{"id":"https://openalex.org/W2055529107","doi":"https://doi.org/10.1155/ijmms/2006/73094","title":"On the biharmonicity of product maps","display_name":"On the biharmonicity of product maps","publication_year":2006,"publication_date":"2006-01-01","ids":{"openalex":"https://openalex.org/W2055529107","doi":"https://doi.org/10.1155/ijmms/2006/73094","mag":"2055529107"},"language":"en","primary_location":{"id":"doi:10.1155/ijmms/2006/73094","is_oa":true,"landing_page_url":"https://doi.org/10.1155/ijmms/2006/73094","pdf_url":"https://downloads.hindawi.com/journals/ijmms/2006/073094.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://downloads.hindawi.com/journals/ijmms/2006/073094.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5026303450","display_name":"L\u00e9onard Todjihounde","orcid":"https://orcid.org/0000-0001-8853-8793"},"institutions":[{"id":"https://openalex.org/I4210119417","display_name":"Universit\u00e9 d'Abomey-Calavi","ror":"https://ror.org/03gzr6j88","country_code":"BJ","type":"education","lineage":["https://openalex.org/I4210119417"]},{"id":"https://openalex.org/I4210123509","display_name":"Institute of Mathematics and Physics","ror":"https://ror.org/035seyp42","country_code":"BJ","type":"facility","lineage":["https://openalex.org/I4210123509"]}],"countries":["BJ"],"is_corresponding":true,"raw_author_name":"Leonard Todjihounde","raw_affiliation_strings":["Institut de Mathematiques et de Sciences Physiques, Universit\u00e9 Nationale du Benin, BP 613, Benin"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut de Mathematiques et de Sciences Physiques, Universit\u00e9 Nationale du Benin, BP 613, Benin","institution_ids":["https://openalex.org/I4210123509","https://openalex.org/I4210119417"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5026303450"],"corresponding_institution_ids":["https://openalex.org/I4210119417","https://openalex.org/I4210123509"],"apc_list":{"value":1025,"currency":"USD","value_usd":1025},"apc_paid":{"value":1025,"currency":"USD","value_usd":1025},"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.14915805,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"2006","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9998000264167786,"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":0.9998000264167786,"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11822","display_name":"Analytic and geometric function theory","score":0.9934999942779541,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/biharmonic-equation","display_name":"Biharmonic equation","score":0.8686034679412842},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8664127588272095},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.7547774910926819},{"id":"https://openalex.org/keywords/harmonic","display_name":"Harmonic","score":0.6288450956344604},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5743057131767273},{"id":"https://openalex.org/keywords/harmonic-map","display_name":"Harmonic map","score":0.565047025680542},{"id":"https://openalex.org/keywords/product-metric","display_name":"Product metric","score":0.44564592838287354},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3619078993797302},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3237023651599884},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1968768835067749},{"id":"https://openalex.org/keywords/metric-space","display_name":"Metric space","score":0.07715487480163574}],"concepts":[{"id":"https://openalex.org/C145784156","wikidata":"https://www.wikidata.org/wiki/Q859808","display_name":"Biharmonic equation","level":3,"score":0.8686034679412842},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8664127588272095},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.7547774910926819},{"id":"https://openalex.org/C127934551","wikidata":"https://www.wikidata.org/wiki/Q1148098","display_name":"Harmonic","level":2,"score":0.6288450956344604},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5743057131767273},{"id":"https://openalex.org/C6270764","wikidata":"https://www.wikidata.org/wiki/Q5659257","display_name":"Harmonic map","level":2,"score":0.565047025680542},{"id":"https://openalex.org/C154497576","wikidata":"https://www.wikidata.org/wiki/Q7247777","display_name":"Product metric","level":3,"score":0.44564592838287354},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3619078993797302},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3237023651599884},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1968768835067749},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.07715487480163574},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1155/ijmms/2006/73094","is_oa":true,"landing_page_url":"https://doi.org/10.1155/ijmms/2006/73094","pdf_url":"https://downloads.hindawi.com/journals/ijmms/2006/073094.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.605.7825","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.605.7825","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.emis.de/journals/HOA/IJMMS/Volume2006/073094.pdf","raw_type":"text"},{"id":"pmh:oai:RePEc:hin:jijmms:073094","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/IJMMS/2006/073094.xml","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"pmh:oai:doaj.org/article:73a5e8ffd5914e6eb25caebc3ae67647","is_oa":true,"landing_page_url":"https://doaj.org/article/73a5e8ffd5914e6eb25caebc3ae67647","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":"International Journal of Mathematics and Mathematical Sciences, Vol 2006 (2006)","raw_type":"article"},{"id":"pmh:oai:hindawi.com:10.1155/IJMMS/2006/73094","is_oa":false,"landing_page_url":"https://doi.org/10.1155/IJMMS/2006/73094","pdf_url":null,"source":{"id":"https://openalex.org/S4306400340","display_name":"Hindawi Journal of Chemistry (Hindawi)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210126990","host_organization_name":"Hindawi (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210126990"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":""}],"best_oa_location":{"id":"doi:10.1155/ijmms/2006/73094","is_oa":true,"landing_page_url":"https://doi.org/10.1155/ijmms/2006/73094","pdf_url":"https://downloads.hindawi.com/journals/ijmms/2006/073094.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320320894","display_name":"Abdus Salam International Centre for Theoretical Physics","ror":"https://ror.org/009gyvm78"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2055529107.pdf","grobid_xml":"https://content.openalex.org/works/W2055529107.grobid-xml"},"referenced_works_count":12,"referenced_works":["https://openalex.org/W1504186549","https://openalex.org/W1632597447","https://openalex.org/W1840740558","https://openalex.org/W1974228548","https://openalex.org/W2036750221","https://openalex.org/W2081119278","https://openalex.org/W2111942807","https://openalex.org/W2123870705","https://openalex.org/W2184095895","https://openalex.org/W2259016291","https://openalex.org/W2892909429","https://openalex.org/W4230656492"],"related_works":["https://openalex.org/W2626911641","https://openalex.org/W2001868403","https://openalex.org/W1977275827","https://openalex.org/W3080936195","https://openalex.org/W2194610970","https://openalex.org/W1491468467","https://openalex.org/W2952428296","https://openalex.org/W3118692344","https://openalex.org/W4311276160","https://openalex.org/W3100204338"],"abstract_inverted_index":{"We":[0,27],"introduce":[1],"the":[2],"warped":[3,10,21,41],"product":[4,11,22,42],"of":[5,32],"maps":[6,23],"defined":[7],"between":[8],"Riemannian":[9],"spaces":[12],"and":[13,17,36],"we":[14],"give":[15],"necessary":[16],"sufficient":[18],"conditions":[19],"for":[20],"to":[24],"be":[25],"(bi)harmonic.":[26],"obtain":[28],"then":[29],"some":[30],"characterizations":[31],"nontrivial":[33],"harmonic":[34],"metrics":[35,39],"nonharmonic":[37],"biharmonic":[38],"on":[40],"spaces.":[43]},"counts_by_year":[{"year":2023,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}