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(2008). The Golden Ratio: The Story of phi, the World\u2019s Most Astonishing Number, Random House, Inc. Crown."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"199","DOI":"10.2996\/kmj\/1138846118","article-title":"Polynomial structures on manifolds","volume":"22","author":"Goldberg","year":"1970","journal-title":"Kodai Math. Sem. Rep."},{"key":"ref_3","first-page":"199","article-title":"On some invariant submanifolds in a Riemannian manifold with golden structure","volume":"53","author":"Hretcanu","year":"2007","journal-title":"An. \u015etiin\u0163 Univ. Al. I. Cuza Ia\u015fi. Mat. (N.S.)"},{"key":"ref_4","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_5","first-page":"179","article-title":"Applications of the golden ratio on Riemannian manifolds","volume":"33","author":"Hretcanu","year":"2009","journal-title":"Turkish J. Math."},{"key":"ref_6","first-page":"693","article-title":"On integrability of golden Riemannian structures","volume":"37","author":"Gezer","year":"2013","journal-title":"Turkish J. Math."},{"key":"ref_7","first-page":"227","article-title":"Prolongations of golden structures to tangent bundles","volume":"16","year":"2014","journal-title":"Differ. Geom. Dyn. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.36890\/iejg.690479","article-title":"A new class of golden Riemannian manifold","volume":"13","author":"Beldjilali","year":"2020","journal-title":"Int. Electron. J. Geom."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1007\/s00009-019-1343-9","article-title":"s-golden manifolds","volume":"16","author":"Gherici","year":"2019","journal-title":"Mediter. J. Math."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Hretcanu, C.E., and Crasmareanu, M. (2023). The (\u03b1, p)-golden metric manifolds and their submanifolds. Mathematics, 11.","DOI":"10.20944\/preprints202306.0082.v1"},{"key":"ref_11","first-page":"967","article-title":"On a study of the totally umbilical semi-invariant submanifolds of golden Riemannian manifolds","volume":"21","author":"Yildirim","year":"2018","journal-title":"Politeknik Dergisi"},{"key":"ref_12","first-page":"1119","article-title":"Invariant submanifolds in golden Riemannian manifolds","volume":"69","year":"2020","journal-title":"Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hretcanu, C.E., and Blaga, A.M. (2021). Warped product submanifolds in locally golden Riemannian manifolds with a slant factor. Mathematics, 9.","DOI":"10.3390\/math9172125"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0002-9947-1969-0251664-4","article-title":"Manifolds of negative curvature","volume":"145","author":"Bishop","year":"1969","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_15","first-page":"177","article-title":"Geometry of warped product CR-submanifolds in Kaehler manifolds, Monatsh","volume":"133","author":"Chen","year":"2001","journal-title":"Math."},{"key":"ref_16","first-page":"125","article-title":"Geometry of warped products as Riemannian submanifolds and related problems","volume":"28","author":"Chen","year":"2002","journal-title":"Soochow J. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"415","DOI":"10.14492\/hokmj\/1350657533","article-title":"Another general inequality for CR-warped products in complex space forms","volume":"32","author":"Chen","year":"2003","journal-title":"Hokkaido Math. J."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/s00605-002-0009-y","article-title":"CR-warped products in complex projective spaces with compact holomorphic factor","volume":"141","author":"Chen","year":"2004","journal-title":"Monatsh. Math."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Chen, B.-Y. (2017). Differential Geometry of Warped Product Manifolds and Submanifolds, World Scientific Publ.","DOI":"10.1142\/10419"},{"key":"ref_20","first-page":"39","article-title":"Golden warped product Riemannian manifolds","volume":"37","author":"Blaga","year":"2018","journal-title":"Lib. Math. (N.S.)"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Duggal, K.L., and Bejancu, A. (1996). Lightlike Submanifold of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-017-2089-2"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1007\/s00009-017-0999-2","article-title":"Lightlike hypersurfaces of a golden semi-Riemannian manifold","volume":"14","author":"Poyraz","year":"2017","journal-title":"Mediterr. J. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.geomphys.2019.03.008","article-title":"Lightlike submanifolds of golden semi-Riemannian manifolds","volume":"141","author":"Poyraz","year":"2019","journal-title":"J. Geom. Phys."},{"key":"ref_24","unstructured":"Ahmad, M., and Qayyoom, M.A. (2022). CR-lightlike submanifolds of a golden semi-Riemannian manifold. arXiv."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Erdo\u011fan, F.E. (2018). Screen transversal lightlike submanifolds of golden semi-Riemannian manifolds. arXiv.","DOI":"10.1063\/1.5047884"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Erdo\u011fan, F.E., Yildirim, C., and Karatas, E. (2018). Transversal lightlike submanifolds of golden semi-Riemannian manifolds. arXiv.","DOI":"10.1063\/1.5047884"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"13","DOI":"10.30755\/NSJOM.12656","article-title":"Screen slant lightlike submanifolds of golden semi-Riemannian manifolds","volume":"54","author":"Kumar","year":"2024","journal-title":"Novi Sad J. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"106","DOI":"10.25092\/baunfbed.679537","article-title":"Semi-invariant lightlike submanifolds of golden semi-Riemannian manifolds","volume":"22","author":"Poyraz","year":"2020","journal-title":"Bal\u0131Kesir \u00dcniversitesi Fen Bilim. Enstit\u00fcs\u00fc Dergisi"},{"key":"ref_29","first-page":"1","article-title":"Screen semi-invariant lightlike submanifolds of a golden semi-Riemannian manifold","volume":"3","author":"Shanker","year":"2022","journal-title":"Bing\u00f6l \u00dcniversitesi Tek. Bilim. Derg."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Erdo\u011fan, F.E., Perkta\u015f, S.Y., Bozda\u011f, \u015e.N., and Acet, B.E. (2023). Lightlike hypersurfaces of meta-golden semi-Riemannian manifolds. Mathematics, 11.","DOI":"10.3390\/math11234798"},{"key":"ref_31","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_32","doi-asserted-by":"crossref","unstructured":"Chen, B.-Y. (2011). Pseudo-Riemannian Geometry, \u03b4-Invariants and Applications, World Scientific Publ.","DOI":"10.1142\/9789814329644"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"465","DOI":"10.2298\/FIL1503465A","article-title":"Some inequalities on submanifolds in statistical manifolds of constant curvature","volume":"29","author":"Aydin","year":"2015","journal-title":"Filomat"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Mihai, I., and Mohammed, M. (2023). Optimal inequalities for submanifolds in trans-Sasakian manifolds endowed with a semi-symmetric metric connection. Symmetry, 15.","DOI":"10.3390\/sym15040877"},{"key":"ref_35","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_36","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."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/BF02413317","article-title":"Mesure de la courbure des surfaces suivant l\u2019id\u00e9e commune","volume":"14","author":"Casorati","year":"1890","journal-title":"Acta Math."},{"key":"ref_38","first-page":"732","article-title":"Some Inequalities for golden Riemannian space forms","volume":"12","author":"Almekhildy","year":"2023","journal-title":"Palest. J. Math."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3906\/mat-2009-8","article-title":"Recent developments in \u03b4-Casorati curvature invariants","volume":"45","author":"Chen","year":"2021","journal-title":"Turkish J. Math."},{"key":"ref_40","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_41","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s00022-020-00544-5","article-title":"Optimization on slant submanifolds of golden Riemannian manifolds using generalized normalized \u03b4-Casorati curvatures","volume":"111","author":"Choudhary","year":"2020","journal-title":"J. Geometry"},{"key":"ref_42","first-page":"A993","article-title":"Sur l\u2019in\u00e9galit\u00e9 de Chen-Willmore","volume":"288","author":"Wintgen","year":"1979","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. A-B"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"6","DOI":"10.36890\/iejg.838446","article-title":"Recent developments in Wintgen inequality and Wintgen ideal submanifolds","volume":"14","author":"Chen","year":"2021","journal-title":"Int. Electron. J. Geom."},{"key":"ref_44","first-page":"1","article-title":"Generalized Wintgen inequality for some submanifolds in golden Riemannian space forms","volume":"25","author":"Choudhary","year":"2020","journal-title":"Balkan J. Geom. Appl."},{"key":"ref_45","first-page":"23","article-title":"Slant submanifolds of golden Riemannian manifolds","volume":"13","author":"Uddin","year":"2019","journal-title":"J. Math. Ext."},{"key":"ref_46","unstructured":"Hretcanu, C. (June, January 28). Submanifolds in Riemannian manifold with golden structure. Proceedings of the Workshop on Finsler Geometry and Its Applications, Balatonf\u00f6ldv\u00e1r, Hungary."},{"key":"ref_47","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_48","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1007\/s10455-015-9475-z","article-title":"Bochner\u2019s technique for statistical structures","volume":"48","author":"Opozda","year":"2015","journal-title":"Ann. Glob. Anal. Geom."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1007\/s00022-021-00590-7","article-title":"Generalized Wintgen inequality for slant submanifolds in metallic Riemannian space forms","volume":"112","author":"Choudhary","year":"2021","journal-title":"J. Geometry"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Choudhary, M.A., Khedher, K.M., Bahad\u0131r, O., and Siddiqi, M.D. (2021). On golden Lorentzian manifolds equipped with generalized symmetric metric connection. Mathematics, 9.","DOI":"10.3390\/math9192430"},{"key":"ref_51","unstructured":"Huylebrouck, D. (2014, January 14\u201319). The meta-golden ratio chi. Proceedings of the Bridges 2014: Mathematics, Music, Art, Architecture, Culture, Seoul, Republic of Korea."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"10491","DOI":"10.1002\/mma.8380","article-title":"Meta-golden Riemannian manifolds","volume":"45","year":"2022","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_53","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_54","first-page":"47","article-title":"Anti-invariant submanifolds of locally decomposable golden Riemannian manifolds","volume":"25","author":"Keles","year":"2020","journal-title":"Balkan J. Geom. Appl."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s40065-020-00307-9","article-title":"Non-invariant submanifolds of locally decomposable golden Riemannian manifolds","volume":"10","year":"2021","journal-title":"Arab. J. Math."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"343","DOI":"10.36890\/iejg.1240437","article-title":"The Stability problem of certain anti-Invariant submanifolds in golden Riemannian manifolds","volume":"16","year":"2023","journal-title":"Int. Electron. J. Geom."},{"key":"ref_57","first-page":"8","article-title":"On submanifolds in a Riemannian manifold with golden structure","volume":"11","author":"Ahmad","year":"2019","journal-title":"Turkish J. Math. Comp. Sci."},{"key":"ref_58","first-page":"1833","article-title":"Semi-invariant submanifolds of golden Riemannian manifolds","volume":"20044","year":"2017","journal-title":"AIP Conf. Proc."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"G\u00f6k, M., Kele\u015f, S., and K\u0131l\u0131\u00e7, E. (2019). Some characterizations of semi-invariant submanifolds of golden Riemannian manifolds. Mathematics, 7.","DOI":"10.3390\/math7121209"},{"key":"ref_60","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_61","first-page":"1","article-title":"Warped product skew semi-invariant submanifolds of locally golden Riemannian manifolds","volume":"44","author":"Ahmad","year":"2022","journal-title":"Honam Math. J."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1017\/S0004972700017925","article-title":"Slant immersions","volume":"41","author":"Chen","year":"1990","journal-title":"Bull. Austral. Math. Soc."},{"key":"ref_63","unstructured":"Chen, B.-Y. (1990). Geometry of Slant Submanifolds, Katholieke Universiteit Leuven."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"29","DOI":"10.5802\/afst.711","article-title":"Slant surfaces with codimension two","volume":"11","author":"Chen","year":"1990","journal-title":"Ann. Fac. Sci. Toulouse Math."},{"key":"ref_65","first-page":"630","article-title":"Pointwise slant submanifolds in almost Hermitian manifolds","volume":"36","author":"Chen","year":"2012","journal-title":"Turkish J. Math."},{"key":"ref_66","first-page":"689","article-title":"CR-submanifolds of a golden Riemannian manifold","volume":"12","author":"Ahmad","year":"2023","journal-title":"Palest. J. Math."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"161","DOI":"10.5556\/j.tkjm.51.2020.2913","article-title":"Warped product submanifolds in metallic Riemannian manifolds","volume":"51","author":"Hretcanu","year":"2020","journal-title":"Tamkang J. Math."},{"key":"ref_68","doi-asserted-by":"crossref","unstructured":"Kupeli, D.N. (1996). Singular Semi-Riemannian Geometry, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-015-8761-7"},{"key":"ref_69","first-page":"211","article-title":"Geometry of lightlike submanifolds of golden semi-Riemannian manifolds","volume":"23","author":"Gupta","year":"2019","journal-title":"J. Int. Acad. Phys. Sci."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"207","DOI":"10.36890\/iejg.893398","article-title":"Screen semi-invariant lightlike submanifolds of golden semi-Riemannian manifolds","volume":"14","author":"Poyraz","year":"2021","journal-title":"Int. Electron. J. Geom."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"3231","DOI":"10.2298\/FIL1910231E","article-title":"On some types of lightlike submanifolds of golden semi-Riemannian manifolds","volume":"33","year":"2019","journal-title":"Filomat"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"13043","DOI":"10.3934\/math.2024636","article-title":"On CR-lightlike submanifolds in a golden semi-Riemannian manifold","volume":"9","author":"Qayyoom","year":"2024","journal-title":"AIMS Math."},{"key":"ref_73","first-page":"121","article-title":"Half lightlike submanifolds of codimension 2","volume":"22","author":"Duggal","year":"1999","journal-title":"Math. J. Toyama Univ."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"99","DOI":"10.46793\/KgJMat2401.099P","article-title":"Half lightlike submanifolds of a golden semi-Riemannian manifold","volume":"48","author":"Poyraz","year":"2024","journal-title":"Kragujevac J. Math."},{"key":"ref_75","first-page":"118","article-title":"Golden generic lightlike submanifolds of a golden semi-Riemannian manifold","volume":"27","author":"Pruthi","year":"2022","journal-title":"Balkan J. Geom. Appl."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1007\/s00009-022-02122-2","article-title":"Screen generic lightlike submanifolds of golden semi-Riemannian manifolds","volume":"19","author":"Yadav","year":"2022","journal-title":"Mediterr. J. Math."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00009-020-01567-7","article-title":"Golden GCR-lightlike submanifolds of golden semi-Riemannian manifolds","volume":"17","author":"Poyraz","year":"2020","journal-title":"Mediterr. J. Math."},{"key":"ref_78","first-page":"107","article-title":"Geometry of golden GCR-lightlike submanifolds of golden semi-Riemannian manifolds","volume":"68","author":"Poyraz","year":"2022","journal-title":"An. S\u00b8 tiint\u00b8. Univ. Al. I. Cuza Ias\u00b8i. Mat. (N.S.)"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"2037","DOI":"10.15672\/hujms.590342","article-title":"Screen pseudo slant lightlike submanifolds of golden semi-Riemannian manifolds","volume":"49","author":"Acet","year":"2020","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_80","first-page":"823","article-title":"Screen semi slant lightlike submanifolds of golden semi-Riemannian manifolds","volume":"15","author":"Yadav","year":"2022","journal-title":"Jordan J. Math. Stat."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"141","DOI":"10.46793\/KgJMat2501.141K","article-title":"Semi-slant lightlike submanifolds of golden semi-Riemannian manifolds","volume":"49","author":"Kumar","year":"2025","journal-title":"Kragujevac. J. Math."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"19526","DOI":"10.3934\/math.2023996","article-title":"Bi-slant lightlike submanifolds of golden semi-Riemannian manifolds","volume":"8","author":"Ahmad","year":"2023","journal-title":"AIMS Math."},{"key":"ref_83","first-page":"175","article-title":"Slant lightlike submanifolds of golden semi-Riemannian manifolds","volume":"75","author":"Yadav","year":"2023","journal-title":"Mat. Vesnik"},{"key":"ref_84","first-page":"813","article-title":"Lightlike hypersurfaces of almost Norden golden semi-Riemannian manifolds","volume":"37","author":"Acet","year":"2022","journal-title":"Facta Univ. Ser. Math. Inform."},{"key":"ref_85","first-page":"152","article-title":"Study on warped product of screen real lightlike submanifolds of a golden semi-Riemannian manifold","volume":"27","author":"Shanker","year":"2022","journal-title":"Balkan J. Geom. Appl."},{"key":"ref_86","unstructured":"Chen, B.-Y. (1998). Strings of Riemannian invariants, inequalities, ideal immersions and their applications. The Third Pacific Rim Geometry Conference (Seoul, 1996), International Press."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"105","DOI":"10.4099\/math1924.26.105","article-title":"Some new obstructions to minimal and Lagrangian isometric immersions","volume":"26","author":"Chen","year":"2000","journal-title":"Jpn. J. Math. (N.S.)"},{"key":"ref_88","first-page":"77","article-title":"A general optimal inequality for arbitrary Riemannian submanifolds","volume":"6","author":"Chen","year":"2005","journal-title":"J. Inequal. Pure Appl. Math."},{"key":"ref_89","doi-asserted-by":"crossref","unstructured":"Decu, S. (2014, January 19\u201321). Optimal inequalities involving Casorati curvature of slant submanifolds in quaternionic space forms. Proceedings of the Riemannian Geometry and Applications\u2013Proceedings RIGA, Bucharest, Romania.","DOI":"10.1186\/1029-242X-2014-452"},{"key":"ref_90","first-page":"79","article-title":"Optimal inequalities characterizing quasi-umbilical submanifolds","volume":"9","author":"Decu","year":"2008","journal-title":"J. Inequal. Pure Appl. Math."},{"key":"ref_91","first-page":"1024","article-title":"Some bounds for Casorati curvatures on Golden Riemannian space forms with SSM connection","volume":"53","author":"Lee","year":"2024","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_92","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. Math."},{"key":"ref_93","first-page":"115","article-title":"A pointwise inequality in submanifold theory","volume":"35","author":"Smet","year":"1999","journal-title":"Arch. Math. (Brno)"},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"87","DOI":"10.2140\/pjm.2008.237.87","article-title":"A proof of the DDVV conjecture and its equality case","volume":"237","author":"Ge","year":"2008","journal-title":"Pac. J. Math."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00009-017-0991-x","article-title":"On the geometry of almost golden Riemannian manifolds","volume":"14","author":"Etayo","year":"2017","journal-title":"Mediter. J. Math."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"123","DOI":"10.29229\/uzmj.2019-4-12","article-title":"Generic submanifolds of golden Riemannian manifolds","volume":"4","author":"Siddiqi","year":"2019","journal-title":"Uzbek Math. J."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"181","DOI":"10.56827\/SEAJMMS.2024.2001.14","article-title":"CR-lightlike warped product in golden semi-Riemannian manifolds","volume":"20","author":"Verma","year":"2024","journal-title":"South East Asian J. Math. Math. Sci."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/s00009-023-02352-y","article-title":"Ricci solitons on golden Riemannian manifolds","volume":"20","author":"Rani","year":"2023","journal-title":"Mediterr. J. 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