{"id":"https://openalex.org/W7204170086","doi":"https://doi.org/10.48550/arxiv.2608.22133","title":"A Metric with Positive Sectional Curvature on $S^2\\times S^3$","display_name":"A Metric with Positive Sectional Curvature on $S^2\\times S^3$","publication_year":2026,"publication_date":"2026-08-22","ids":{"openalex":"https://openalex.org/W7204170086","doi":"https://doi.org/10.48550/arxiv.2608.22133"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.22133","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.22133","pdf_url":"https://arxiv.org/pdf/2608.22133","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":"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/2608.22133","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5058903960","display_name":"Shengtao Guo","orcid":"https://orcid.org/0000-0002-3404-1738","observed_orcids":["https://orcid.org/0000-0002-3404-1738"]},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Guo, Shengtao","raw_affiliation_strings":["Department of Biostatistics , Harvard T.H. Chan School of Public Health"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Biostatistics , Harvard T.H. Chan School of Public Health","institution_ids":["https://openalex.org/I136199984"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5011349573","display_name":"Ethan X. Fang","orcid":"https://orcid.org/0000-0003-3762-9155","observed_orcids":["https://orcid.org/0000-0003-3762-9155"]},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Fang, Ethan X.","raw_affiliation_strings":["Department of Biostatistics , Harvard T.H. Chan School of Public Health"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Biostatistics , Harvard T.H. Chan School of Public Health","institution_ids":["https://openalex.org/I136199984"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5150919206","display_name":"Junwei Lu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Lu, Junwei","raw_affiliation_strings":["Department of Biostatistics , Harvard T.H. Chan School of Public Health"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Biostatistics , Harvard T.H. Chan School of Public Health","institution_ids":["https://openalex.org/I136199984"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I136199984"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":5,"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.7753999829292297,"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":0.7753999829292297,"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":0.2102999985218048,"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":0.010099999606609344,"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/positive-sectional-curvature","display_name":"positive sectional curvature","score":0.984000027179718},{"id":"https://openalex.org/keywords/riemannian-metrics","display_name":"Riemannian metrics","score":0.7879999876022339},{"id":"https://openalex.org/keywords/flat-tori","display_name":"flat tori","score":0.7080000042915344},{"id":"https://openalex.org/keywords/second-fundamental-form","display_name":"second fundamental form","score":0.7839999794960022},{"id":"https://openalex.org/keywords/gauss-equation","display_name":"Gauss equation","score":0.8209999799728394},{"id":"https://openalex.org/keywords/ai-assisted-discovery","display_name":"AI-assisted discovery","score":0.5090000033378601}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8294000029563904},{"id":"https://openalex.org/C76453154","wikidata":"https://www.wikidata.org/wiki/Q788494","display_name":"Metric connection","level":5,"score":0.6520000100135803},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.5874999761581421},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.5814999938011169},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.5443000197410583},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5246000289916992},{"id":"https://openalex.org/C129049296","wikidata":"https://www.wikidata.org/wiki/Q548675","display_name":"Levi-Civita connection","level":5,"score":0.5045999884605408},{"id":"https://openalex.org/C151300846","wikidata":"https://www.wikidata.org/wiki/Q125539","display_name":"Submanifold","level":2,"score":0.4959000051021576},{"id":"https://openalex.org/C42448751","wikidata":"https://www.wikidata.org/wiki/Q855622","display_name":"Sectional curvature","level":4,"score":0.4943000078201294},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.4684999883174896},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4410000145435333},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.426800012588501},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.38269999623298645},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.3479999899864197},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.34540000557899475},{"id":"https://openalex.org/C74050887","wikidata":"https://www.wikidata.org/wiki/Q848368","display_name":"Rotation (mathematics)","level":2,"score":0.33629998564720154},{"id":"https://openalex.org/C12520029","wikidata":"https://www.wikidata.org/wiki/Q1147161","display_name":"Scalar curvature","level":3,"score":0.3246000111103058},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.31679999828338623},{"id":"https://openalex.org/C4306028","wikidata":"https://www.wikidata.org/wiki/Q2185349","display_name":"Fundamental theorem of Riemannian geometry","level":4,"score":0.3149000108242035},{"id":"https://openalex.org/C167204820","wikidata":"https://www.wikidata.org/wiki/Q912058","display_name":"Simply connected space","level":2,"score":0.3041999936103821},{"id":"https://openalex.org/C177547787","wikidata":"https://www.wikidata.org/wiki/Q6034319","display_name":"Injective metric space","level":3,"score":0.2856999933719635},{"id":"https://openalex.org/C203763787","wikidata":"https://www.wikidata.org/wiki/Q371029","display_name":"Null (SQL)","level":2,"score":0.28540000319480896},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.2851000130176544},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.2847999930381775},{"id":"https://openalex.org/C191226916","wikidata":"https://www.wikidata.org/wiki/Q5384710","display_name":"Equivalence of metrics","level":4,"score":0.2775000035762787},{"id":"https://openalex.org/C41261874","wikidata":"https://www.wikidata.org/wiki/Q3855712","display_name":"Intrinsic metric","level":4,"score":0.2662999927997589},{"id":"https://openalex.org/C16140857","wikidata":"https://www.wikidata.org/wiki/Q945953","display_name":"Gaussian curvature","level":3,"score":0.25999999046325684},{"id":"https://openalex.org/C109546454","wikidata":"https://www.wikidata.org/wiki/Q3798604","display_name":"Information geometry","level":4,"score":0.25839999318122864},{"id":"https://openalex.org/C47422493","wikidata":"https://www.wikidata.org/wiki/Q2755127","display_name":"Wedge (geometry)","level":2,"score":0.25760000944137573},{"id":"https://openalex.org/C183517385","wikidata":"https://www.wikidata.org/wiki/Q855112","display_name":"Riemann curvature tensor","level":3,"score":0.25429999828338623},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.2540999948978424},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.2538999915122986}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.22133","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.22133","pdf_url":"https://arxiv.org/pdf/2608.22133","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.22133","doi":"https://doi.org/10.48550/arxiv.2608.22133","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.22133","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:2608.22133","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.22133","pdf_url":"https://arxiv.org/pdf/2608.22133","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204170086.pdf","grobid_xml":"https://content.openalex.org/works/W7204170086.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W1897034207","https://openalex.org/W7203884121"],"related_works":["https://openalex.org/W7212693265","https://openalex.org/W7203884121","https://openalex.org/W7204126347","https://openalex.org/W3010194318","https://openalex.org/W2162459162","https://openalex.org/W1996921179","https://openalex.org/W2051680479","https://openalex.org/W7100250173","https://openalex.org/W7096902701","https://openalex.org/W7212319814"],"abstract_inverted_index":{"We":[0,13,63],"prove":[1],"that":[2,112],"$S^2\\times":[3,22],"S^3$":[4],"admits":[5],"a":[6,17,32,44,73,84,107],"Riemannian":[7],"metric":[8,48,67,141],"with":[9],"positive":[10,108],"sectional":[11],"curvature.":[12],"view":[14],"it":[15],"as":[16,114],"principal":[18],"circle":[19,81],"bundle":[20],"over":[21],"S^2$.":[23],"A":[24,118],"diagonal":[25],"Cheeger":[26],"deformation":[27],"of":[28,56,72,88,97,131],"the":[29,39,53,57,69,80,89,98,115,123,129,143,148],"base":[30],"and":[31,142],"connection":[33,47],"whose":[34,49],"curvature":[35,110],"form":[36,92],"vanishes":[37],"on":[38,93],"remaining":[40],"flat":[41,99],"tori":[42],"yield":[43],"nonnegatively":[45],"curved":[46],"zero-curvature":[50,132],"planes":[51,59,133],"are":[52,145],"horizontal":[54,95],"lifts":[55,96],"tangent":[58],"to":[60,128,137],"those":[61],"tori.":[62,100],"then":[64],"perturb":[65],"this":[66,105,135],"by":[68,147],"real":[70],"part":[71],"global":[74],"complex-valued":[75],"symmetric":[76],"$2$-tensor.":[77],"Differentiation":[78],"along":[79],"fibers":[82,116],"produces":[83],"trace-free":[85],"first":[86],"variation":[87],"second":[90],"fundamental":[91],"local":[94],"The":[101,140],"Gauss":[102],"equation":[103],"converts":[104],"into":[106],"second-order":[109],"term":[111],"dominates":[113],"shrink.":[117],"quantitative":[119],"lower":[120],"bound":[121],"for":[122],"Hessian":[124],"in":[125],"directions":[126],"normal":[127],"set":[130],"extends":[134],"positivity":[136],"nearby":[138],"planes.":[139],"proof":[144],"discovered":[146],"Odin":[149],"Automatic":[150],"AI":[151],"Research":[152],"Agent.":[153]},"counts_by_year":[{"year":2026,"cited_by_count":5}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-26T00:00:00"}