{"id":"https://openalex.org/W7222999744","doi":"https://doi.org/10.48550/arxiv.2610.25035","title":"Saddle Codazzi tensor on spheres and hemisphere rigidity","display_name":"Saddle Codazzi tensor on spheres and hemisphere rigidity","publication_year":2026,"publication_date":"2026-10-18","ids":{"openalex":"https://openalex.org/W7222999744","doi":"https://doi.org/10.48550/arxiv.2610.25035"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.25035","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.25035","pdf_url":"https://arxiv.org/pdf/2610.25035","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/2610.25035","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5136469370","display_name":"Jian Ge","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I183067930","display_name":"Shanghai Jiao Tong University","ror":"https://ror.org/0220qvk04","country_code":"CN","type":"education","lineage":["https://openalex.org/I183067930"]},{"id":"https://openalex.org/I25254941","display_name":"Beijing Normal University","ror":"https://ror.org/022k4wk35","country_code":"CN","type":"education","lineage":["https://openalex.org/I25254941"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ge, Jian","raw_affiliation_strings":["School of Mathematical Sciences , Laboratory of Mathematics and Complex Sys- tems , Beijing Normal University , Beijing 100875 , P. R. China","School of Mathematical Sciences , Shanghai Jiao Tong University , Shanghai 200240 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , Laboratory of Mathematics and Complex Sys- tems , Beijing Normal University , Beijing 100875 , P. R. China","institution_ids":["https://openalex.org/I25254941"]},{"raw_affiliation_string":"School of Mathematical Sciences , Shanghai Jiao Tong University , Shanghai 200240 , China","institution_ids":["https://openalex.org/I183067930"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5052180126","display_name":"Mijia Lai","orcid":"https://orcid.org/0000-0002-8316-6043","observed_orcids":["https://orcid.org/0000-0002-8316-6043"]},"institutions":[{"id":"https://openalex.org/I25254941","display_name":"Beijing Normal University","ror":"https://ror.org/022k4wk35","country_code":"CN","type":"education","lineage":["https://openalex.org/I25254941"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Lai, Mijia","raw_affiliation_strings":["School of Mathematical Sciences , Laboratory of Mathematics and Complex Sys- tems , Beijing Normal University , Beijing 100875 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , Laboratory of Mathematics and Complex Sys- tems , Beijing Normal University , Beijing 100875 , P. R. China","institution_ids":["https://openalex.org/I25254941"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5159156239","display_name":"Yuntian Zhu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I25254941","display_name":"Beijing Normal University","ror":"https://ror.org/022k4wk35","country_code":"CN","type":"education","lineage":["https://openalex.org/I25254941"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhu, Yuntian","raw_affiliation_strings":["School of Mathematical Sciences , Laboratory of Mathematics and Complex Sys- tems , Beijing Normal University , Beijing 100875 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , Laboratory of Mathematics and Complex Sys- tems , Beijing Normal University , Beijing 100875 , P. R. China","institution_ids":["https://openalex.org/I25254941"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"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":0.5503000020980835,"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":0.5503000020980835,"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":0.444599986076355,"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.0010999999940395355,"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/second-fundamental-form","display_name":"second fundamental form","score":0.7289999723434448},{"id":"https://openalex.org/keywords/round-spheres","display_name":"round spheres","score":0.652999997138977},{"id":"https://openalex.org/keywords/sectional-curvature-lower-bound","display_name":"sectional curvature lower bound","score":0.7950000166893005},{"id":"https://openalex.org/keywords/riemannian-manifolds-with-boundary","display_name":"Riemannian manifolds with boundary","score":0.8379999995231628}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7211999893188477},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6182000041007996},{"id":"https://openalex.org/C72422203","wikidata":"https://www.wikidata.org/wiki/Q7392545","display_name":"SPHERES","level":2,"score":0.47909998893737793},{"id":"https://openalex.org/C160343418","wikidata":"https://www.wikidata.org/wiki/Q185256","display_name":"Rigidity (electromagnetism)","level":2,"score":0.4745999872684479},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.47200000286102295},{"id":"https://openalex.org/C2777127463","wikidata":"https://www.wikidata.org/wiki/Q10862618","display_name":"Saddle","level":2,"score":0.45730000734329224},{"id":"https://openalex.org/C119830904","wikidata":"https://www.wikidata.org/wiki/Q233220","display_name":"Second fundamental form","level":4,"score":0.42239999771118164},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4187000095844269},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.4133000075817108},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.3831999897956848},{"id":"https://openalex.org/C59831887","wikidata":"https://www.wikidata.org/wiki/Q7887158","display_name":"Unit tangent bundle","level":4,"score":0.35670000314712524},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3549000024795532},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.35429999232292175},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.3540000021457672},{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.3310999870300293},{"id":"https://openalex.org/C2779593128","wikidata":"https://www.wikidata.org/wiki/Q632814","display_name":"Riemannian manifold","level":2,"score":0.30820000171661377},{"id":"https://openalex.org/C42448751","wikidata":"https://www.wikidata.org/wiki/Q855622","display_name":"Sectional curvature","level":4,"score":0.2922999858856201},{"id":"https://openalex.org/C82457910","wikidata":"https://www.wikidata.org/wiki/Q740207","display_name":"Isometry (Riemannian geometry)","level":2,"score":0.29030001163482666},{"id":"https://openalex.org/C9136319","wikidata":"https://www.wikidata.org/wiki/Q362640","display_name":"Covariant transformation","level":2,"score":0.2874999940395355},{"id":"https://openalex.org/C183517385","wikidata":"https://www.wikidata.org/wiki/Q855112","display_name":"Riemann curvature tensor","level":3,"score":0.2867000102996826},{"id":"https://openalex.org/C191948623","wikidata":"https://www.wikidata.org/wiki/Q1307800","display_name":"Unit sphere","level":2,"score":0.2865999937057495},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.2671000063419342},{"id":"https://openalex.org/C16171025","wikidata":"https://www.wikidata.org/wiki/Q863349","display_name":"Singularity","level":2,"score":0.2662999927997589},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.2637999951839447},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2549999952316284}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.25035","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.25035","pdf_url":"https://arxiv.org/pdf/2610.25035","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.2610.25035","doi":"https://doi.org/10.48550/arxiv.2610.25035","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.25035","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:2610.25035","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.25035","pdf_url":"https://arxiv.org/pdf/2610.25035","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":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7222999744.pdf","grobid_xml":"https://content.openalex.org/works/W7222999744.grobid-xml"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W2074427892","https://openalex.org/W2083716502","https://openalex.org/W2136260648","https://openalex.org/W2963715068","https://openalex.org/W3003750512","https://openalex.org/W3003827842","https://openalex.org/W3045890590","https://openalex.org/W3212448312","https://openalex.org/W4237656740","https://openalex.org/W4388022674"],"related_works":["https://openalex.org/W1672914790","https://openalex.org/W7120462873","https://openalex.org/W3003827842","https://openalex.org/W7024434690","https://openalex.org/W2083716502","https://openalex.org/W82698651","https://openalex.org/W7223200594","https://openalex.org/W2962749847","https://openalex.org/W3132902033","https://openalex.org/W2962699266"],"abstract_inverted_index":{"Let":[0],"$(M^n,g)$,":[1],"$n\\geq":[2,132],"3$,":[3],"be":[4,18],"a":[5],"compact":[6],"connected":[7],"Riemannian":[8],"manifold":[9],"with":[10,92,197],"smooth":[11,79,110,156],"boundary.":[12],"We":[13],"ask":[14],"whether":[15],"$M$":[16,203],"must":[17],"isometric":[19,31,205],"to":[20,32,61,65,126,176,206],"the":[21,33,55,59,71,127,139,151,159,163,168,183,191,207],"standard":[22,34,160,208],"hemisphere":[23,128],"if":[24,199,202],"$\\sec_g\\geq":[25],"1$,":[26],"$\\partial":[27],"M":[28],"$":[29],"is":[30,42,204],"unit":[35,102],"sphere,":[36],"and":[37,167,186,200],"its":[38],"second":[39,152],"fundamental":[40,153],"form":[41],"Codazzi.":[43],"Our":[44],"approach":[45],"centers":[46],"on":[47,51,83,95],"saddle":[48,56,80,111,141],"Codazzi":[49,81,112,142],"tensors":[50,113,143],"round":[52,164],"spheres,":[53],"where":[54],"condition":[57],"requires":[58],"restriction":[60],"every":[62,78],"tangent":[63],"two-plane":[64],"have":[66],"nonpositive":[67],"determinant.":[68],"By":[69],"analyzing":[70],"associated":[72],"support":[73],"maps,":[74],"we":[75,137,189],"prove":[76,190],"that":[77],"tensor":[82],"$\\mathbb":[84,96],"S^m$":[85],"vanishes":[86],"identically":[87],"for":[88,131],"$m\\geq":[89],"3$.":[90],"Together":[91],"Panina's":[93,146],"examples":[94],"S^2$,":[97],"this":[98],"shows":[99],"that,":[100],"among":[101],"spheres":[103],"of":[104,155,158],"dimension":[105,117,135,179],"at":[106],"least":[107],"two,":[108],"nontrivial":[109,140],"exist":[114],"precisely":[115],"in":[116,178],"two.":[118],"The":[119],"vanishing":[120],"theorem":[121],"yields":[122],"an":[123],"affirmative":[124],"answer":[125],"rigidity":[129,177],"question":[130],"4$.":[133],"In":[134],"three,":[136],"realize":[138],"provided":[144],"by":[145],"construction,":[147],"after":[148],"scaling,":[149],"as":[150],"forms":[154],"perturbations":[157],"hemisphere,":[161],"preserving":[162],"boundary":[165,187],"metric":[166],"sectional":[169],"curvature":[170,185],"lower":[171],"bound.":[172],"This":[173],"produces":[174],"counterexamples":[175],"three.":[180],"Nevertheless,":[181],"under":[182],"same":[184],"assumptions,":[188],"sharp":[192],"inradius":[193],"estimate":[194],"$r_{\\mathrm{in}}(M)\\leq":[195],"\u03c0/2$,":[196],"equality":[198],"only":[201],"hemisphere.":[209]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-21T00:00:00"}
