{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:44:30Z","timestamp":1760060670410,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T00:00:00Z","timestamp":1757548800000},"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>This study focuses on submanifolds embedded in a conformal Sasakian space form (CSSF) equipped with a quarter-symmetric metric connection (QSMC). Utilizing the framework of generalized normalized \u03b4-Casorati curvature (GNDCC) alongside scalar curvature, we derive sharp optimal inequalities that characterize the intrinsic and extrinsic geometry of the submanifolds. Additionally, we examine the geometric behavior of these submanifolds under conformal deformations of the ambient manifold. To substantiate the theoretical developments, we construct an explicit example of a conformal Sasakian manifold that is not Sasakian, thereby confirming the validity and applicability of the derived results.<\/jats:p>","DOI":"10.3390\/sym17091514","type":"journal-article","created":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T07:51:46Z","timestamp":1757577106000},"page":"1514","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Sharp Curvature Inequalities for Submanifolds in Conformal Sasakian Space Forms Equipped with Quarter-Symmetric Metric 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":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohd","family":"Aslam","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Information Technology, Maulana Azad National Urdu University, Hyderabad 500032, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pooja","family":"Bansal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ram Lal Anand College, University of Delhi, New Delhi 110021, India"}],"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":[[2025,9,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"113","DOI":"10.5817\/AM2016-2-113","article-title":"Ricci and Scalar curvatures of submanifolds of a CSSF","volume":"52","author":"Abedi","year":"2016","journal-title":"Arch. Math. Brno Tomus"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/BF02413317","article-title":"Mesure de la courbure des surface suivant 1\u2019idea commune. Ses rapports avec les mesures de coubure gaussienne et moyenne","volume":"14","author":"Casorati","year":"1999","journal-title":"Acta Math."},{"key":"ref_3","first-page":"85","article-title":"Optimal inequalities involving Casorati curvatures","volume":"49","author":"Descu","year":"2007","journal-title":"Bull. Transylv. Univ. Brasov. Ser. B"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.difgeo.2018.12.006","article-title":"Classification of Casorati ideal Lagrangian submanifolds in complex space forms","volume":"63","author":"Aquib","year":"2019","journal-title":"Differ. Geom. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Aquib, M. (2025). An Optimal Inequality for Warped Product Pointwise Semi-Slant Submanifolds in Complex Space Forms. Axioms, 14.","DOI":"10.3390\/axioms14030213"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1007\/s11040-018-9297-x","article-title":"On Casorati curvatures of submanifolds in pointwise Kenmotsu space forms","volume":"22","author":"Lone","year":"2019","journal-title":"Math. Phys. Anal. Geom."},{"key":"ref_7","unstructured":"Ghasoiu, V. (2011). Inequalities for the Casorati curvatures of the submanifolds in complex space form, Riemannian geometry and application. Proceeding RIGA, Univ. Buchuresti. [2011 ed.]."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1209","DOI":"10.1016\/j.jmaa.2018.05.060","article-title":"An optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvature","volume":"465","year":"2018","journal-title":"J. Math. Anal. Appl."},{"key":"ref_9","first-page":"26","article-title":"Horizontally conformal submersions from CR-submanifolds of locally conformal Kaehler manifolds","volume":"17","year":"2020","journal-title":"Med. J. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1007\/BFb0088694","article-title":"Conformal chnages of almost contact metric structures","volume":"792","author":"Vaisman","year":"1980","journal-title":"Geom. Differ. Geom. Lect. Notes Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"195","DOI":"10.24033\/bsmf.1460","article-title":"Sur les structures infinitesimales regulieres","volume":"83","author":"Libermann","year":"1955","journal-title":"Bull Soc. Math. Fr."},{"key":"ref_12","unstructured":"Banaru, M. (2001, January 27\u201331). A new characterization of the Gray-Herneella classes of almost Hermitian manifold. Proceedings of the 8th International Conference on Differential Geometry and Its Applications, Opava, Czech Republic."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"103512","DOI":"10.1016\/j.mex.2025.103512","article-title":"Quasi Sasakian Manifold Endowed with Vanishing Pseudo Quasi Conformal Curvature Tensor","volume":"15","author":"AlHusseini","year":"2025","journal-title":"MethodsX"},{"key":"ref_14","first-page":"383","article-title":"Pseudo-quasi Conformal Curvature Tensor of Quasi Sasakian Manifold with Generalized Sasakian Space Forms","volume":"19","author":"AlHusseini","year":"2025","journal-title":"Malays. J. Math. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"253","DOI":"10.5556\/j.tkjm.52.2021.3276","article-title":"A study of New Class of Almost Contact Metric Manifold of Kenmotsu Type","volume":"52","author":"Abood","year":"2021","journal-title":"Tamkang J. Math."},{"key":"ref_16","first-page":"269","article-title":"On the Conharmonic Curvature Tensor of A Locally Conformal Cosymplectic Manifold","volume":"35","author":"Abood","year":"2020","journal-title":"Commun. Kor. Math. Soc."},{"key":"ref_17","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_18","doi-asserted-by":"crossref","unstructured":"Aquib, M. (2025). Casorati-Type Inequalities for Submanifolds in S-Space Forms with Semi-Symmetric Connection. Symmetry, 17.","DOI":"10.3390\/sym17071100"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"955","DOI":"10.1016\/j.jmaa.2015.06.011","article-title":"Multiple warped products with a quarter-symmetric connection","volume":"431","author":"Qu","year":"2015","journal-title":"J. Meth. Anal. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1950118","DOI":"10.1142\/S0219887819501184","article-title":"Chen inequalities for submanifolds of complex space forms and Sasakian space forms with quarter-symmetric connection","volume":"16","author":"Wang","year":"2019","journal-title":"Int. J. Geom. Meth. Mod. Phy."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Blair, D.E. (1976). Contact Manifolds in Riemannian Geometry, Springer. Lecture Notes in Mathematics.","DOI":"10.1007\/BFb0079307"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Yano, K., and Kon, M. (1984). Structures on Manifolds, World Scientific.","DOI":"10.1142\/0067"},{"key":"ref_23","first-page":"2","article-title":"Generalized Wintgen inequality for bi-slant submanifolds in conformal Sasakian space form with quarter-symmetric connection","volume":"31","author":"Aslam","year":"2025","journal-title":"Arab. J. Math. Sci."},{"key":"ref_24","first-page":"5","article-title":"Geometry of submanifolds 1, The first Casorati curvature indicatrices","volume":"37","author":"Verstralelen","year":"2013","journal-title":"Kragujevac J. Math."},{"key":"ref_25","first-page":"2009","article-title":"Casorati curvatures","volume":"50","author":"Kowalczyk","year":"2018","journal-title":"Bull Transilvania Univ. Brasov Ser. III"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"691","DOI":"10.11650\/tjm.19.2015.4832","article-title":"Inequalities for generalized normalized \u03b4-Casorati curvatures of slant submanifolds in quaternion space forms","volume":"19","author":"Lee","year":"2015","journal-title":"Taiwan. J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Blair, D.E. (2002). Riemannian Geometry of Contact and Simplectic Manifolds, Birkhouser Boston, Inc.. Progress in Mathmatics.","DOI":"10.1007\/978-1-4757-3604-5"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1514\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:43:37Z","timestamp":1760035417000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1514"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,11]]},"references-count":27,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["sym17091514"],"URL":"https:\/\/doi.org\/10.3390\/sym17091514","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,9,11]]}}}