{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:56:03Z","timestamp":1760057763076,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,24]],"date-time":"2025-02-24T00:00:00Z","timestamp":1740355200000},"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-DDRSP2503"],"award-info":[{"award-number":["IMSIU-DDRSP2503"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we study Einstein doubly warped product Poisson manifolds. First, we provide necessary and sufficient conditions for a doubly warped product manifold (M=Bf\u00d7bF,g,\u03a0), equipped with a Poisson structure \u03a0 to be a contravariant Einstein manifold. Additionally, under certain conditions on the base space B, we prove that if M is an Einstein doubly warped product Poisson manifold with non-positive scalar curvature, then M is simply a singly warped product Poisson manifold. We also investigate the existence and non-existence of the warping function on the base space B associated with constant scalar curvature on M, assuming that the fiber space F has constant scalar curvature.<\/jats:p>","DOI":"10.3390\/sym17030342","type":"journal-article","created":{"date-parts":[[2025,2,25]],"date-time":"2025-02-25T03:40:12Z","timestamp":1740454812000},"page":"342","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Einstein Doubly Warped Product Poisson Manifolds"],"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":[[2025,2,24]]},"reference":[{"key":"ref_1","first-page":"266","article-title":"M\u00e9moire sur la variation des constantes arbitraires dans les questions de m\u00e9canique","volume":"15","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. 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