{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,17]],"date-time":"2025-12-17T23:08:25Z","timestamp":1766012905715,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T00:00:00Z","timestamp":1682035200000},"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 paper, we initiate the study of a generalized soliton on a Riemannian manifold, we find a characterization for the Euclidean space, and in the compact case, we find a sufficient condition under which it reduces to a quasi-Einstein manifold. We also find sufficient conditions under which a compact generalized soliton reduces to an Einstein manifold. Note that Ricci solitons being self-similar solutions of the heat flow, this topic is related to the symmetry in the geometry of Riemannian manifolds. Moreover, generalized solitons being generalizations of Ricci solitons are naturally related to symmetry.<\/jats:p>","DOI":"10.3390\/sym15040954","type":"journal-article","created":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T10:11:25Z","timestamp":1682071885000},"page":"954","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Note on Generalized Solitons"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1729-4805","authenticated-orcid":false,"given":"Amira","family":"Ishan","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Taif University, Taif 21944, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3700-8164","authenticated-orcid":false,"given":"Sharief","family":"Deshmukh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"255","DOI":"10.4310\/jdg\/1214436922","article-title":"Three-manifolds with positive Ricci curvature","volume":"17","author":"Hamilton","year":"1982","journal-title":"J. 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Pura Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.difgeo.2016.07.008","article-title":"Generalized quasi Yamabe gradient solitons","volume":"49","author":"Neto","year":"2016","journal-title":"Differ. Geom. Its Appl."},{"key":"ref_10","unstructured":"Petersen, P. (2006). Riemannian Geometry, Springer. [2nd ed.]. Graduate Texts in Mathematics."},{"key":"ref_11","first-page":"757","article-title":"Ricci almost solitons","volume":"5","author":"Pigola","year":"2011","journal-title":"Ann. Della Sc. Norm. Super. Pisa-Cl. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"449","DOI":"10.2140\/pjm.2011.254.449","article-title":"On noncompact \u03c4-quasi-Einstein metrics","volume":"254","author":"Wang","year":"2011","journal-title":"Pac. J. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1007\/s00013-013-0533-0","article-title":"On conformal solutions of the Yamabe flow","volume":"101","author":"Barbosa","year":"2013","journal-title":"Arch. Math."},{"key":"ref_14","unstructured":"Yano, K. (1970). Integral Formulas in Riemannian Geometry, Marcel Dekker Inc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/4\/954\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:20:46Z","timestamp":1760124046000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/4\/954"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,21]]},"references-count":14,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["sym15040954"],"URL":"https:\/\/doi.org\/10.3390\/sym15040954","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,4,21]]}}}