{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:26:46Z","timestamp":1760059606279,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,6,28]],"date-time":"2025-06-28T00:00:00Z","timestamp":1751068800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Imam Mohammad Ibn Saud Islamic University (IMSIU)","award":["IMSIU-DDRSP2502"],"award-info":[{"award-number":["IMSIU-DDRSP2502"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we extend the study of contravariant Einstein-like metrics to Poisson doubly warped product manifolds (PDWPMs). We derive the necessary and sufficient conditions under which the base and fiber manifolds of a PDWPM inherit Einstein-like structures from the total space. As applications, we construct Einstein-like Poisson doubly warped product structures belonging to classes A, B, and P in various spacetime models, including generalizations of Reissner\u2013Nordstr\u00f6m, standard static, and Robertson\u2013Walker spacetimes.<\/jats:p>","DOI":"10.3390\/sym17071021","type":"journal-article","created":{"date-parts":[[2025,6,30]],"date-time":"2025-06-30T03:54:28Z","timestamp":1751255668000},"page":"1021","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Contravariant Einstein-like Doubly Warped Metrics: Theory and Applications"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-7359-484X","authenticated-orcid":false,"given":"Foued","family":"Aloui","sequence":"first","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"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,28]]},"reference":[{"key":"ref_1","first-page":"266","article-title":"M\u00e9moire sur la variation des constantes arbitraires dans les questions de m\u00e9canique","volume":"8","author":"Poisson","year":"1809","journal-title":"J. \u00c9c. Polytech."},{"key":"ref_2","first-page":"253","article-title":"Les vari\u00e9t\u00e9s de Poisson et leurs alg\u00e8bres de Lie associ\u00e9es","volume":"12","author":"Lichnerowicz","year":"1977","journal-title":"J. Diff. Geom."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Vaisman, I. (1994). Lectures on the Geometry of Poisson Manifolds, Birkh\u00e4user. Progress in Mathematics.","DOI":"10.1007\/978-3-0348-8495-2"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"303","DOI":"10.4310\/jdg\/1214341648","article-title":"Connections in Poisson geometry I. Holonomy and invariants","volume":"54","author":"Fernandes","year":"2000","journal-title":"J. Differ. Geom."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"115002","DOI":"10.1088\/1361-6382\/aa6e5b","article-title":"Contravariant gravity on Poisson manifolds and Einstein gravity","volume":"34","author":"Kaneko","year":"2017","journal-title":"Class. Quantum Gravity"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Aloui, F., and Al-Dayel, I. (2025). Einstein doubly warped product Poisson manifolds. 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