{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T08:21:31Z","timestamp":1762071691116,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,10,27]],"date-time":"2022-10-27T00:00:00Z","timestamp":1666828800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007446","name":"Deanship of Scientific Research at King Khalid University","doi-asserted-by":"publisher","award":["R.G.P.2\/199\/43"],"award-info":[{"award-number":["R.G.P.2\/199\/43"]}],"id":[{"id":"10.13039\/501100007446","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This research article endeavors to discuss the attributes of Riemannian submersions under the canonical variation in terms of the conformal \u03b7-Ricci soliton and gradient conformal \u03b7-Ricci soliton with a potential vector field \u03b6. Additionally, we estimate the various conditions for which the target manifold of Riemannian submersion under the canonical variation is a conformal \u03b7-Ricci soliton with a Killing vector field and a \u03c6(Ric)-vector field. Moreover, we deduce the generalized Liouville equation for Riemannian submersion under the canonical variation satisfying by a last multiplier \u03a8 of the vertical potential vector field \u03b6 and show that the base manifold of Riemanian submersion under canonical variation is an \u03b7 Einstein for gradient conformal \u03b7-Ricci soliton with a scalar concircular field \u03b3 on base manifold. Finally, we illustrate an example of Riemannian submersions between Riemannian manifolds, which verify our results.<\/jats:p>","DOI":"10.3390\/axioms11110594","type":"journal-article","created":{"date-parts":[[2022,10,27]],"date-time":"2022-10-27T20:37:58Z","timestamp":1666903078000},"page":"594","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Conformal \u03b7-Ricci Solitons on Riemannian Submersions under Canonical Variation"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1713-6831","authenticated-orcid":false,"given":"Mohd. Danish","family":"Siddiqi","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ali Hussain","family":"Alkhaldi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 9004, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6554-1228","authenticated-orcid":false,"given":"Meraj Ali","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3895-7548","authenticated-orcid":false,"given":"Aliya Naaz","family":"Siddiqui","sequence":"additional","affiliation":[{"name":"Division of Mathematics, School of Basic & Applied Sciences, Galgotias University, Greater Noida, Uttar Pradesh 203201, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1090\/conm\/071\/954419","article-title":"The Ricci flow on surfaces, Mathematics and General Relativity (University of California: Santa Cruz, CA, USA, 1986)","volume":"71","author":"Hamilton","year":"1988","journal-title":"Contemp. 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