{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:34:20Z","timestamp":1760060060916,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,7,29]],"date-time":"2025-07-29T00:00:00Z","timestamp":1753747200000},"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 paper explores novel inequalities for statistical submanifolds within the framework of the Norden golden-like statistical manifold. By leveraging the intrinsic properties of statistical manifolds and the structural richness of Norden golden geometry, we establish fundamental relationships between the intrinsic and extrinsic invariants of submanifolds. The methodology involves deriving generalized Chen-type and \u03b4(2,2) curvature inequalities using curvature tensor analysis and dual affine connections. A concrete example is provided to verify the theoretical framework. The novelty of this work lies in extending classical curvature inequalities to a newly introduced statistical structure, thereby opening new perspectives in the study of geometric inequalities in information geometry and related mathematical physics contexts.<\/jats:p>","DOI":"10.3390\/sym17081206","type":"journal-article","created":{"date-parts":[[2025,7,29]],"date-time":"2025-07-29T09:31:45Z","timestamp":1753781505000},"page":"1206","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Inequality Constraints on Statistical Submanifolds of Norden-Golden-like Statistical Manifold"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-2258-991X","authenticated-orcid":false,"given":"Amit Kumar","family":"Rai","sequence":"first","affiliation":[{"name":"Department of Applied Sciences, GTBIT, GGSIPU, G-8 Area, Rajouri Garden, New Delhi 110064, India"}]},{"given":"Majid Ali","family":"Choudhary","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Sciences, Maulana Azad National Urdu University, Hyderabad 500032, India"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-0901-6123","authenticated-orcid":false,"given":"Mohammed","family":"Nisar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Sciences, Maulana Azad National Urdu University, Hyderabad 500032, India"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-7359-484X","authenticated-orcid":false,"given":"Foued","family":"Aloui","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"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1631","DOI":"10.1016\/S0960-0779(97)00001-5","article-title":"On characterization of the onset to chaos","volume":"8","author":"Spinadel","year":"1997","journal-title":"Chaos Solitons Fractals"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1016\/S0362-546X(98)00123-0","article-title":"The metallic means family and multifractal spectra","volume":"36","author":"Spinadel","year":"1999","journal-title":"Nonlinear Anal."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Amari, S. (1985). Differential-Geometrical Methods in Statistics, Springer.","DOI":"10.1007\/978-1-4612-5056-2"},{"key":"ref_4","unstructured":"Nomizu, K., Katsumi, N., and Sasaki, T. (1994). Affine Differential Geometry: Geometry of Affine Immersions, Cambridge University Press."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1016\/j.laa.2016.02.021","article-title":"A sectional curvature for statistical structures","volume":"497","author":"Opozda","year":"2016","journal-title":"Linear Algebra Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"8925","DOI":"10.2298\/FIL2425925C","article-title":"Several fundamental inequalities for submanifolds immersed in Riemannian manifolds equipped with golden structure","volume":"38","author":"Choudhary","year":"2024","journal-title":"Filomat"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"9607","DOI":"10.2298\/FIL2427607C","article-title":"Inequalities for golden Lorentzian manifolds with gsm U-connection","volume":"38","author":"Choudhary","year":"2024","journal-title":"Filomat"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Choudhary, M.A., and Mihai, I. (2023). Inequalities for the generalized normalized \u03b4-Casorati curvatures of submanifolds in golden Riemannian manifolds. Axioms, 12.","DOI":"10.3390\/axioms12100952"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1155","DOI":"10.2298\/FIL2304155C","article-title":"Some basic inequalities on golden Riemannian product manifolds with constant curvatures","volume":"37","author":"Choudhary","year":"2023","journal-title":"Filomat"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5443","DOI":"10.2298\/FIL2316443C","article-title":"On some inequalities for metallic Riemannian space forms","volume":"37","author":"Choudhary","year":"2023","journal-title":"Filomat"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Macsim, G., Mihai, A., and Mihai, I. (2020). \u03b4(2,2)-invariant for Lagrangian submanifolds in quaternionic space forms. Mathematics, 8.","DOI":"10.3390\/math8040480"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Mihai, A., and Mihai, I. (2020). The \u03b4(2,2)-invariant on statistical submanifold in Hessian manifolds of constant Hessian curvature. Entropy, 22.","DOI":"10.3390\/e22020164"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3585","DOI":"10.1007\/s13398-019-00718-0","article-title":"Chen-Ricci inequality for warped products in Kenmotsu space forms and its applications","volume":"113","author":"Mustafa","year":"2019","journal-title":"Rev. Real Acad. Cienc. Exactas F\u00eds. Nat. Ser. A Mat."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"163","DOI":"10.4064\/cm108-1-15","article-title":"Chen\u2019s inequality in the Lagrangian case","volume":"108","author":"Oprea","year":"2007","journal-title":"Colloq. Math."},{"key":"ref_15","first-page":"128","article-title":"Statistical manifold with almost complex structures and its statistical submersions","volume":"65","author":"Takano","year":"2004","journal-title":"Tensor New Ser."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"6213","DOI":"10.3390\/e17096213","article-title":"Statistical manifold with almost quaternionic structures and quaternionic K\u00e4hler-like statistical submersions","volume":"17","year":"2015","journal-title":"Entropy"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1950129","DOI":"10.1142\/S0219887819501299","article-title":"On some inequalities for statistical submanifold of quaternion K\u00e4hler-like statistical space forms","volume":"16","author":"Aquib","year":"2019","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s00025-019-1091-y","article-title":"A Chen first inequality for statistical submanifold in Hessian manifolds of constant Hessian curvature","volume":"74","author":"Chen","year":"2019","journal-title":"Results Math."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Aytimur, H., Kon, M., Mihai, A., \u00d6zg\u00fcr, C., and Takano, K. (2019). Chen inequalities for statistical submanifold of K\u00e4hler-like statistical manifold. Mathematics, 7.","DOI":"10.3390\/math7121202"},{"key":"ref_20","unstructured":"Nielsen, F., and Barbaresco, F. (2019). Chen inequalities for statistical submanifold in Sasakian statistical manifold. Geometric Science of Information, GSI 2019, Springer. Lecture Notes in Computer Science."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s00022-018-0418-2","article-title":"Generalized normalized \u03b4-Casorati curvature for statistical submanifold in quaternion K\u00e4hler-like statistical space forms","volume":"109","author":"Aquib","year":"2018","journal-title":"J. Geom."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Alkhaldi, A.H., Aquib, M., Siddiqui, A.N., and Shahid, M.H. (2018). Pinching theorems for statistical submanifold in Sasaki-like statistical space forms. Entropy, 20.","DOI":"10.3390\/e20090690"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"465","DOI":"10.2298\/FIL1503465A","article-title":"Some inequalities on submanifolds in statistical manifold of constant curvature","volume":"29","author":"Aydin","year":"2015","journal-title":"Filomat"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Mihai, A., and Mihai, I. (2018). Curvature invariants for statistical submanifold of Hessian manifolds of constant Hessian curvature. Mathematics, 6.","DOI":"10.3390\/math6030044"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3571","DOI":"10.2298\/FIL2111571S","article-title":"On CR-statistical submanifolds of holomorphic statistical manifolds","volume":"35","author":"Siddiqui","year":"2021","journal-title":"Filomat"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Yano, K., and Kon, M. (1984). Structures on Manifolds, World Scientific.","DOI":"10.1142\/0067"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Chen, B.Y. (2017). Differential Geometry of Warped Product Manifolds and Submanifolds, World Scientific.","DOI":"10.1142\/10419"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1229","DOI":"10.1016\/j.chaos.2008.04.007","article-title":"Golden differential geometry","volume":"38","author":"Crasmareanu","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/s00009-022-02094-3","article-title":"Golden Riemannian manifolds having constant sectional curvatures and their submanifolds","volume":"19","year":"2022","journal-title":"Mediterr. J. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"226","DOI":"10.32323\/ujma.439013","article-title":"Sasakian statistical manifolds with semi-symmetric metric connection","volume":"1","author":"Kazan","year":"2018","journal-title":"Univ. J. Math. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"393","DOI":"10.46793\/KgJMat2403.393K","article-title":"Geometric inequalities for statistical submanifolds in cosymplectic statistical manifolds","volume":"48","author":"Kazaz","year":"2024","journal-title":"Kragujev. J. Math."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Uddin, S., Peyghan, E., Nourmohammadifar, L., and Bossly, R. (2023). On nearly Sasakian and nearly K\u00e4hler statistical manifolds. Mathematics, 11.","DOI":"10.20944\/preprints202305.1356.v1"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"\u015eahin, F., \u015eahin, B., and Erdo\u011fan, F.E. (2023). Norden golden manifolds with constant sectional curvature and their submanifolds. Mathematics, 11.","DOI":"10.3390\/math11153301"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3941","DOI":"10.1007\/s40840-020-00905-y","article-title":"Classification of almost Norden golden manifolds","volume":"43","author":"Etayo","year":"2020","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"905","DOI":"10.1016\/S1874-5741(00)80012-6","article-title":"Affine differential geometry","volume":"Volume 1","author":"Dillen","year":"2000","journal-title":"Handbook of Differential Geometry"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1007\/BF00050660","article-title":"Fundamental equations for statistical submanifold with applications to the Bartlett correction","volume":"41","author":"Vos","year":"1989","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1515\/math-2022-0017","article-title":"Basic inequalities for statistical submanifolds in golden-like statistical manifolds","volume":"20","author":"Lone","year":"2022","journal-title":"Open Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1206\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:17:50Z","timestamp":1760033870000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1206"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,29]]},"references-count":37,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["sym17081206"],"URL":"https:\/\/doi.org\/10.3390\/sym17081206","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,7,29]]}}}