{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T04:34:53Z","timestamp":1787027693565,"version":"build-2736575974"},"reference-count":32,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,30]],"date-time":"2023-01-30T00:00:00Z","timestamp":1675036800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University Researchers","award":["PNURSP2023R27"],"award-info":[{"award-number":["PNURSP2023R27"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The present paper deals with the investigations of a Kenmotsu manifold satisfying certain curvature conditions endowed with \u2605-\u03b7-Ricci solitons. First we find some necessary conditions for such a manifold to be \u03c6-Einstein. Then, we study the notion of \u2605-\u03b7-Ricci soliton on this manifold and prove some significant results related to this notion. Finally, we construct a nontrivial example of three-dimensional Kenmotsu manifolds to verify some of our results.<\/jats:p>","DOI":"10.3390\/axioms12020140","type":"journal-article","created":{"date-parts":[[2023,1,30]],"date-time":"2023-01-30T02:01:18Z","timestamp":1675044078000},"page":"140","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Certain Curvature Conditions on Kenmotsu Manifolds and \u2605-\u03b7-Ricci Solitons"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3238-6484","authenticated-orcid":false,"given":"Halil","family":"Yolda\u015f","sequence":"first","affiliation":[{"name":"Department of Management Information Systems, Adana Alparslan T\u00fcrke\u015f Science and Technology University, 01250 Adana, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1175-6423","authenticated-orcid":false,"given":"Abdul","family":"Haseeb","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, P.O. Box 2097, Jazan 45142, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2116-7382","authenticated-orcid":false,"given":"Fatemah","family":"Mofarreh","sequence":"additional","affiliation":[{"name":"Mathematical Science Department, Faculty of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 13415, Riyadh 11546, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,30]]},"reference":[{"key":"ref_1","first-page":"247","article-title":"On almost-analytic vectors in almost Kahlerian manifolds","volume":"11","author":"Tachibana","year":"1959","journal-title":"Tohoku Math. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"473","DOI":"10.3836\/tjm\/1244208866","article-title":"Real hypersurfaces of complex space forms in terms of Ricci \ud83d\udfc9-tensor","volume":"25","author":"Hamada","year":"2002","journal-title":"Tokyo J. Math."},{"key":"ref_3","first-page":"445","article-title":"The \ud83d\udfc9-Ricci tensor for hypersurfaces in CPn and CHn","volume":"34","author":"Ivey","year":"2011","journal-title":"Tokyo J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"7939654","DOI":"10.1155\/2022\/7939654","article-title":"\u2217-Ricci tensor on \u03b1-cosymplectic manifolds","volume":"2022","author":"Amruthalakshmi","year":"2022","journal-title":"Adv. Math. Phys."},{"key":"ref_5","first-page":"178","article-title":"\ud83d\udfc9-Ricci tensor on normal metric contact pair manifolds","volume":"27","author":"Unal","year":"2022","journal-title":"Balkan J. Geom. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1447","DOI":"10.1515\/ms-2017-0321","article-title":"\ud83d\udfc9-Ricci solitons and gradient almost \ud83d\udfc9-Ricci solitons on Kenmotsu manifolds","volume":"69","author":"Venkatesha","year":"2019","journal-title":"Math. Slovaca"},{"key":"ref_7","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."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1016\/j.geomphys.2014.09.004","article-title":"\ud83d\udfc9-Ricci solitons of real hypersurface in non-flat complex space forms","volume":"76","author":"Kaimakamis","year":"2014","journal-title":"J. Geom. Phy."},{"key":"ref_9","first-page":"163","article-title":"\ud83d\udfc9-\u03b7-Ricci soliton within the framework of Sasakian manifold","volume":"18","author":"Dey","year":"2020","journal-title":"J. Dyn. Syst. Geom. Theor."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"433","DOI":"10.3836\/tjm\/1502179275","article-title":"Real hypersurfaces with \ud83d\udfc9-Ricci solitons of non-flat complex space forms","volume":"41","author":"Chen","year":"2018","journal-title":"Tokyo J. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"874","DOI":"10.1515\/math-2019-0056","article-title":"\ud83d\udfc9-Ricci soliton on (\u03ba,\u03bc)\u2032-almost Kenmotsu manifold","volume":"17","author":"Dai","year":"2019","journal-title":"Open Math."},{"key":"ref_12","unstructured":"Dey, S., Sarkar, S., and Bhattacharyya, A. (2021). \ud83d\udfc9-\u03b7-Ricci soliton and contact geometry. Ric. Mat., 1\u201319."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"59","DOI":"10.15330\/cmp.11.1.59-69","article-title":"Ricci soliton and Ricci almost soliton within the framework of Kenmotsu manifold","volume":"11","author":"Ghosh","year":"2019","journal-title":"Carpathian Math. Publ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1850120","DOI":"10.1142\/S0219887818501207","article-title":"\ud83d\udfc9-Ricci soliton within the framework of Sasakian and (\u03ba,\u03bc)-contact manifold","volume":"15","author":"Ghosh","year":"2018","journal-title":"Int. J. Geom. Methods Modern Phys."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Haseeb, A., Bilal, M., Chaubey, S.K., and Khan, M.N.I. (2022). Geometry of indefinite Kenmotsu manifolds as \u2217\u03b7-Ricci-Yamabe solitons. Axioms, 11.","DOI":"10.3390\/axioms11090461"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"167","DOI":"10.46793\/KgJMat2402.167H","article-title":"Lorentzian para-Sasakian manifolds and \u2217-Ricci solitons","volume":"48","author":"Haseeb","year":"2024","journal-title":"Krag. J. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1007\/s00022-019-0490-2","article-title":"\u03b7-Ricci solitons in \u03f5-Kenmotsu manifolds","volume":"110","author":"Haseeb","year":"2019","journal-title":"J. Geom."},{"key":"ref_18","first-page":"105","article-title":"3-dimensional trans-Sasakian manifolds admitting \u2217-Ricci solitons","volume":"15","author":"Haseeb","year":"2022","journal-title":"Jordan J. Math. Stat."},{"key":"ref_19","first-page":"5718736","article-title":"Sasakian manifolds admitting \u2217-\u03b7-Ricci-Yamabe solitons","volume":"2022","author":"Haseeb","year":"2022","journal-title":"Adv. Math. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"6605127","DOI":"10.1155\/2022\/6605127","article-title":"LP-Kenmotsu manifolds admitting \u03b7-Ricci solitons and spacetime","volume":"2022","author":"Li","year":"2022","journal-title":"J. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"241","DOI":"10.5486\/PMD.2018.8245","article-title":"\ud83d\udfc9-Ricci solitons on Sasakian 3-manifolds","volume":"93","author":"Majhi","year":"2018","journal-title":"Publ. Math. Debrecen"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Patra, D.S., Ali, A., and Mofarreh, F. (2021). Geometry of almost contact metrics as almost \ud83d\udfc9-Ricci solitons. arXiv.","DOI":"10.1007\/s13366-020-00539-y"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2150189","DOI":"10.1142\/S0219887821501899","article-title":"On Kenmotsu manifolds admitting \u03b7-Ricci-Yamabe solitons","volume":"18","year":"2021","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_24","first-page":"146","article-title":"Remarks on some soliton types with certain vector fields","volume":"3","year":"2022","journal-title":"Fundam. Contemp. Math. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"93","DOI":"10.2748\/tmj\/1178241594","article-title":"A class of almost contact Riemannian manifolds","volume":"24","author":"Kenmotsu","year":"1972","journal-title":"Tohoku Math. J."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Blair, D.E. (1976). Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, Springer.","DOI":"10.1007\/BFb0079307"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Yano, K., and Kon, M. (1984). Structures on Manifolds, Series in Mathematics, Springer.","DOI":"10.1142\/0067"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/s00009-010-0049-9","article-title":"On \u03c6-Einstein contact Riemannian manifolds","volume":"7","author":"Cho","year":"2011","journal-title":"Mediterr. J. Math."},{"key":"ref_29","unstructured":"Mishra, R.S. (1984). Structures on a Differentiable Manifold and Their Applications, Chandrama Prakashan."},{"key":"ref_30","first-page":"97","article-title":"Curvature tensors and their relativistic significance","volume":"19","author":"Pokhariyal","year":"1971","journal-title":"Yokohama Math. J."},{"key":"ref_31","first-page":"195","article-title":"Concircular geometry I. concircular transformations","volume":"16","author":"Yano","year":"1940","journal-title":"Proc. Imp. Acad."},{"key":"ref_32","first-page":"89","article-title":"Almost conformal Ricci solitons on f-Kenmotsu manifolds","volume":"5","author":"Hui","year":"2019","journal-title":"Khayyam J. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/140\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:19:36Z","timestamp":1760120376000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/140"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,30]]},"references-count":32,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["axioms12020140"],"URL":"https:\/\/doi.org\/10.3390\/axioms12020140","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,30]]}}}