{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T03:07:31Z","timestamp":1780456051811,"version":"3.54.1"},"reference-count":29,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2016,10,27]],"date-time":"2016-10-27T00:00:00Z","timestamp":1477526400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we prove some optimal inequalities involving the intrinsic scalar curvature and the extrinsic Casorati curvature of submanifolds in a generalized complex space form with a semi-symmetric non-metric connection and a generalized Sasakian space form with a semi-symmetric non-metric connection. Moreover, we show that in both cases, the equalities hold if and only if submanifolds are invariantly quasi-umbilical.<\/jats:p>","DOI":"10.3390\/sym8110113","type":"journal-article","created":{"date-parts":[[2016,10,27]],"date-time":"2016-10-27T10:17:52Z","timestamp":1477563472000},"page":"113","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Optimal Inequalities for the Casorati Curvatures of Submanifolds in Generalized Space Forms Endowed with Semi-Symmetric Non-Metric Connections"],"prefix":"10.3390","volume":"8","author":[{"given":"Guoqing","family":"He","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Science, AnHui Normal University, Wuhu 241000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hairong","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Science, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Liang","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, AnHui Normal University, Wuhu 241000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2016,10,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1007\/BF01236084","article-title":"Some pinching and classiffication theorems for minimal submanifolds","volume":"60","author":"Chen","year":"1993","journal-title":"Arch. 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