{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T17:34:26Z","timestamp":1769103266877,"version":"3.49.0"},"reference-count":49,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T00:00:00Z","timestamp":1701043200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In the current research, we develop optimal inequalities for submanifolds in trans-Sasakian manifolds or (\u03b1,\u03b2)-type almost contact manifolds endowed with the Schouten\u2013Van Kampen connection (SVK-connection), including generalized normalized \u03b4-Casorati Curvatures (\u03b4-CC). We also discuss submanifolds on which the equality situations occur. Lastly, we provided an example derived from this research.<\/jats:p>","DOI":"10.3390\/axioms12121082","type":"journal-article","created":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T03:32:03Z","timestamp":1701055923000},"page":"1082","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Optimal Inequalities on (\u03b1,\u03b2)-Type Almost Contact Manifold with the Schouten\u2013Van Kampen Connection"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1713-6831","authenticated-orcid":false,"given":"Mohd Danish","family":"Siddiqi","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, P.O. Box 277, Jazan 4512, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6499-097X","authenticated-orcid":false,"given":"Ali H.","family":"Hakami","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, P.O. Box 277, Jazan 4512, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1007\/BF01236084","article-title":"Some pinching and classification theorems for minimal submanifolds","volume":"60","author":"Chen","year":"1993","journal-title":"Arch. 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