{"id":"https://openalex.org/W4414767973","doi":"https://doi.org/10.3390/axioms14100749","title":"Weakly B-Symmetric Warped Product Manifolds with Applications","display_name":"Weakly B-Symmetric Warped Product Manifolds with Applications","publication_year":2025,"publication_date":"2025-10-02","ids":{"openalex":"https://openalex.org/W4414767973","doi":"https://doi.org/10.3390/axioms14100749"},"language":"en","primary_location":{"id":"doi:10.3390/axioms14100749","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14100749","pdf_url":null,"source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://doi.org/10.3390/axioms14100749","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5037263664","display_name":"Bang\u2010Yen Chen","orcid":"https://orcid.org/0000-0002-1270-094X"},"institutions":[{"id":"https://openalex.org/I87216513","display_name":"Michigan State University","ror":"https://ror.org/05hs6h993","country_code":"US","type":"education","lineage":["https://openalex.org/I87216513"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Bang-Yen Chen","raw_affiliation_strings":["Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824, USA"],"raw_orcid":"https://orcid.org/0000-0002-1270-094X","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824, USA","institution_ids":["https://openalex.org/I87216513"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5053149818","display_name":"Sameh Shenawy","orcid":null},"institutions":[{"id":"https://openalex.org/I890898191","display_name":"Modern Academy","ror":"https://ror.org/02exeb428","country_code":"EG","type":"education","lineage":["https://openalex.org/I890898191"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"Sameh Shenawy","raw_affiliation_strings":["Basic Science Department, Modern Academy for Engineering and Technology, Maadi 4411602, Egypt"],"raw_orcid":"https://orcid.org/0000-0003-3548-4239","affiliations":[{"raw_affiliation_string":"Basic Science Department, Modern Academy for Engineering and Technology, Maadi 4411602, Egypt","institution_ids":["https://openalex.org/I890898191"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5008191077","display_name":"Uday Chand De","orcid":"https://orcid.org/0000-0002-8990-4609"},"institutions":[{"id":"https://openalex.org/I106542073","display_name":"University of Calcutta","ror":"https://ror.org/01e7v7w47","country_code":"IN","type":"education","lineage":["https://openalex.org/I106542073"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Uday Chand De","raw_affiliation_strings":["Department of Pure Mathematics, University of Calcutta 35, Ballygaunge Circular Road Kolkata, West Bengal 700019, India"],"raw_orcid":"https://orcid.org/0000-0002-8990-4609","affiliations":[{"raw_affiliation_string":"Department of Pure Mathematics, University of Calcutta 35, Ballygaunge Circular Road Kolkata, West Bengal 700019, India","institution_ids":["https://openalex.org/I106542073"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5104365571","display_name":"S. Saleem Ahmed","orcid":null},"institutions":[{"id":"https://openalex.org/I66513531","display_name":"Fayoum University","ror":"https://ror.org/023gzwx10","country_code":"EG","type":"education","lineage":["https://openalex.org/I66513531"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"Safaa Ahmed","raw_affiliation_strings":["DMathematics Department, Faculty of Science, Fayoum University, Fayoum 63514, Egypt"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"DMathematics Department, Faculty of Science, Fayoum University, Fayoum 63514, Egypt","institution_ids":["https://openalex.org/I66513531"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5092690151","display_name":"Hanan Alohali","orcid":"https://orcid.org/0009-0005-0052-4976"},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Hanan Alohali","raw_affiliation_strings":["Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia"],"raw_orcid":"https://orcid.org/0009-0005-0052-4976","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia","institution_ids":["https://openalex.org/I28022161"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5037263664"],"corresponding_institution_ids":["https://openalex.org/I87216513"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.3661514,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"14","issue":"10","first_page":"749","last_page":"749"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9945999979972839,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9945999979972839,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10719","display_name":"3D Shape Modeling and Analysis","score":0.9869999885559082,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.986299991607666,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/geodesic","display_name":"Geodesic","score":0.6399000287055969},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.5205000042915344},{"id":"https://openalex.org/keywords/curvature","display_name":"Curvature","score":0.453900009393692},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.42890000343322754},{"id":"https://openalex.org/keywords/differential-geometry","display_name":"Differential geometry","score":0.40849998593330383},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.3903999924659729},{"id":"https://openalex.org/keywords/extension","display_name":"Extension (predicate logic)","score":0.3716000020503998},{"id":"https://openalex.org/keywords/image-warping","display_name":"Image warping","score":0.3684999942779541},{"id":"https://openalex.org/keywords/scalar-curvature","display_name":"Scalar curvature","score":0.3601999878883362}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.680899977684021},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.6399000287055969},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5605999827384949},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.5205000042915344},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.453900009393692},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.42890000343322754},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.40849998593330383},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.3903999924659729},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.3716000020503998},{"id":"https://openalex.org/C157202957","wikidata":"https://www.wikidata.org/wiki/Q1659609","display_name":"Image warping","level":2,"score":0.3684999942779541},{"id":"https://openalex.org/C12520029","wikidata":"https://www.wikidata.org/wiki/Q1147161","display_name":"Scalar curvature","level":3,"score":0.3601999878883362},{"id":"https://openalex.org/C146846114","wikidata":"https://www.wikidata.org/wiki/Q901448","display_name":"Einstein","level":2,"score":0.359499990940094},{"id":"https://openalex.org/C95857938","wikidata":"https://www.wikidata.org/wiki/Q658429","display_name":"Vector bundle","level":2,"score":0.3440000116825104},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3418999910354614},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.33629998564720154},{"id":"https://openalex.org/C2779668893","wikidata":"https://www.wikidata.org/wiki/Q2250422","display_name":"Constant curvature","level":3,"score":0.33309999108314514},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.3328999876976013},{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.3059000074863434},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.3025999963283539},{"id":"https://openalex.org/C203934109","wikidata":"https://www.wikidata.org/wiki/Q7322952","display_name":"Ricci-flat manifold","level":4,"score":0.3000999987125397},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.2867000102996826},{"id":"https://openalex.org/C12089564","wikidata":"https://www.wikidata.org/wiki/Q1195879","display_name":"Ricci curvature","level":3,"score":0.2842000126838684},{"id":"https://openalex.org/C128107574","wikidata":"https://www.wikidata.org/wiki/Q182003","display_name":"Injective function","level":2,"score":0.28279998898506165},{"id":"https://openalex.org/C146710177","wikidata":"https://www.wikidata.org/wiki/Q1154996","display_name":"Fiber bundle","level":3,"score":0.2694000005722046},{"id":"https://openalex.org/C2776196297","wikidata":"https://www.wikidata.org/wiki/Q17138781","display_name":"Twist","level":2,"score":0.25290000438690186}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/axioms14100749","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14100749","pdf_url":null,"source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:1c178faad1d74a6f95fdff4b0ee8b26d","is_oa":true,"landing_page_url":"https://doaj.org/article/1c178faad1d74a6f95fdff4b0ee8b26d","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 14, Iss 10, p 749 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms14100749","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14100749","pdf_url":null,"source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W1607146642","https://openalex.org/W1622953872","https://openalex.org/W1676326315","https://openalex.org/W2009780691","https://openalex.org/W2034237196","https://openalex.org/W2037236360","https://openalex.org/W2056730219","https://openalex.org/W2228243317","https://openalex.org/W2562468920","https://openalex.org/W2570949444","https://openalex.org/W2894844131","https://openalex.org/W2909039210","https://openalex.org/W2940713130","https://openalex.org/W2994849527","https://openalex.org/W3097388198","https://openalex.org/W3098693533","https://openalex.org/W3099575820","https://openalex.org/W3105920530","https://openalex.org/W3159004522","https://openalex.org/W3213766375","https://openalex.org/W4280538903","https://openalex.org/W4283159122","https://openalex.org/W4312902550","https://openalex.org/W4377024844","https://openalex.org/W4391589163","https://openalex.org/W4400357305","https://openalex.org/W4402521335","https://openalex.org/W4403943888"],"related_works":[],"abstract_inverted_index":{"This":[0],"work":[1],"presents":[2],"a":[3,13],"comprehensive":[4],"study":[5],"of":[6,16,34,90,104,127,156],"weakly":[7,95],"B-symmetric":[8,96],"warped":[9,55,157],"product":[10,56,158],"manifolds":[11,70,159],"(WBS)n,":[12],"natural":[14],"extension":[15],"several":[17],"classical":[18],"curvature-restricted":[19,146],"geometries":[20],"including":[21,130],"B-flat,":[22],"B-parallel,":[23],"and":[24,48,68,87,118,133,153],"B-recurrent":[25],"manifolds.":[26],"We":[27],"begin":[28],"by":[29],"formulating":[30],"the":[31,35,41,45,88,112,115,125],"fundamental":[32],"properties":[33],"B-tensor":[36],"B(X,Y)=aS(X,Y)+brg(X,Y),":[37],"where":[38],"S":[39],"is":[40,58,98],"Ricci":[42],"tensor,":[43],"r":[44],"scalar":[46],"curvature,":[47],"a,b":[49],"are":[50,77,122],"smooth":[51],"non-vanishing":[52],"functions.":[53],"The":[54,94],"structure":[57],"then":[59],"exploited":[60],"to":[61,108],"obtain":[62],"explicit":[63],"curvature":[64],"identities":[65],"for":[66,79],"base":[67],"fiber":[69],"under":[71],"various":[72],"geometric":[73,152],"constraints.":[74],"Detailed":[75],"characterizations":[76],"established":[78],"Einstein":[80],"conditions,":[81],"Codazzi-type":[82],"tensors,":[83,86],"cyclic":[84],"parallel":[85],"behavior":[89],"geodesic":[91],"vector":[92,105],"fields.":[93],"condition":[97],"analyzed":[99],"through":[100],"all":[101],"possible":[102],"projections":[103],"fields,":[106],"leading":[107],"sharp":[109],"criteria":[110],"describing":[111],"interaction":[113],"between":[114],"warping":[116],"function":[117],"curvature.":[119],"Several":[120],"applications":[121],"discussed":[123],"in":[124,137],"context":[126],"Lorentzian":[128],"geometry,":[129],"perfect":[131],"fluid":[132],"generalized":[134],"Robertson\u2013Walker":[135],"spacetimes":[136],"general":[138],"relativity.":[139],"These":[140],"results":[141],"not":[142],"only":[143],"unify":[144],"different":[145],"frameworks":[147],"but":[148],"also":[149],"reveal":[150],"new":[151],"physical":[154],"implications":[155],"endowed":[160],"with":[161],"weak":[162],"B-symmetry.":[163]},"counts_by_year":[],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-03T00:00:00"}