-title":"The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986)","volume":"71","author":"Hamilton","year":"1988","journal-title":"Contemp. Math. Am. Math. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2631","DOI":"10.3906\/mat-1902-38","article-title":"Ricci\u2013Yamabe maps for Riemannian flow and their volume variation and volume entropy","volume":"43","author":"Crasmareanu","year":"2019","journal-title":"Turk. J. Math."},{"key":"ref_6","unstructured":"Siddiqi, M.D., and Akyol, M.A. (2020). \u03b7-Ricci\u2013Yamabe solitons on Riemannian submersions from Riemannian manifolds. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1090\/conm\/071\/954419","article-title":"The Ricci flow on surfaces","volume":"71","author":"Hamilton","year":"1988","journal-title":"Contemp. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"20","DOI":"10.2307\/1969989","article-title":"The imbedding problem for Riemannian manifolds","volume":"63","author":"Nash","year":"1956","journal-title":"Ann. Math."},{"key":"ref_9","first-page":"458","article-title":"The fundamental equations of a submersion","volume":"13","year":"1966","journal-title":"Mich. Math. J."},{"key":"ref_10","first-page":"715","article-title":"Pseudo-Riemannian almost product manifolds and submersion","volume":"16","author":"Gray","year":"1967","journal-title":"J. Math. Mech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1007\/BF01457962","article-title":"\u00dcber die Abbildungen der dreidimensional Sph\u00e4re auf die Kugelf\u00e4che","volume":"104","author":"Hopf","year":"1931","journal-title":"Math. Ann."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Hakami, A.H., and Siddiqi, M.D. (2023). Properties of Anti-Invariant Submersions and Some Applications to Number Theory. Mathematics, 11.","DOI":"10.3390\/math11153368"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hakami, A.H., Siddiqi, M.D., Bahadir, O., and Khan, T. (2023). Aspects of Submanifolds on (\u03b1, \u03b2)-Type Almost Contact Manifolds with Quasi-Hemi-Slant Factor. Symmetry, 15.","DOI":"10.3390\/sym15061270"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1317","DOI":"10.1088\/0264-9381\/4\/5\/026","article-title":"Kaluza-Klein theory with scalar fields and generalized Hopf manifolds","volume":"4","author":"Ianus","year":"1987","journal-title":"Class. Quantum Gravity"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Rassias, G. (1991). the Mathematical Heritage of C.F. Gauss, World Scientific.","DOI":"10.1142\/1086"},{"key":"ref_16","unstructured":"Bourguignon, J.P., and Lawson, H.B. (1989). A Mathematician\u2019s Visit to Kaluza-Klein Theory, Rendiconti del Seminario Matematico-Politecnico di Torino."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/BF01942061","article-title":"Stability and isolation phenomena for Yang-mills fields","volume":"79","author":"Bourguignon","year":"1981","journal-title":"Commun. Math. Phys."},{"key":"ref_18","first-page":"324","article-title":"G, \u2014G\u2032 Riemannian submersions and nonlinear gauge field equations of general relativity","volume":"Volume 57","author":"Rassias","year":"1983","journal-title":"Global Analysis-Analysis on Manifolds: Dedicated to Marston Morse"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"\u1e62ahin, B. (2017). Riemannian Submersions, Riemannian Maps in Hermitian Geometry and Their Applications, Academic Press.","DOI":"10.1016\/B978-0-12-804391-2.50003-8"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Falcitelli, M., Ianus, S., and Pastore, A.M. (2004). Riemannian Submersions and Related Topics, World Scientific.","DOI":"10.1142\/9789812562333"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Meri\u00e7, \u1e62.E., and K\u0131l\u0131\u00e7, E. (2019). Riemannian submersions whose total manifolds admit a Ricci soliton. Int. J. Geom. Meth. Mod. Phys., 16.","DOI":"10.1142\/S0219887819501962"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Bejan, C.L., Meri\u00e7, \u1e62.E., and K\u0131l\u0131\u00e7, E. (2021). Contact-Complex Riemannian submersions. Mathematics, 9.","DOI":"10.3390\/math9232996"},{"key":"ref_23","first-page":"733","article-title":"Almost Hermitian submersions whose total manifolds admit a Ricci soliton","volume":"42","year":"2020","journal-title":"Honam Math. J."},{"key":"ref_24","first-page":"295","article-title":"E. Some remarks on Riemannian submersions admitting an almost Yamabe soliton","volume":"10","year":"2020","journal-title":"Adiyman Univ. J. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Siddiqi, M.D., Mofarreh, F., Akyol, M.A., and Hakami, A.H. (2023). \u03b7-Ricci\u2013Yamabe Solitons along Riemannian Submersions. Axioms, 12.","DOI":"10.3390\/axioms12080796"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Siddiqi, M.D., Alkhaldi, A.H., Khan, M.A., and Siddiqui, A.N. (2022). Conformal \u03b7-Ricci Solitons on Riemannian Submersions under Canonical Variation. Axioms, 11.","DOI":"10.3390\/axioms11110594"},{"key":"ref_27","first-page":"24","article-title":"Almost \u03b7-Ricci-Bourguignon solitons on submersions from Riemannian submersions","volume":"27","author":"Chaubey","year":"2022","journal-title":"Balk. J. Geom. Its Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"17335","DOI":"10.3934\/math.2023886","article-title":"Some notes on the tangent bundle with a Ricci quarter-symmetric metric connection","volume":"8","author":"Li","year":"2023","journal-title":"AIMS Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"16278","DOI":"10.3934\/math.2023833","article-title":"Zermelo\u2019s navigation problem for some special surfaces of rotation","volume":"8","author":"Li","year":"2023","journal-title":"AIMS Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"13875","DOI":"10.3934\/math.2023709","article-title":"Investigation of ruled surfaces and their singularities according to Blaschke frame in Euclidean 3-space","volume":"8","author":"Li","year":"2023","journal-title":"AIMS Math."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Li, Y., and Caliskan, A. (2023). Quaternionic Shape Operator and Rotation Matrix on Ruled Surfaces. Axioms, 12.","DOI":"10.3390\/axioms12050486"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Li, Y., Srivastava, S.K., Mofarreh, F., Kumar, A., and Ali, A. (2023). Ricci Soliton of CR-Warped Product Manifolds and Their Classifications. Symmetry, 15.","DOI":"10.3390\/sym15050976"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"22256","DOI":"10.3934\/math.20231135","article-title":"On simultaneous characterizations of partner-ruled surfaces in Minkowski 3-space","volume":"8","author":"Li","year":"2023","journal-title":"AIMS Math."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Li, Y., Gupta, M.K., Sharma, S., and Chaubey, S.K. (2023). On Ricci Curvature of a Homogeneous Generalized Matsumoto Finsler Space. Mathematics, 11.","DOI":"10.3390\/math11153365"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Li, Y., Bhattacharyya, S., Azami, S., Saha, A., and Hui, S.K. (2023). Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications. Mathematics, 11.","DOI":"10.2139\/ssrn.4347476"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Li, Y., Kumara, H.A., Siddesha, M.S., and Naik, D.M. (2023). Characterization of Ricci Almost Soliton on Lorentzian Manifolds. Symmetry, 15.","DOI":"10.2139\/ssrn.4339908"},{"key":"ref_37","first-page":"181","article-title":"Laplacian and Riemannian submersions with totally geodesic fibers","volume":"26","author":"Bergery","year":"1982","journal-title":"Ill J. Math."},{"key":"ref_38","first-page":"385","article-title":"\u03c6(Ric)-vector field in Riemannian spaces","volume":"94","author":"Hinterleither","year":"2008","journal-title":"Arch. Math."},{"key":"ref_39","unstructured":"Gardner, M. (1984). The Sixth Book of Mathematical Games from Scientific American, University of Chicago Press."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/0270-0255(80)90048-2","article-title":"Riemannian submersions and Instantons","volume":"1","author":"Watson","year":"1980","journal-title":"Math. Model."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"135","DOI":"10.2748\/tmj\/1178240982","article-title":"\u03b4-Commuting mappings and Betti numbers","volume":"27","author":"Watson","year":"1975","journal-title":"Tohoku Math. J."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1090\/S0002-9939-1960-0112151-4","article-title":"A sufficient condition that a mapping of Riemannian manifolds be a fiber bundle","volume":"11","author":"Hermann","year":"1960","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_43","unstructured":"Hirzebruch, F. (1995). The Signature Theorem, Topological Methods in Algebraic Geometry, Classics in Mathematics, Springer."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Li, Y., and G\u00fcler, E. (2023). A Hypersurfaces of Revolution Family in the Five-Dimensional Pseudo-Euclidean Space E25. Mathematics, 11.","DOI":"10.3390\/math11153427"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Li, Y., and Mak, M. (2023). Framed Natural Mates of Framed Curves in Euclidean 3-Space. Mathematics, 11.","DOI":"10.3390\/math11163571"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"20220610","DOI":"10.1515\/math-2022-0610","article-title":"Geometric classifications of k-almost Ricci solitons admitting paracontact metrices","volume":"21","author":"Li","year":"2023","journal-title":"Open Math."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"24957","DOI":"10.3934\/math.20231273","article-title":"Hypersurfaces of revolution family supplying in pseudo-Euclidean space","volume":"8","author":"Li","year":"2023","journal-title":"AIMS Math."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Li, Y., Abolarinwa, A., Alkhaldi, A., and Ali, A. (2022). Some Inequalities of Hardy Type Related to Witten-Laplace Operator on Smooth Metric Measure Spaces. Mathematics, 10.","DOI":"10.3390\/math10234580"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"2226","DOI":"10.3934\/math.2023115","article-title":"The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space","volume":"8","author":"Li","year":"2023","journal-title":"AIMS Math."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s00009-023-02396-0","article-title":"Kenmotsu Metric as Conformal \u03b7-Ricci Soliton","volume":"20","author":"Li","year":"2023","journal-title":"Mediterr. J. Math."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11005-021-01493-z","article-title":"D; De, U.C; Relativistic perfect fluid spacetimes and Ricci\u2013Yamabe solitons","volume":"112","author":"Siddiqi","year":"2022","journal-title":"Lett. Math. Phys."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"340","DOI":"10.2298\/FIL2202397S","article-title":"D; De, U.C; Deshmukh, S; Estimation of almost Ricci\u2013Yamabe solitons on static spacetimes","volume":"36","author":"Siddiqi","year":"2022","journal-title":"Filomat"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"114","DOI":"10.15672\/hujms.1052831","article-title":"Differential Geometric Approach of Betchow-Da Rios Soliton Equation","volume":"52","author":"Li","year":"2023","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"2386","DOI":"10.3934\/math.2023123","article-title":"Primitivoids of curves in Minkowski plane","volume":"8","author":"Li","year":"2023","journal-title":"AIMS Math."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"115266","DOI":"10.1016\/j.cam.2023.115266","article-title":"Neural network interpolation operators of multivariate functions","volume":"431","author":"Wang","year":"2023","journal-title":"J. Comput. Anal. Math."},{"key":"ref_56","first-page":"126781","article-title":"Rates of approximation by neural network interpolation operators","volume":"41","author":"Qian","year":"2022","journal-title":"Appl. Math. Comput."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/10\/1841\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:01:12Z","timestamp":1760130072000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/10\/1841"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,28]]},"references-count":56,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["sym15101841"],"URL":"https:\/\/doi.org\/10.3390\/sym15101841","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,9,28]]}}}