{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T09:14:10Z","timestamp":1772183650196,"version":"3.50.1"},"reference-count":29,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T00:00:00Z","timestamp":1772150400000},"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","award":["IMSIU-DDRSP2602"],"award-info":[{"award-number":["IMSIU-DDRSP2602"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This work presents a pair of sharp geometric inequalities that connect the normalized scalar curvature with the generalized normalized \u03b4-Casorati curvature for \u03b8-slant submanifolds immersed in quaternionic space forms endowed with a quarter-symmetric metric connection (QSMC). Alongside establishing these estimates, we rigorously describe the geometric conditions under which equality is achieved. The results not only generalize prior findings related to Casorati curvature but also offer new insights into the extrinsic geometry of submanifolds under non-standard connections. To conclude, we propose several open problems that invite further exploration in this direction.<\/jats:p>","DOI":"10.3390\/axioms15030164","type":"journal-article","created":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T08:31:19Z","timestamp":1772181079000},"page":"164","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Curvature Pinching Conditions in Quaternionic Manifolds Under Quarter-Symmetric Metric Connections"],"prefix":"10.3390","volume":"15","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":[{"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":[[2026,2,27]]},"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. Math."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Chen, B.-Y. (2011). Pseudo-Riemannian Geometry, \u03b4-Invariants and Applications, World Scientific Publishing Co. Pte. 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