{"id":"https://openalex.org/W4367181767","doi":"https://doi.org/10.3390/axioms12050432","title":"On an Indefinite Metric on a Four-Dimensional Riemannian Manifold","display_name":"On an Indefinite Metric on a Four-Dimensional Riemannian Manifold","publication_year":2023,"publication_date":"2023-04-27","ids":{"openalex":"https://openalex.org/W4367181767","doi":"https://doi.org/10.3390/axioms12050432"},"language":"en","primary_location":{"id":"doi:10.3390/axioms12050432","doi":"https://doi.org/10.3390/axioms12050432","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12050432","pdf_url":"https://www.mdpi.com/2075-1680/12/5/432/pdf?version=1682583053","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,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-2","vabb-shw"],"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"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/12/5/432/pdf?version=1682583053","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5016144457","display_name":"Dimitar Razpopov","orcid":"https://orcid.org/0000-0002-9851-2439","observed_orcids":["https://orcid.org/0000-0002-9851-2439"]},"institutions":[{"id":"https://openalex.org/I4210144807","display_name":"Agricultural University Plovdiv","ror":"https://ror.org/04e5bwf87","country_code":"BG","type":"education","lineage":["https://openalex.org/I4210144807"]}],"countries":["BG"],"is_corresponding":false,"raw_author_name":"Dimitar Razpopov","raw_affiliation_strings":["Department of Mathematics and Informatics, Faculty of Economics, Agricultural University of Plovdiv, 12 Mendeleev Blvd., 4000 Plovdiv, Bulgaria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Informatics, Faculty of Economics, Agricultural University of Plovdiv, 12 Mendeleev Blvd., 4000 Plovdiv, Bulgaria","institution_ids":["https://openalex.org/I4210144807"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034564202","display_name":"Georgi Dzhelepov","orcid":"https://orcid.org/0000-0002-7062-1908","observed_orcids":["https://orcid.org/0000-0002-7062-1908"]},"institutions":[{"id":"https://openalex.org/I4210144807","display_name":"Agricultural University Plovdiv","ror":"https://ror.org/04e5bwf87","country_code":"BG","type":"education","lineage":["https://openalex.org/I4210144807"]}],"countries":["BG"],"is_corresponding":false,"raw_author_name":"Georgi Dzhelepov","raw_affiliation_strings":["Department of Mathematics and Informatics, Faculty of Economics, Agricultural University of Plovdiv, 12 Mendeleev Blvd., 4000 Plovdiv, Bulgaria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Informatics, Faculty of Economics, Agricultural University of Plovdiv, 12 Mendeleev Blvd., 4000 Plovdiv, Bulgaria","institution_ids":["https://openalex.org/I4210144807"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5039748971","display_name":"\u0130va Dokuzova","orcid":"https://orcid.org/0000-0002-9099-6969","observed_orcids":["https://orcid.org/0000-0002-9099-6969"]},"institutions":[{"id":"https://openalex.org/I192878625","display_name":"Plovdiv University","ror":"https://ror.org/0545p3742","country_code":"BG","type":"education","lineage":["https://openalex.org/I192878625"]}],"countries":["BG"],"is_corresponding":true,"raw_author_name":"Iva Rumenova Dokuzova","raw_affiliation_strings":["Department of Algebra and Geometry, Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria"],"raw_orcid":"https://orcid.org/0000-0002-9099-6969","affiliations":[{"raw_affiliation_string":"Department of Algebra and Geometry, Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria","institution_ids":["https://openalex.org/I192878625"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5039748971"],"corresponding_institution_ids":["https://openalex.org/I192878625"],"apc_list":{"value":1600,"currency":"CHF","value_usd":1779},"apc_paid":{"value":1600,"currency":"CHF","value_usd":1779},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"12","issue":"5","first_page":"432","last_page":"432"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.9990000128746033,"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"}},"topics":[{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.9990000128746033,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.0003000000142492354,"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/T12504","display_name":"Mathematics and Applications","score":0.0003000000142492354,"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/indefinite-metric","display_name":"indefinite metric","score":0.9909999966621399},{"id":"https://openalex.org/keywords/isometries","display_name":"isometries","score":0.5989999771118164},{"id":"https://openalex.org/keywords/tangent-space","display_name":"tangent space","score":0.7889999747276306},{"id":"https://openalex.org/keywords/circulation","display_name":"circulation","score":0.9179999828338623},{"id":"https://openalex.org/keywords/green-s-formula","display_name":"Green's formula","score":0.8989999890327454}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7623013854026794},{"id":"https://openalex.org/C121705324","wikidata":"https://www.wikidata.org/wiki/Q1510587","display_name":"Pseudo-Riemannian manifold","level":4,"score":0.7374276518821716},{"id":"https://openalex.org/C157157409","wikidata":"https://www.wikidata.org/wiki/Q909601","display_name":"Tangent space","level":2,"score":0.618181586265564},{"id":"https://openalex.org/C2779593128","wikidata":"https://www.wikidata.org/wiki/Q632814","display_name":"Riemannian manifold","level":2,"score":0.6013400554656982},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5789322853088379},{"id":"https://openalex.org/C1432948","wikidata":"https://www.wikidata.org/wiki/Q5741935","display_name":"Hermitian manifold","level":4,"score":0.5283017158508301},{"id":"https://openalex.org/C127104897","wikidata":"https://www.wikidata.org/wiki/Q5454858","display_name":"Fisher information metric","level":4,"score":0.482530802488327},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.4747110903263092},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.46215158700942993},{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.4509914219379425},{"id":"https://openalex.org/C147663694","wikidata":"https://www.wikidata.org/wiki/Q1144631","display_name":"Curl (programming language)","level":2,"score":0.4371762275695801},{"id":"https://openalex.org/C47890412","wikidata":"https://www.wikidata.org/wiki/Q1179296","display_name":"Tangent vector","level":3,"score":0.4370593726634979},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.4274788796901703},{"id":"https://openalex.org/C129049296","wikidata":"https://www.wikidata.org/wiki/Q548675","display_name":"Levi-Civita connection","level":5,"score":0.41309306025505066},{"id":"https://openalex.org/C12089564","wikidata":"https://www.wikidata.org/wiki/Q1195879","display_name":"Ricci curvature","level":3,"score":0.2965623140335083},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.280889630317688},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.26547372341156006},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.20115333795547485},{"id":"https://openalex.org/C177547787","wikidata":"https://www.wikidata.org/wiki/Q6034319","display_name":"Injective metric space","level":3,"score":0.20096439123153687},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.15660125017166138},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","level":1,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/axioms12050432","doi":"https://doi.org/10.3390/axioms12050432","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12050432","pdf_url":"https://www.mdpi.com/2075-1680/12/5/432/pdf?version=1682583053","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,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-2","vabb-shw"],"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":"doi:10.20944/preprints202303.0223.v1","doi":"https://doi.org/10.20944/preprints202303.0223.v1","is_oa":true,"landing_page_url":"https://doi.org/10.20944/preprints202303.0223.v1","pdf_url":"https://www.preprints.org/frontend/manuscript/6af419a65d522467e33cfe2ade32b4de/download_pub","source":{"id":"https://openalex.org/S6309402219","display_name":"Preprints.org","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},{"id":"pmh:oai:doaj.org/article:b9e23f2f17b84770b1479b4cbfa62950","doi":null,"is_oa":true,"landing_page_url":"https://doaj.org/article/b9e23f2f17b84770b1479b4cbfa62950","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,"listed_in":[],"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 12, Iss 5, p 432 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms12050432","doi":"https://doi.org/10.3390/axioms12050432","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12050432","pdf_url":"https://www.mdpi.com/2075-1680/12/5/432/pdf?version=1682583053","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,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-2","vabb-shw"],"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":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F3160440111","display_name":"Agriculture University Plovdiv","ror":"https://ror.org/04e5bwf87"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4367181767.pdf","grobid_xml":"https://content.openalex.org/works/W4367181767.grobid-xml"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W268610678","https://openalex.org/W606278518","https://openalex.org/W1578014057","https://openalex.org/W1615179988","https://openalex.org/W1941572028","https://openalex.org/W1974932178","https://openalex.org/W2005487810","https://openalex.org/W2026892575","https://openalex.org/W2114475304","https://openalex.org/W2149698301","https://openalex.org/W2530726151","https://openalex.org/W2951073747","https://openalex.org/W2979849844","https://openalex.org/W3193885405","https://openalex.org/W4206196235","https://openalex.org/W4224298688","https://openalex.org/W4312202504","https://openalex.org/W4366991569","https://openalex.org/W4390991754","https://openalex.org/W6609538339","https://openalex.org/W6698315271","https://openalex.org/W6828123280"],"related_works":["https://openalex.org/W4324025946","https://openalex.org/W2899636749","https://openalex.org/W4390991754","https://openalex.org/W4315705763","https://openalex.org/W2795104329","https://openalex.org/W2611110384","https://openalex.org/W1820909989","https://openalex.org/W1664265147","https://openalex.org/W3188185937","https://openalex.org/W7095466370"],"abstract_inverted_index":{"Our":[0],"research":[1],"focuses":[2],"on":[3,10,60,86],"the":[4,21,38,48,61,70,73,83,87,92,97,117,120,125,130,133,141,157,162],"tangent":[5,58,84],"space":[6,59,85],"of":[7,30,72,96,119,132,135,156],"a":[8,11,17,100,106,110,146],"point":[9],"four-dimensional":[12],"Riemannian":[13],"manifold.":[14,62,88],"Besides":[15],"having":[16],"positive":[18],"definite":[19],"metric,":[20,49,75],"manifold":[22],"is":[23,36,43,54,153],"endowed":[24],"with":[25,47],"an":[26,52,66,154],"additional":[27,41],"tensor":[28],"structure":[29,42],"type":[31],"(1,1),":[32],"whose":[33],"fourth":[34],"power":[35],"minus":[37],"identity.":[39],"The":[40],"skew-circulant":[44],"and":[45,94],"compatible":[46],"such":[50,104],"that":[51],"isometry":[53],"induced":[55],"in":[56,79,82,161],"every":[57],"Both":[63],"structures":[64],"define":[65,109],"indefinite":[67,74],"metric.":[68],"With":[69],"help":[71],"we":[76,108,115,144],"determine":[77],"circles":[78],"different":[80],"two-planes":[81],"We":[89],"also":[90],"calculate":[91],"length":[93],"area":[95],"circles.":[98],"On":[99],"smooth":[101],"closed":[102],"curve,":[103,126],"as":[105,127,129],"circle,":[107],"vector":[111,121,137],"force":[112,122,138],"field.":[113],"Further,":[114],"obtain":[116],"circulation":[118],"field":[123,139],"along":[124],"well":[128],"flux":[131],"curl":[134],"this":[136],"across":[140],"curve.":[142],"Finally,":[143],"find":[145],"relation":[147],"between":[148],"these":[149],"two":[150],"values,":[151],"which":[152],"analog":[155],"well-known":[158],"Green\u2019s":[159],"formula":[160],"Euclidean":[163],"space.":[164]},"counts_by_year":[],"updated_date":"2026-11-09T13:22:03.289127","created_date":"2023-04-28T00:00:00"}
