{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:06:38Z","timestamp":1760148398707,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,25]],"date-time":"2023-04-25T00:00:00Z","timestamp":1682380800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12101168","LQ22A010014","R. G. P.1\/87\/43","PNURSP2023R27"],"award-info":[{"award-number":["12101168","LQ22A010014","R. G. P.1\/87\/43","PNURSP2023R27"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["12101168","LQ22A010014","R. G. P.1\/87\/43","PNURSP2023R27"],"award-info":[{"award-number":["12101168","LQ22A010014","R. G. P.1\/87\/43","PNURSP2023R27"]}]},{"name":"King Khalid University, Saudi Arabia","award":["12101168","LQ22A010014","R. G. P.1\/87\/43","PNURSP2023R27"],"award-info":[{"award-number":["12101168","LQ22A010014","R. G. P.1\/87\/43","PNURSP2023R27"]}]},{"DOI":"10.13039\/501100004242","name":"Princess Nourah bint Abdulrahman University","doi-asserted-by":"publisher","award":["12101168","LQ22A010014","R. G. P.1\/87\/43","PNURSP2023R27"],"award-info":[{"award-number":["12101168","LQ22A010014","R. G. P.1\/87\/43","PNURSP2023R27"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we derived an equality for CR-warped product in a complex space form which forms the relationship between the gradient and Laplacian of the warping function and second fundamental form. We derived the necessary conditions of a CR-warped product submanifolds in Ka\u00a8hler manifold to be an Einstein manifold in the impact of gradient Ricci soliton. Some classification of CR-warped product submanifolds in the Ka\u00a8hler manifold by using the Euler\u2013Lagrange equation, Dirichlet energy and Hamiltonian is given. We also derive some characterizations of Einstein warped product manifolds under the impact of Ricci Curvature and Divergence of Hessian tensor.<\/jats:p>","DOI":"10.3390\/sym15050976","type":"journal-article","created":{"date-parts":[[2023,4,25]],"date-time":"2023-04-25T04:27:02Z","timestamp":1682396822000},"page":"976","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["Ricci Soliton of CR-Warped Product Manifolds and Their Classifications"],"prefix":"10.3390","volume":"15","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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5992-1902","authenticated-orcid":false,"given":"Sachin Kumar","family":"Srivastava","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Central University of Himachal Pradesh, Dharamshala 176215, Himachal Pradesh, India"}]},{"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, Riyadh 11546, Saudi Arabia"}]},{"given":"Anuj","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Central University of Himachal Pradesh, Dharamshala 176215, Himachal Pradesh, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6053-3031","authenticated-orcid":false,"given":"Akram","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,25]]},"reference":[{"key":"ref_1","first-page":"255","article-title":"The Ricci flow in dimension three","volume":"17","author":"Hamilton","year":"1982","journal-title":"J. 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