{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,3]],"date-time":"2025-10-03T00:45:50Z","timestamp":1759452350154,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,2]],"date-time":"2025-10-02T00:00:00Z","timestamp":1759363200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ongoing Research Funding Program","award":["ORF\u20142025-860"],"award-info":[{"award-number":["ORF\u20142025-860"]}]},{"name":"King Saud University, Riyadh, Saudi Arabia"}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Axioms"],"abstract":"<jats:p>This work presents a comprehensive study of weakly B-symmetric warped product manifolds (WBS)n, a natural extension of several classical curvature-restricted geometries including B-flat, B-parallel, and B-recurrent manifolds. We begin by formulating the fundamental properties of the B-tensor B(X,Y)=aS(X,Y)+brg(X,Y), where S is the Ricci tensor, r the scalar curvature, and a,b are smooth non-vanishing functions. The warped product structure is then exploited to obtain explicit curvature identities for base and fiber manifolds under various geometric constraints. Detailed characterizations are established for Einstein conditions, Codazzi-type tensors, cyclic parallel tensors, and the behavior of geodesic vector fields. The weakly B-symmetric condition is analyzed through all possible projections of vector fields, leading to sharp criteria describing the interaction between the warping function and curvature. Several applications are discussed in the context of Lorentzian geometry, including perfect fluid and generalized Robertson\u2013Walker spacetimes in general relativity. These results not only unify different curvature-restricted frameworks but also reveal new geometric and physical implications of warped product manifolds endowed with weak B-symmetry.<\/jats:p>","DOI":"10.3390\/axioms14100749","type":"journal-article","created":{"date-parts":[[2025,10,2]],"date-time":"2025-10-02T15:07:48Z","timestamp":1759417668000},"page":"749","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Weakly B-Symmetric Warped Product Manifolds with Applications"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1270-094X","authenticated-orcid":false,"given":"Bang-Yen","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3548-4239","authenticated-orcid":false,"given":"Sameh","family":"Shenawy","sequence":"additional","affiliation":[{"name":"Basic Science Department, Modern Academy for Engineering and Technology, Maadi 4411602, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8990-4609","authenticated-orcid":false,"given":"Uday Chand","family":"De","sequence":"additional","affiliation":[{"name":"Department of Pure Mathematics, University of Calcutta 35, Ballygaunge Circular Road Kolkata, West Bengal 700019, India"}]},{"given":"Safaa","family":"Ahmed","sequence":"additional","affiliation":[{"name":"DMathematics Department, Faculty of Science, Fayoum University, Fayoum 63514, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-0052-4976","authenticated-orcid":false,"given":"Hanan","family":"Alohali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"895","DOI":"10.2298\/FIL2103895M","article-title":"On weakly B-symmetric pseudo-Riemannian manifolds and their applications","volume":"35","author":"Mantica","year":"2021","journal-title":"Filomat"},{"key":"ref_2","first-page":"1271","article-title":"On weakly cyclic generalized B-symmetric manifolds","volume":"20","author":"Ali","year":"2023","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"615","DOI":"10.4134\/CKMS.2010.25.4.615","article-title":"Pseudo symmetric and pseudo Ricci symmetric warped product manifolds","volume":"25","author":"De","year":"2010","journal-title":"Commun. Korean Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/s10474-014-0462-9","article-title":"On weakly cyclic \u03c7-symmetric manifolds","volume":"146","author":"De","year":"2015","journal-title":"Acta Math. Hung."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"541","DOI":"10.4310\/jdg\/1664378619","article-title":"Weakly symmetric pseudo-Riemannian nilmanifolds","volume":"121","author":"Wolf","year":"2022","journal-title":"J. Differ. Geom."},{"key":"ref_6","first-page":"6","article-title":"On generalized weakly symmetric Kenmotsu manifolds","volume":"39","author":"Baishya","year":"2021","journal-title":"Bol. Soc. Paran. Mat."},{"key":"ref_7","first-page":"223","article-title":"On weakly symmetric and special weakly Ricci symmetric LP-Sasakian manifolds admitting semi-symmetric semi-metric connection","volume":"38","author":"Anitha","year":"2021","journal-title":"Facta Univ. Ser. Math. Inform."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Chen, B.-Y., and Blaga, A.M. (2021). Geometric inequalities for warped products in Riemannian manifolds. Mathematics, 9.","DOI":"10.3390\/math9090923"},{"key":"ref_9","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_10","doi-asserted-by":"crossref","first-page":"3395","DOI":"10.1007\/s40840-019-00874-x","article-title":"Concircular curvature on warped product manifolds and applications","volume":"43","author":"De","year":"2020","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_11","first-page":"168","article-title":"Pseudo-projective tensor on sequential warped products","volume":"20","year":"2023","journal-title":"Mediterr. J. Math."},{"key":"ref_12","first-page":"639","article-title":"Gradient \u03c1-Einstein solitons and applications (on doubly warped products)","volume":"38","year":"2023","journal-title":"Facta Univ. Ser. Math. Inform."},{"key":"ref_13","unstructured":"G\u00fcler, S., and \u00dcnal, B. (2024). Ricci\u2013Bourguignon solitons on sequential warped products. arXiv."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1550065","DOI":"10.1142\/S0129167X15500652","article-title":"2-Killing vector fields on warped product manifolds","volume":"26","author":"Shenawy","year":"2015","journal-title":"Int. J. Math."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Uddin, S., Chen, B.-Y., and Bossly, R. (2023). Geometry of CR-slant warped products in nearly K\u00e4hler manifolds. Mathematics, 11.","DOI":"10.20944\/preprints202305.1002.v1"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"239","DOI":"10.21099\/tkbjm\/1496162996","article-title":"Constant scalar curvatures on warped product manifolds","volume":"20","author":"Ehrlich","year":"1996","journal-title":"Tsukuba J. Math."},{"key":"ref_17","first-page":"169","article-title":"On warped product manifolds","volume":"9","year":"1995","journal-title":"Filomat"},{"key":"ref_18","first-page":"159","article-title":"Ricci\u2013Bourguignon solitons on sequential warped product manifolds","volume":"20","author":"Kaya","year":"2024","journal-title":"J. Math. Phys. Anal. Geom."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.2298\/FIL2403023A","article-title":"Hyperbolic Ricci solitons on sequential warped product manifolds","volume":"38","author":"Kaya","year":"2024","journal-title":"Filomat"},{"key":"ref_20","first-page":"353","article-title":"On quasi-Einstein warped products","volume":"58","author":"Sular","year":"2012","journal-title":"An. \u015etiin\u0163. Univ. Al. I. Cuza Ia\u015fi Ser. I Mat."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"733","DOI":"10.11568\/kjm.2015.23.4.733","article-title":"The existence of some metrics on Riemannian warped product manifolds with fiber manifold of class (B)","volume":"23","author":"Jung","year":"2015","journal-title":"Korean J. Math."},{"key":"ref_22","unstructured":"O\u2019Neill, B. (1983). Semi-Riemannian Geometry with Applications to Relativity, Academic Press. Pure and Applied Mathematics."},{"key":"ref_23","unstructured":"Bhattacharyya, A., Ganguly, D., and Basu, N. (2021). Characteristics of conformal Ricci soliton on warped product spaces. arXiv."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"114561","DOI":"10.1016\/j.chaos.2024.114561","article-title":"2-Killing vector fields on multiply warped product manifolds","volume":"180","author":"Blaga","year":"2024","journal-title":"Chaos Solitons Fractals"},{"key":"ref_25","unstructured":"De, U.C., Shenawy, S., and \u00dcnal, B. (2015). Sequential warped products: Curvature and Killing vector fields. arXiv."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1950124","DOI":"10.1142\/S021988781950124X","article-title":"Generalized quasi-Einstein GRW space-times","volume":"16","author":"De","year":"2019","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1007\/s44198-022-00066-5","article-title":"Perfect fluid spacetimes and gradient solitons","volume":"29","author":"De","year":"2022","journal-title":"J. Nonlinear Math. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1730001","DOI":"10.1142\/S021988781730001X","article-title":"Generalized Robertson\u2013Walker spacetimes: A survey","volume":"14","author":"Mantica","year":"2017","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_29","first-page":"1930002","article-title":"Perfect fluid spacetimes with harmonic generalized curvature tensor","volume":"16","author":"Mantica","year":"2019","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1950008","DOI":"10.1142\/S0219887819500087","article-title":"Cosmological perfect fluids in f(R) gravity","volume":"16","author":"Capozziello","year":"2019","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_31","first-page":"1143","article-title":"The equation of state at ultrahigh densities and its relativistic limitations","volume":"14","year":"1962","journal-title":"Sov. Phys. JETP"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1016\/j.na.2004.07.035","article-title":"Special standard static space-times","volume":"59","author":"Dobarro","year":"2004","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_33","first-page":"75","article-title":"Hessian tensor and standard static space-times","volume":"Volume 435","author":"Plaue","year":"2008","journal-title":"Advances in Lorentzian Geometry"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Shenawy, S., Mantica, C.A., Molinari, L.G., and Bin Turki, N. (2022). A Note on Generalized Quasi-Einstein and (\u03bb,n+m)-Einstein Manifolds with Harmonic Conformal Tensor. Mathematics, 10.","DOI":"10.3390\/math10101731"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/10\/749\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,2]],"date-time":"2025-10-02T15:21:31Z","timestamp":1759418491000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/10\/749"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,2]]},"references-count":34,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["axioms14100749"],"URL":"https:\/\/doi.org\/10.3390\/axioms14100749","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,2]]}}}