{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T08:43:43Z","timestamp":1782895423974,"version":"3.54.5"},"reference-count":35,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,9]],"date-time":"2025-07-09T00:00:00Z","timestamp":1752019200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at 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>The primary aim of this paper is to establish two sharp geometric inequalities concerning submanifolds of S-space forms equipped with semi-symmetric metric connections (SSMCs). Specifically, we derive new inequalities involving the generalized normalized \u03b4-Casorati curvatures \u03b4c(t;q1+q2\u22121) and \u03b4^c(t;q1+q2\u22121) for bi-slant submanifolds. The cases in which equality holds are thoroughly examined, offering a deeper understanding of the geometric structure underlying such submanifolds. In addition, we present several immediate applications that highlight the relevance of our findings, and we support the article with illustrative examples.<\/jats:p>","DOI":"10.3390\/sym17071100","type":"journal-article","created":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T07:38:27Z","timestamp":1752133107000},"page":"1100","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Casorati-Type Inequalities for Submanifolds in S-Space Forms with Semi-Symmetric Connection"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9343-0725","authenticated-orcid":false,"given":"Md","family":"Aquib","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":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00009-025-02851-0","article-title":"The \u03bb-Wave Front Between a Smooth Surface and Its Gauss Map","volume":"22","author":"Li","year":"2025","journal-title":"Mediterr. J. Math."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Chen, B.Y. (2011). Pseudo-Riemannian Geometry, \u03b4-Invariants and Applications, World Scientific.","DOI":"10.1142\/9789814329644"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"95","DOI":"10.2140\/pjm.1983.106.95","article-title":"Normal curvature of surfaces in space forms","volume":"106","author":"Guadalupe","year":"1983","journal-title":"Pac. J. 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