{"id":"https://openalex.org/W4392662976","doi":"https://doi.org/10.3390/axioms13030183","title":"Chen\u2013Ricci Inequality for Isotropic Submanifolds in Locally Metallic Product Space Forms","display_name":"Chen\u2013Ricci Inequality for Isotropic Submanifolds in Locally Metallic Product Space Forms","publication_year":2024,"publication_date":"2024-03-11","ids":{"openalex":"https://openalex.org/W4392662976","doi":"https://doi.org/10.3390/axioms13030183"},"language":"en","primary_location":{"id":"doi:10.3390/axioms13030183","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13030183","pdf_url":"https://www.mdpi.com/2075-1680/13/3/183/pdf?version=1710207043","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://www.mdpi.com/2075-1680/13/3/183/pdf?version=1710207043","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100376055","display_name":"Yanlin Li","orcid":"https://orcid.org/0000-0003-1614-3228"},"institutions":[{"id":"https://openalex.org/I163151501","display_name":"Hangzhou Normal University","ror":"https://ror.org/014v1mr15","country_code":"CN","type":"education","lineage":["https://openalex.org/I163151501"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yanlin Li","raw_affiliation_strings":["School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China"],"raw_orcid":"https://orcid.org/0000-0003-1614-3228","affiliations":[{"raw_affiliation_string":"School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China","institution_ids":["https://openalex.org/I163151501"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5030826658","display_name":"Meraj Ali Khan","orcid":"https://orcid.org/0000-0001-6554-1228"},"institutions":[{"id":"https://openalex.org/I240666556","display_name":"Imam Mohammad ibn Saud Islamic University","ror":"https://ror.org/05gxjyb39","country_code":"SA","type":"education","lineage":["https://openalex.org/I240666556"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Meraj Ali Khan","raw_affiliation_strings":["Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0001-6554-1228","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia","institution_ids":["https://openalex.org/I240666556"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5094119424","display_name":"MD Aquib","orcid":null},"institutions":[{"id":"https://openalex.org/I240666556","display_name":"Imam Mohammad ibn Saud Islamic University","ror":"https://ror.org/05gxjyb39","country_code":"SA","type":"education","lineage":["https://openalex.org/I240666556"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"MD Aquib","raw_affiliation_strings":["Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia","institution_ids":["https://openalex.org/I240666556"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5006916462","display_name":"Ibrahim Al-Dayel","orcid":"https://orcid.org/0000-0002-5901-2511"},"institutions":[{"id":"https://openalex.org/I240666556","display_name":"Imam Mohammad ibn Saud Islamic University","ror":"https://ror.org/05gxjyb39","country_code":"SA","type":"education","lineage":["https://openalex.org/I240666556"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Ibrahim Al-Dayel","raw_affiliation_strings":["Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-5901-2511","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia","institution_ids":["https://openalex.org/I240666556"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5049511450","display_name":"Maged Z. Youssef","orcid":"https://orcid.org/0000-0002-0365-1891"},"institutions":[{"id":"https://openalex.org/I240666556","display_name":"Imam Mohammad ibn Saud Islamic University","ror":"https://ror.org/05gxjyb39","country_code":"SA","type":"education","lineage":["https://openalex.org/I240666556"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Maged Zakaria Youssef","raw_affiliation_strings":["Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia","institution_ids":["https://openalex.org/I240666556"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5030826658"],"corresponding_institution_ids":["https://openalex.org/I240666556"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":9.0375,"has_fulltext":true,"cited_by_count":11,"citation_normalized_percentile":{"value":0.98478915,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":98,"max":99},"biblio":{"volume":"13","issue":"3","first_page":"183","last_page":"183"},"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":0.982699990272522,"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/T11830","display_name":"Point processes and geometric inequalities","score":0.9793000221252441,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/submanifold","display_name":"Submanifold","score":0.8719147443771362},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6535115838050842},{"id":"https://openalex.org/keywords/space-form","display_name":"Space form","score":0.626762866973877},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.6202570796012878},{"id":"https://openalex.org/keywords/isotropy","display_name":"Isotropy","score":0.6183546781539917},{"id":"https://openalex.org/keywords/geodesic","display_name":"Geodesic","score":0.5867732763290405},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5586881637573242},{"id":"https://openalex.org/keywords/constant-curvature","display_name":"Constant curvature","score":0.5453306436538696},{"id":"https://openalex.org/keywords/einstein","display_name":"Einstein","score":0.5414777398109436},{"id":"https://openalex.org/keywords/totally-geodesic","display_name":"Totally geodesic","score":0.5370138883590698},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.49264079332351685},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.48957833647727966},{"id":"https://openalex.org/keywords/second-fundamental-form","display_name":"Second fundamental form","score":0.4478340446949005},{"id":"https://openalex.org/keywords/mean-curvature","display_name":"Mean curvature","score":0.41811513900756836},{"id":"https://openalex.org/keywords/curvature","display_name":"Curvature","score":0.4086323082447052},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.2526056170463562},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.20110702514648438},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.15611532330513},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.11298277974128723}],"concepts":[{"id":"https://openalex.org/C151300846","wikidata":"https://www.wikidata.org/wiki/Q125539","display_name":"Submanifold","level":2,"score":0.8719147443771362},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6535115838050842},{"id":"https://openalex.org/C2776341046","wikidata":"https://www.wikidata.org/wiki/Q4381552","display_name":"Space form","level":3,"score":0.626762866973877},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.6202570796012878},{"id":"https://openalex.org/C184050105","wikidata":"https://www.wikidata.org/wiki/Q273163","display_name":"Isotropy","level":2,"score":0.6183546781539917},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.5867732763290405},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5586881637573242},{"id":"https://openalex.org/C2779668893","wikidata":"https://www.wikidata.org/wiki/Q2250422","display_name":"Constant curvature","level":3,"score":0.5453306436538696},{"id":"https://openalex.org/C146846114","wikidata":"https://www.wikidata.org/wiki/Q901448","display_name":"Einstein","level":2,"score":0.5414777398109436},{"id":"https://openalex.org/C3019300132","wikidata":"https://www.wikidata.org/wiki/Q3237377","display_name":"Totally geodesic","level":2,"score":0.5370138883590698},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.49264079332351685},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.48957833647727966},{"id":"https://openalex.org/C119830904","wikidata":"https://www.wikidata.org/wiki/Q233220","display_name":"Second fundamental form","level":4,"score":0.4478340446949005},{"id":"https://openalex.org/C175017881","wikidata":"https://www.wikidata.org/wiki/Q1318998","display_name":"Mean curvature","level":3,"score":0.41811513900756836},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.4086323082447052},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.2526056170463562},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.20110702514648438},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.15611532330513},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.11298277974128723},{"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/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/axioms13030183","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13030183","pdf_url":"https://www.mdpi.com/2075-1680/13/3/183/pdf?version=1710207043","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:f8c1cd70b93244d79d642aee733df4f8","is_oa":true,"landing_page_url":"https://doaj.org/article/f8c1cd70b93244d79d642aee733df4f8","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 13, Iss 3, p 183 (2024)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms13030183","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13030183","pdf_url":"https://www.mdpi.com/2075-1680/13/3/183/pdf?version=1710207043","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":[{"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10","score":0.4000000059604645}],"awards":[{"id":"https://openalex.org/G8870555227","display_name":null,"funder_award_id":"IMSIU-RP23078","funder_id":"https://openalex.org/F4320321420","funder_display_name":"Imam Mohammed Ibn Saud Islamic University"}],"funders":[{"id":"https://openalex.org/F4320321420","display_name":"Imam Mohammed Ibn Saud Islamic University","ror":"https://ror.org/05gxjyb39"},{"id":"https://openalex.org/F4320325415","display_name":"Al-Imam Muhammad Ibn Saud Islamic University","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4392662976.pdf","grobid_xml":"https://content.openalex.org/works/W4392662976.grobid-xml"},"referenced_works_count":31,"referenced_works":["https://openalex.org/W1487237407","https://openalex.org/W1598279643","https://openalex.org/W1997827312","https://openalex.org/W2000179513","https://openalex.org/W2004238057","https://openalex.org/W2031917130","https://openalex.org/W2051047346","https://openalex.org/W2063646760","https://openalex.org/W2071263744","https://openalex.org/W2115131806","https://openalex.org/W2150254557","https://openalex.org/W2508798694","https://openalex.org/W2576511095","https://openalex.org/W2798167706","https://openalex.org/W2949806910","https://openalex.org/W2992739134","https://openalex.org/W3048164324","https://openalex.org/W3171530165","https://openalex.org/W3205526445","https://openalex.org/W3205651275","https://openalex.org/W4211063922","https://openalex.org/W4226160701","https://openalex.org/W4239545531","https://openalex.org/W4240085759","https://openalex.org/W4242801507","https://openalex.org/W4254024075","https://openalex.org/W4285791510","https://openalex.org/W4378714901","https://openalex.org/W6684586453","https://openalex.org/W6771179625","https://openalex.org/W6835743930"],"related_works":["https://openalex.org/W2389509973","https://openalex.org/W1991759242","https://openalex.org/W2356258394","https://openalex.org/W2148312433","https://openalex.org/W4295599407","https://openalex.org/W3013913646","https://openalex.org/W4210680389","https://openalex.org/W2374106464","https://openalex.org/W3013174408","https://openalex.org/W2079355117"],"abstract_inverted_index":{"In":[0],"this":[1],"article,":[2],"we":[3,14,33,47,66,96],"study":[4],"isotropic":[5,57],"submanifolds":[6,21,39,58,71],"in":[7,40],"locally":[8,41],"metallic":[9,42],"product":[10,43,75],"space":[11,44],"forms.":[12],"Firstly,":[13],"establish":[15],"the":[16,24,28,35,49,70],"Chen\u2013Ricci":[17],"inequality":[18,29],"for":[19,56],"such":[20],"and":[22,46],"determine":[23],"conditions":[25],"under":[26],"which":[27],"becomes":[30],"equality.":[31],"Additionally,":[32],"explore":[34],"minimality":[36],"of":[37,76],"Lagrangian":[38],"forms,":[45],"apply":[48],"result":[50],"to":[51,87],"create":[52],"a":[53,74,88],"classification":[54],"theorem":[55],"whose":[59],"mean":[60],"curvature":[61],"is":[62],"constant.":[63],"More":[64],"specifically,":[65],"have":[67],"demonstrated":[68],"that":[69],"are":[72,85],"either":[73],"two":[77],"Einstein":[78,81],"manifolds":[79],"with":[80],"constants,":[82],"or":[83],"they":[84],"isometric":[86],"totally":[89],"geodesic":[90],"submanifold.":[91],"To":[92],"support":[93],"our":[94],"findings,":[95],"provide":[97],"several":[98],"examples.":[99]},"counts_by_year":[{"year":2024,"cited_by_count":11}],"updated_date":"2026-06-15T08:34:33.830935","created_date":"2025-10-10T00:00:00"}