{"id":"https://openalex.org/W7214030816","doi":"https://doi.org/10.48550/arxiv.2609.23597","title":"Rigidity of Einstein Manifolds under Two Eigenvalue Conditions on the Curvature Operator of the Second Kind","display_name":"Rigidity of Einstein Manifolds under Two Eigenvalue Conditions on the Curvature Operator of the Second Kind","publication_year":2026,"publication_date":"2026-09-20","ids":{"openalex":"https://openalex.org/W7214030816","doi":"https://doi.org/10.48550/arxiv.2609.23597"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.23597","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.23597","pdf_url":"https://arxiv.org/pdf/2609.23597","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.23597","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5012054727","display_name":"Haiqing Cheng","orcid":"https://orcid.org/0009-0007-8029-9266","observed_orcids":["https://orcid.org/0009-0007-8029-9266"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Cheng, Haiqing","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5122926391","display_name":"Kui Wang","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wang, Kui","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"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":0.8690999746322632,"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":0.8690999746322632,"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":0.11940000206232071,"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/T12351","display_name":"Geometry and complex manifolds","score":0.005799999926239252,"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/rigidity-theorems","display_name":"rigidity theorems","score":0.6970000267028809},{"id":"https://openalex.org/keywords/einstein-manifolds","display_name":"Einstein manifolds","score":0.9309999942779541},{"id":"https://openalex.org/keywords/eigenvalue-conditions","display_name":"eigenvalue conditions","score":0.6169999837875366},{"id":"https://openalex.org/keywords/closed-manifolds","display_name":"closed manifolds","score":0.8140000104904175},{"id":"https://openalex.org/keywords/flat-manifolds","display_name":"flat manifolds","score":0.7860000133514404},{"id":"https://openalex.org/keywords/spherical-space-forms","display_name":"spherical space forms","score":0.9760000109672546},{"id":"https://openalex.org/keywords/ricci-curvature","display_name":"Ricci curvature","score":0.621999979019165}],"concepts":[{"id":"https://openalex.org/C160343418","wikidata":"https://www.wikidata.org/wiki/Q185256","display_name":"Rigidity (electromagnetism)","level":2,"score":0.755299985408783},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7443000078201294},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6234999895095825},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.6079000234603882},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6035000085830688},{"id":"https://openalex.org/C146846114","wikidata":"https://www.wikidata.org/wiki/Q901448","display_name":"Einstein","level":2,"score":0.5575000047683716},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.5490000247955322},{"id":"https://openalex.org/C2779780104","wikidata":"https://www.wikidata.org/wiki/Q1309313","display_name":"Einstein manifold","level":4,"score":0.5461999773979187},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.5177000164985657},{"id":"https://openalex.org/C2779668893","wikidata":"https://www.wikidata.org/wiki/Q2250422","display_name":"Constant curvature","level":3,"score":0.49900001287460327},{"id":"https://openalex.org/C74349738","wikidata":"https://www.wikidata.org/wiki/Q1517914","display_name":"Closed manifold","level":3,"score":0.40400001406669617},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.39579999446868896},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.36329999566078186},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.3513999879360199},{"id":"https://openalex.org/C119830904","wikidata":"https://www.wikidata.org/wiki/Q233220","display_name":"Second fundamental form","level":4,"score":0.33059999346733093},{"id":"https://openalex.org/C175017881","wikidata":"https://www.wikidata.org/wiki/Q1318998","display_name":"Mean curvature","level":3,"score":0.3163999915122986},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.3086000084877014},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.3046000003814697},{"id":"https://openalex.org/C12520029","wikidata":"https://www.wikidata.org/wiki/Q1147161","display_name":"Scalar curvature","level":3,"score":0.30329999327659607},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.30329999327659607},{"id":"https://openalex.org/C12089564","wikidata":"https://www.wikidata.org/wiki/Q1195879","display_name":"Ricci curvature","level":3,"score":0.2921000123023987},{"id":"https://openalex.org/C58677928","wikidata":"https://www.wikidata.org/wiki/Q911268","display_name":"D'Alembert operator","level":2,"score":0.27059999108314514},{"id":"https://openalex.org/C42448751","wikidata":"https://www.wikidata.org/wiki/Q855622","display_name":"Sectional curvature","level":4,"score":0.26989999413490295}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.23597","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.23597","pdf_url":"https://arxiv.org/pdf/2609.23597","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},{"id":"doi:10.48550/arxiv.2609.23597","doi":"https://doi.org/10.48550/arxiv.2609.23597","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.23597","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.23597","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.23597","pdf_url":"https://arxiv.org/pdf/2609.23597","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320321605","display_name":"Government of Jiangsu Province","ror":"https://ror.org/004svx814"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214030816.pdf","grobid_xml":"https://content.openalex.org/works/W7214030816.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W7117560322","https://openalex.org/W7123631585"],"related_works":["https://openalex.org/W7117560322","https://openalex.org/W7124982272","https://openalex.org/W4415219900","https://openalex.org/W7123631585","https://openalex.org/W7220978900","https://openalex.org/W7214949569","https://openalex.org/W4391680297","https://openalex.org/W7118322616","https://openalex.org/W2052373956","https://openalex.org/W2963841328"],"abstract_inverted_index":{"We":[0],"prove":[1,118],"a":[2,35,126],"rigidity":[3],"theorem":[4],"for":[5,26],"closed":[6],"Einstein":[7],"manifolds":[8],"of":[9,20,51,55],"dimension":[10],"$n\\ge6$":[11],"under":[12],"two":[13],"lower":[14],"bounds":[15],"on":[16],"the":[17,21,49,52,56,68,76,84,91,97,120],"curvature":[18,53],"operator":[19,54],"second":[22,57,85],"kind.":[23],"More":[24],"precisely,":[25],"$L_1,":[27],"L_2\\ge0$":[28],"and":[29,62,100,115],"some":[30],"real":[31],"number":[32],"$\u03b1>1$":[33],"in":[34,71,94],"suitable":[36],"range,":[37],"we":[38,117],"consider":[39],"\\[":[40],"\u03bb_1\\ge-L_1\\bar\u03bb,":[41],"\\qquad":[42],"\\frac1\u03b1\\sum_{j=1}^\u03b1\u03bb_j":[43],"\\ge-L_2\\bar\u03bb.":[44],"\\]":[45],"Here":[46],"$\u03bb_1\\le\\cdots\\le\u03bb_N$":[47],"are":[48],"eigenvalues":[50],"kind":[58],"$\\mathring":[59],"R$,":[60],"$N=(n-1)(n+2)/2$,":[61],"$\\bar\u03bb=N^{-1}\\sum_{j=1}^N\u03bb_j$.":[63],"Let":[64],"$\u03b8(n,":[65],"\u03b1)$":[66],"be":[67],"constant":[69],"appearing":[70],"\\cite[Theorem~1.1]{CW26};":[72],"see":[73],"(2).":[74],"In":[75],"parameter":[77],"range":[78],"considered":[79],"here,":[80],"$L_2>\u03b8(n,":[81],"\u03b1)$,":[82],"so":[83],"condition":[86,93,99,102],"is":[87,122],"strictly":[88],"weaker":[89],"than":[90],"corresponding":[92],"\\cite{CW26},":[95],"while":[96],"first":[98],"that":[101,119],"do":[103],"not":[104],"imply":[105],"each":[106],"other.":[107],"Under":[108],"an":[109],"additional":[110],"explicit":[111],"relation":[112],"between":[113],"$L_1$":[114],"$L_2$,":[116],"manifold":[121],"either":[123],"flat":[124],"or":[125],"spherical":[127],"space":[128],"form.":[129]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-23T00:00:00"}
