{"id":"https://openalex.org/W2332445708","doi":"https://doi.org/10.2307/1969203","title":"On the Curvatura Integra in a Riemannian Manifold","display_name":"On the Curvatura Integra in a Riemannian Manifold","publication_year":1945,"publication_date":"1945-10-01","ids":{"openalex":"https://openalex.org/W2332445708","doi":"https://doi.org/10.2307/1969203","mag":"2332445708"},"language":"en","primary_location":{"id":"doi:10.2307/1969203","doi":"https://doi.org/10.2307/1969203","is_oa":false,"landing_page_url":"https://doi.org/10.2307/1969203","pdf_url":null,"source":{"id":"https://openalex.org/S45918994","display_name":"Annals of Mathematics","issn_l":"0003-486X","issn":["0003-486X","1939-8980"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","cwts-core","dongbi-a","jufo-3","norway-2","poland-200","russia-white-list-1"],"host_organization":"https://openalex.org/P4310316273","host_organization_name":"Princeton University","host_organization_lineage":["https://openalex.org/P4310316273"],"host_organization_lineage_names":["Princeton University"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Annals of Mathematics","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5109052943","display_name":"Shiing-Shen Chern","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Shiing-shen Chern","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5109052943"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":6.4576,"has_fulltext":false,"cited_by_count":285,"citation_normalized_percentile":{"value":0.95597484,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"46","issue":"4","first_page":"674","last_page":"674"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.9829000234603882,"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"}},"topics":[{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.9829000234603882,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.010599999688565731,"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.002400000113993883,"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/riemannian-manifolds","display_name":"Riemannian manifolds","score":0.9169999957084656},{"id":"https://openalex.org/keywords/gauss-bonnet-theorem","display_name":"Gauss-Bonnet theorem","score":0.8230000138282776},{"id":"https://openalex.org/keywords/unit-tangent-bundle","display_name":"unit tangent bundle","score":0.9700000286102295},{"id":"https://openalex.org/keywords/differential-forms","display_name":"differential forms","score":0.722000002861023},{"id":"https://openalex.org/keywords/euler-characteristic","display_name":"Euler characteristic","score":0.9570000171661377},{"id":"https://openalex.org/keywords/submanifolds","display_name":"submanifolds","score":0.9449999928474426},{"id":"https://openalex.org/keywords/topological-invariants","display_name":"topological invariants","score":0.8420000076293945},{"id":"https://openalex.org/keywords/intrinsic-geometry","display_name":"intrinsic geometry","score":0.3230000138282776}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8596010208129883},{"id":"https://openalex.org/C2779593128","wikidata":"https://www.wikidata.org/wiki/Q632814","display_name":"Riemannian manifold","level":2,"score":0.6568154096603394},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.5354056358337402},{"id":"https://openalex.org/C12520029","wikidata":"https://www.wikidata.org/wiki/Q1147161","display_name":"Scalar curvature","level":3,"score":0.4595884084701538},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.370892733335495},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3660394251346588},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.31165188550949097},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.15269649028778076},{"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":1,"locations":[{"id":"doi:10.2307/1969203","doi":"https://doi.org/10.2307/1969203","is_oa":false,"landing_page_url":"https://doi.org/10.2307/1969203","pdf_url":null,"source":{"id":"https://openalex.org/S45918994","display_name":"Annals of Mathematics","issn_l":"0003-486X","issn":["0003-486X","1939-8980"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","cwts-core","dongbi-a","jufo-3","norway-2","poland-200","russia-white-list-1"],"host_organization":"https://openalex.org/P4310316273","host_organization_name":"Princeton University","host_organization_lineage":["https://openalex.org/P4310316273"],"host_organization_lineage_names":["Princeton University"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Annals of Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":2,"referenced_works":["https://openalex.org/W1965989024","https://openalex.org/W2325765688"],"related_works":["https://openalex.org/W4223616807","https://openalex.org/W2087112780","https://openalex.org/W4293156355","https://openalex.org/W4226246021","https://openalex.org/W4289750046","https://openalex.org/W2883447785","https://openalex.org/W1616296999","https://openalex.org/W2546668543","https://openalex.org/W4302389195","https://openalex.org/W2952208990"],"abstract_inverted_index":{"In":[0],"a":[1,114,136,144,147,159,188],"previous":[2],"paper":[3,113],"[1]":[4],"we":[5,102],"have":[6],"given":[7],"an":[8,93],"intrinsic":[9,94],"proof":[10,38,82],"of":[11,14,21,27,36,46,64,72,76,97,117,143,162,170,187,190],"the":[12,24,37,44,55,61,70,86,118,140,167,171,191,198],"formula":[13,26],"Allendoerfer-Weil":[15],"which":[16,50],"generalizes":[17],"to":[18,40,54,92,139,166,180,204,207],"Riemannian":[19,56,62,127,148],"manifolds":[20],"n":[22,30,66,89,99],"dimensions":[23],"classical":[25],"Gauss-Bonnet":[28],"for":[29,121],"=":[31],"2.":[32],"The":[33,174],"main":[34],"idea":[35],"is":[39,51,90,164],"draw":[41],"into":[42],"consideration":[43],"manifold":[45,63,71,192],"unit":[47,78],"tangent":[48,79],"vectors":[49],"intrinsically":[52],"associated":[53],"manifold.":[57],"Denoting":[58],"by":[59,68,104],"R'":[60,163],"dimension":[65,73],"and":[67,124,193,200],"M2'":[69],"2n":[74],"1":[75,100],"its":[77,156],"vectors,":[80],"our":[81],"has":[83],"led,":[84],"in":[85,106,111,146],"case":[87],"that":[88,131,155],"even,":[91],"differential":[95,115,133,194],"form":[96,116,134],"degree":[98],"(which":[101],"denoted":[103],"II)":[105],"M2'.":[107],"We":[108,129],"shall":[109],"introduce":[110],"this":[112,132,208],"same":[119],"nature":[120],"both":[122],"even":[123],"odd":[125],"dimensional":[126],"manifolds.":[128],"find":[130],"bears":[135],"close":[137],"relation":[138],"Curvatura":[141],"Integra":[142],"submanifold":[145,161,189],"manifold,":[149],"because":[150],"it":[151],"will":[152],"be":[153,177,205],"proved":[154],"integral":[157],"over":[158,179],"closed":[160],"equal":[165],"Euler-Poincare":[168],"characteristic":[169],"submanif":[172],"old.":[173],"method":[175],"can":[176],"carried":[178],"deduce":[181],"relations":[182],"between":[183],"relative":[184],"topological":[185],"invariants":[186,195],"derived":[196],"from":[197],"imbedding,":[199],"some":[201],"remarks":[202],"are":[203],"added":[206],"effect.":[209]},"counts_by_year":[{"year":2026,"cited_by_count":8},{"year":2025,"cited_by_count":10},{"year":2024,"cited_by_count":16},{"year":2023,"cited_by_count":10},{"year":2022,"cited_by_count":11},{"year":2021,"cited_by_count":12},{"year":2020,"cited_by_count":12},{"year":2019,"cited_by_count":10},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":6},{"year":2016,"cited_by_count":6},{"year":2015,"cited_by_count":10},{"year":2014,"cited_by_count":6},{"year":2013,"cited_by_count":7},{"year":2012,"cited_by_count":2}],"updated_date":"2026-10-12T09:40:04.860316","created_date":"2025-10-10T00:00:00"}
