{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:52:56Z","timestamp":1760143976742,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,11]],"date-time":"2024-03-11T00:00:00Z","timestamp":1710115200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002713","name":"Imam Mohammed Ibn Saud Islamic University","doi-asserted-by":"publisher","award":["IMSIU-RP23078"],"award-info":[{"award-number":["IMSIU-RP23078"]}],"id":[{"id":"10.13039\/501100002713","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, we study isotropic submanifolds in locally metallic product space forms. Firstly, we establish the Chen\u2013Ricci inequality for such submanifolds and determine the conditions under which the inequality becomes equality. Additionally, we explore the minimality of Lagrangian submanifolds in locally metallic product space forms, and we apply the result to create a classification theorem for isotropic submanifolds whose mean curvature is constant. More specifically, we have demonstrated that the submanifolds are either a product of two Einstein manifolds with Einstein constants, or they are isometric to a totally geodesic submanifold. To support our findings, we provide several examples.<\/jats:p>","DOI":"10.3390\/axioms13030183","type":"journal-article","created":{"date-parts":[[2024,3,11]],"date-time":"2024-03-11T11:05:10Z","timestamp":1710155110000},"page":"183","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Chen\u2013Ricci Inequality for Isotropic Submanifolds in Locally Metallic Product Space Forms"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1614-3228","authenticated-orcid":false,"given":"Yanlin","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China"}],"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 and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MD","family":"Aquib","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5901-2511","authenticated-orcid":false,"given":"Ibrahim","family":"Al-Dayel","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maged Zakaria","family":"Youssef","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1007\/PL00000420","article-title":"On Ricci curvature of isotropic and lagrangian submanifolds in complex space forms","volume":"74","author":"Chen","year":"2000","journal-title":"Arch. Math."},{"key":"ref_2","first-page":"10","article-title":"Chen-Ricci inequalities with a quarter symmetric connection in generalized space forms","volume":"2021","author":"Aquib","year":"2021","journal-title":"Adv. Math. 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