{"id":"https://openalex.org/W4366991569","doi":"https://doi.org/10.3390/sym15050976","title":"Ricci Soliton of CR-Warped Product Manifolds and Their Classifications","display_name":"Ricci Soliton of CR-Warped Product Manifolds and Their Classifications","publication_year":2023,"publication_date":"2023-04-25","ids":{"openalex":"https://openalex.org/W4366991569","doi":"https://doi.org/10.3390/sym15050976"},"language":"en","primary_location":{"id":"doi:10.3390/sym15050976","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15050976","pdf_url":"https://www.mdpi.com/2073-8994/15/5/976/pdf?version=1682408728","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/15/5/976/pdf?version=1682408728","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"],"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/A5100627111","display_name":"S. K. Srivastava","orcid":"https://orcid.org/0000-0001-5992-1902"},"institutions":[{"id":"https://openalex.org/I180952551","display_name":"Central University of Himachal Pradesh","ror":"https://ror.org/04v5nzb91","country_code":"IN","type":"education","lineage":["https://openalex.org/I180952551"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Sachin Kumar Srivastava","raw_affiliation_strings":["Department of Mathematics, Central University of Himachal Pradesh, Dharamshala 176215, Himachal Pradesh, India"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Central University of Himachal Pradesh, Dharamshala 176215, Himachal Pradesh, India","institution_ids":["https://openalex.org/I180952551"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5039522245","display_name":"Fatemah Mofarreh","orcid":"https://orcid.org/0000-0002-2116-7382"},"institutions":[{"id":"https://openalex.org/I106778892","display_name":"Princess Nourah bint Abdulrahman University","ror":"https://ror.org/05b0cyh02","country_code":"SA","type":"education","lineage":["https://openalex.org/I106778892"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Fatemah Mofarreh","raw_affiliation_strings":["Mathematical Science Department, Faculty of Science, Princess Nourah bint Abdulrahman University, Riyadh 11546, Saudi Arabia"],"affiliations":[{"raw_affiliation_string":"Mathematical Science Department, Faculty of Science, Princess Nourah bint Abdulrahman University, Riyadh 11546, Saudi Arabia","institution_ids":["https://openalex.org/I106778892"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5020313232","display_name":"Anuj Kumar","orcid":"https://orcid.org/0000-0002-6018-7861"},"institutions":[{"id":"https://openalex.org/I180952551","display_name":"Central University of Himachal Pradesh","ror":"https://ror.org/04v5nzb91","country_code":"IN","type":"education","lineage":["https://openalex.org/I180952551"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Anuj Kumar","raw_affiliation_strings":["Department of Mathematics, Central University of Himachal Pradesh, Dharamshala 176215, Himachal Pradesh, India"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Central University of Himachal Pradesh, Dharamshala 176215, Himachal Pradesh, India","institution_ids":["https://openalex.org/I180952551"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5053272822","display_name":"Akram Ali","orcid":"https://orcid.org/0000-0002-6053-3031"},"institutions":[{"id":"https://openalex.org/I82952536","display_name":"King Khalid University","ror":"https://ror.org/052kwzs30","country_code":"SA","type":"education","lineage":["https://openalex.org/I82952536"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Akram Ali","raw_affiliation_strings":["Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia","institution_ids":["https://openalex.org/I82952536"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5053272822"],"corresponding_institution_ids":["https://openalex.org/I82952536"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":15.7945,"has_fulltext":true,"cited_by_count":30,"citation_normalized_percentile":{"value":0.99775244,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":100},"biblio":{"volume":"15","issue":"5","first_page":"976","last_page":"976"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":1.0,"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":1.0,"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/T12351","display_name":"Geometry and complex manifolds","score":0.9991000294685364,"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"}},{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.9968000054359436,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/ricci-curvature","display_name":"Ricci curvature","score":0.6327321529388428},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6038373708724976},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.5260978937149048},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5093463659286499},{"id":"https://openalex.org/keywords/hessian-matrix","display_name":"Hessian matrix","score":0.4969065487384796},{"id":"https://openalex.org/keywords/einstein","display_name":"Einstein","score":0.48177430033683777},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.45772773027420044},{"id":"https://openalex.org/keywords/ricci-flat-manifold","display_name":"Ricci-flat manifold","score":0.4429278075695038},{"id":"https://openalex.org/keywords/riemann-curvature-tensor","display_name":"Riemann curvature tensor","score":0.426119863986969},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.4130752384662628},{"id":"https://openalex.org/keywords/curvature","display_name":"Curvature","score":0.40189847350120544},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.36422884464263916},{"id":"https://openalex.org/keywords/scalar-curvature","display_name":"Scalar curvature","score":0.3548746705055237},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10298529267311096},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.06967571377754211}],"concepts":[{"id":"https://openalex.org/C12089564","wikidata":"https://www.wikidata.org/wiki/Q1195879","display_name":"Ricci curvature","level":3,"score":0.6327321529388428},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6038373708724976},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.5260978937149048},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5093463659286499},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.4969065487384796},{"id":"https://openalex.org/C146846114","wikidata":"https://www.wikidata.org/wiki/Q901448","display_name":"Einstein","level":2,"score":0.48177430033683777},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.45772773027420044},{"id":"https://openalex.org/C203934109","wikidata":"https://www.wikidata.org/wiki/Q7322952","display_name":"Ricci-flat manifold","level":4,"score":0.4429278075695038},{"id":"https://openalex.org/C183517385","wikidata":"https://www.wikidata.org/wiki/Q855112","display_name":"Riemann curvature tensor","level":3,"score":0.426119863986969},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.4130752384662628},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.40189847350120544},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.36422884464263916},{"id":"https://openalex.org/C12520029","wikidata":"https://www.wikidata.org/wiki/Q1147161","display_name":"Scalar curvature","level":3,"score":0.3548746705055237},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10298529267311096},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.06967571377754211},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym15050976","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15050976","pdf_url":"https://www.mdpi.com/2073-8994/15/5/976/pdf?version=1682408728","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:e7bbf16078f54bac8c382ffa6142c62d","is_oa":true,"landing_page_url":"https://doaj.org/article/e7bbf16078f54bac8c382ffa6142c62d","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 15, Iss 5, p 976 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/15/5/976/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym15050976","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym15050976","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15050976","pdf_url":"https://www.mdpi.com/2073-8994/15/5/976/pdf?version=1682408728","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1231421488","display_name":null,"funder_award_id":"under","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2053653470","display_name":null,"funder_award_id":"PNURSP","funder_id":"https://openalex.org/F4320322484","funder_display_name":"Princess Nourah Bint Abdulrahman University"},{"id":"https://openalex.org/G2087396116","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2600515145","display_name":null,"funder_award_id":"12101168","funder_id":"https://openalex.org/F4320324433","funder_display_name":"King Khalid University"},{"id":"https://openalex.org/G260937657","display_name":null,"funder_award_id":"12101168","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2893605509","display_name":null,"funder_award_id":"Saudi Arabia.","funder_id":"https://openalex.org/F4320322484","funder_display_name":"Princess Nourah Bint Abdulrahman University"},{"id":"https://openalex.org/G3085993365","display_name":null,"funder_award_id":"(Grant No.","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3317480652","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G367687827","display_name":null,"funder_award_id":"12101168","funder_id":"https://openalex.org/F4320322484","funder_display_name":"Princess Nourah Bint Abdulrahman University"},{"id":"https://openalex.org/G4010074732","display_name":null,"funder_award_id":"Riyadh","funder_id":"https://openalex.org/F4320322484","funder_display_name":"Princess Nourah Bint Abdulrahman University"},{"id":"https://openalex.org/G4020255992","display_name":null,"funder_award_id":"Project","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5167091242","display_name":null,"funder_award_id":"No. 1","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5249852646","display_name":null,"funder_award_id":"PNURSP2023R27","funder_id":"https://openalex.org/F4320322484","funder_display_name":"Princess Nourah Bint Abdulrahman University"},{"id":"https://openalex.org/G5823439880","display_name":null,"funder_award_id":"210116","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5913933531","display_name":null,"funder_award_id":"LQ22A010014","funder_id":"https://openalex.org/F4320338464","funder_display_name":"Natural Science Foundation of Zhejiang Province"},{"id":"https://openalex.org/G5929343061","display_name":null,"funder_award_id":"PNURSP2023R2","funder_id":"https://openalex.org/F4320322484","funder_display_name":"Princess Nourah Bint Abdulrahman University"},{"id":"https://openalex.org/G5994120800","display_name":null,"funder_award_id":"Natural","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6772716045","display_name":null,"funder_award_id":"PNURSP2023","funder_id":"https://openalex.org/F4320322484","funder_display_name":"Princess Nourah Bint Abdulrahman University"},{"id":"https://openalex.org/G7508974387","display_name":null,"funder_award_id":"1210116","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7726157001","display_name":null,"funder_award_id":"Grant No.","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320322484","display_name":"Princess Nourah Bint Abdulrahman University","ror":"https://ror.org/05b0cyh02"},{"id":"https://openalex.org/F4320324433","display_name":"King Khalid University","ror":"https://ror.org/052kwzs30"},{"id":"https://openalex.org/F4320338464","display_name":"Natural Science Foundation of Zhejiang Province","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4366991569.pdf","grobid_xml":"https://content.openalex.org/works/W4366991569.grobid-xml"},"referenced_works_count":37,"referenced_works":["https://openalex.org/W1546446866","https://openalex.org/W1553604161","https://openalex.org/W1586829048","https://openalex.org/W1972682939","https://openalex.org/W2020384646","https://openalex.org/W2042459012","https://openalex.org/W2076228339","https://openalex.org/W2141776582","https://openalex.org/W2302778078","https://openalex.org/W2478532420","https://openalex.org/W2555613349","https://openalex.org/W2570949444","https://openalex.org/W2912495810","https://openalex.org/W2920807242","https://openalex.org/W2940768037","https://openalex.org/W2951507084","https://openalex.org/W3021918379","https://openalex.org/W3091843802","https://openalex.org/W3100477042","https://openalex.org/W3115675213","https://openalex.org/W4200254982","https://openalex.org/W4220722392","https://openalex.org/W4241145230","https://openalex.org/W4249279429","https://openalex.org/W4280561449","https://openalex.org/W4281917972","https://openalex.org/W4312202504","https://openalex.org/W4312435947","https://openalex.org/W4312909758","https://openalex.org/W4313824547","https://openalex.org/W4317503052","https://openalex.org/W4318473121","https://openalex.org/W4319317882","https://openalex.org/W4322768285","https://openalex.org/W4365397491","https://openalex.org/W6787489780","https://openalex.org/W6849920934"],"related_works":["https://openalex.org/W2254577414","https://openalex.org/W2604503417","https://openalex.org/W2744264588","https://openalex.org/W2017266758","https://openalex.org/W4386710603","https://openalex.org/W2736492506","https://openalex.org/W115373153","https://openalex.org/W2375842186","https://openalex.org/W4307537658","https://openalex.org/W1998114779"],"abstract_inverted_index":{"In":[0],"this":[1],"article,":[2],"we":[3],"derived":[4,33],"an":[5,47],"equality":[6],"for":[7],"CR-warped":[8,39,60],"product":[9,40,61,86],"in":[10,42,50,63],"a":[11,38],"complex":[12],"space":[13],"form":[14],"which":[15],"forms":[16],"the":[17,20,25,34,51,64,69,89],"relationship":[18],"between":[19],"gradient":[21,54],"and":[22,28,74,94],"Laplacian":[23],"of":[24,37,53,59,83,91,96],"warping":[26],"function":[27],"second":[29],"fundamental":[30],"form.":[31],"We":[32,78],"necessary":[35],"conditions":[36],"submanifolds":[41,62],"Ka\u00a8hler":[43,65],"manifold":[44,49,66],"to":[45],"be":[46],"Einstein":[48,84],"impact":[52,90],"Ricci":[55,92],"soliton.":[56],"Some":[57],"classification":[58],"by":[67],"using":[68],"Euler\u2013Lagrange":[70],"equation,":[71],"Dirichlet":[72],"energy":[73],"Hamiltonian":[75],"is":[76],"given.":[77],"also":[79],"derive":[80],"some":[81],"characterizations":[82],"warped":[85],"manifolds":[87],"under":[88],"Curvature":[93],"Divergence":[95],"Hessian":[97],"tensor.":[98]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":24}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2023-04-27T00:00:00"}
