{"id":"https://openalex.org/W7224371576","doi":"https://doi.org/10.5281/zenodo.23765514","title":"Riemannian three-manifolds as hypersurfaces of Euclidean four-space: the realizability operator, its symbol, and its linearisation at the round sphere","display_name":"Riemannian three-manifolds as hypersurfaces of Euclidean four-space: the realizability operator, its symbol, and its linearisation at the round sphere","publication_year":2026,"publication_date":"2026-10-29","ids":{"openalex":"https://openalex.org/W7224371576","doi":"https://doi.org/10.5281/zenodo.23765514"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23765514","doi":"https://doi.org/10.5281/zenodo.23765514","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23765514","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23765514","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5149201535","display_name":"Jay Thrussell","orcid":"https://orcid.org/0009-0004-7707-7871","observed_orcids":["https://orcid.org/0009-0004-7707-7871"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Thrussell, Jay","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0004-7707-7871","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5149201535"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":1,"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":0.899399995803833,"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":0.899399995803833,"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":0.0869000032544136,"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":0.0044999998062849045,"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/gauss-equation","display_name":"Gauss equation","score":0.7089999914169312},{"id":"https://openalex.org/keywords/second-fundamental-form","display_name":"second fundamental form","score":0.6880000233650208},{"id":"https://openalex.org/keywords/principal-symbol","display_name":"principal symbol","score":0.7509999871253967},{"id":"https://openalex.org/keywords/linearization","display_name":"linearization","score":0.4650000035762787},{"id":"https://openalex.org/keywords/codazzi-equation","display_name":"Codazzi equation","score":0.6629999876022339},{"id":"https://openalex.org/keywords/round-spheres","display_name":"round spheres","score":0.8809999823570251},{"id":"https://openalex.org/keywords/shape-operator","display_name":"shape operator","score":0.5230000019073486}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.829800009727478},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.6298999786376953},{"id":"https://openalex.org/C114410712","wikidata":"https://www.wikidata.org/wiki/Q973321","display_name":"Hypersurface","level":2,"score":0.5623000264167786},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5306000113487244},{"id":"https://openalex.org/C119830904","wikidata":"https://www.wikidata.org/wiki/Q233220","display_name":"Second fundamental form","level":4,"score":0.5210999846458435},{"id":"https://openalex.org/C128107574","wikidata":"https://www.wikidata.org/wiki/Q182003","display_name":"Injective function","level":2,"score":0.4959999918937683},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.46700000762939453},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4390000104904175},{"id":"https://openalex.org/C47556283","wikidata":"https://www.wikidata.org/wiki/Q1058314","display_name":"Diffeomorphism","level":2,"score":0.41920000314712524},{"id":"https://openalex.org/C2776378722","wikidata":"https://www.wikidata.org/wiki/Q3454417","display_name":"Realizability","level":2,"score":0.4187999963760376},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.3817000091075897},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.37470000982284546},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.34389999508857727},{"id":"https://openalex.org/C134400042","wikidata":"https://www.wikidata.org/wiki/Q2372244","display_name":"Symbol (formal)","level":2,"score":0.33730000257492065},{"id":"https://openalex.org/C74050887","wikidata":"https://www.wikidata.org/wiki/Q848368","display_name":"Rotation (mathematics)","level":2,"score":0.3312999904155731},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.32989999651908875},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.3206999897956848},{"id":"https://openalex.org/C82457910","wikidata":"https://www.wikidata.org/wiki/Q740207","display_name":"Isometry (Riemannian geometry)","level":2,"score":0.311599999666214},{"id":"https://openalex.org/C7979169","wikidata":"https://www.wikidata.org/wiki/Q1142143","display_name":"Cotangent bundle","level":3,"score":0.296999990940094},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.2892000079154968},{"id":"https://openalex.org/C2779593128","wikidata":"https://www.wikidata.org/wiki/Q632814","display_name":"Riemannian manifold","level":2,"score":0.2831999957561493},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.28189998865127563},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.2734000086784363},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.27149999141693115},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.2685999870300293},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.2669999897480011},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.2651999890804291},{"id":"https://openalex.org/C116858840","wikidata":"https://www.wikidata.org/wiki/Q1340800","display_name":"Endomorphism","level":2,"score":0.26420000195503235},{"id":"https://openalex.org/C16977076","wikidata":"https://www.wikidata.org/wiki/Q1589551","display_name":"Principal curvature","level":4,"score":0.2639999985694885},{"id":"https://openalex.org/C206236147","wikidata":"https://www.wikidata.org/wiki/Q3555165","display_name":"Covariance and contravariance of vectors","level":2,"score":0.2540999948978424}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.5281/zenodo.23765514","doi":"https://doi.org/10.5281/zenodo.23765514","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23765514","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.23765515","doi":"https://doi.org/10.5281/zenodo.23765515","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23765515","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.23773768","doi":"https://doi.org/10.5281/zenodo.23773768","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23773768","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.23811178","doi":"https://doi.org/10.5281/zenodo.23811178","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23811178","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.23869205","doi":"https://doi.org/10.5281/zenodo.23869205","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23869205","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"doi:10.5281/zenodo.23765514","doi":"https://doi.org/10.5281/zenodo.23765514","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23765514","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":1,"referenced_works":["https://openalex.org/W7167693451"],"related_works":["https://openalex.org/W7213624631","https://openalex.org/W7203630594","https://openalex.org/W7204698706","https://openalex.org/W7204712579","https://openalex.org/W1502824797","https://openalex.org/W7095990660","https://openalex.org/W2804059522","https://openalex.org/W4281728984","https://openalex.org/W2970978119","https://openalex.org/W2081796218"],"abstract_inverted_index":{"##":[0],"Abstract":[1],"For":[2,326],"a":[3,20,86,180,230,249,307,347,374,392,408,466,476],"Riemannian":[4],"three-manifold":[5,251],"$(\\Sigma,":[6],"g)$":[7],"with":[8,46,211,282,372,394,407],"negative":[9],"Einstein":[10],"endomorphism":[11],"$\\mathbb":[12,25],"M":[13,26,31],"=":[14,27,41,56,59,384],"-G$,":[15],"the":[16,69,98,109,113,118,121,136,167,173,223,235,239,243,252,259,266,276,296,311,318,337,343,353,362,378,382,399,411,419,427,433,442,449,459],"Gauss":[17],"equation":[18],"of":[19,22,72,101,120,238,255,342,355,377,398,410,426,430,432,435,444,452],"hypersurface":[21],"$E^4$":[23],"reads":[24],"\\mathrm{adj}\\,K$.":[28],"On":[29,248,273],"$\\{\\det\\mathbb":[30],">":[32],"0\\}$":[33],"it":[34,405],"is":[35,51,63,108,133,141,152,177,262,278,365,388,463,473],"solved":[36],"identically":[37],"by":[38,329,346,359,391,475],"$\\tilde":[39],"K":[40],"\\sqrt{\\det\\mathbb":[42],"M}\\,\\mathbb":[43],"M^{-1}$.":[44],"Realizability":[45],"nondegenerate":[47,96,205,267,460],"second":[48,315],"fundamental":[49],"form":[50,336,451],"therefore":[52],"one":[53,225,429],"equation,":[54,428],"$F(g)":[55],"\\mathrm{Cod}(\\tilde":[57],"K(g))":[58],"0$.":[60,165],"This":[61],"reduction":[62],"classical":[64],"(Beez,":[65],"Killing,":[66],"Thomas),":[67],"and":[68,80,127,140,179,227,293,454],"zero":[70,253],"set":[71,254,313],"$F$":[73,84,102,132],"has":[74,103],"been":[75],"characterized":[76],"intrinsically":[77],"(Rivertz;":[78],"Agaoka":[79],"Hashinaga).":[81],"We":[82,350,417,438],"study":[83],"as":[85,441],"nonlinear":[87],"operator":[88],"on":[89,128,154,458],"metrics.":[90],"-":[91,147,162,187,246,269,324,414],"**The":[92,188,270,415],"symbol.**":[93],"At":[94],"every":[95,125,204],"point":[97],"principal":[99],"symbol":[100,119,151,240],"rank":[104],"two.":[105],"Its":[106],"kernel":[107,308],"diffeomorphism":[110],"directions":[111],"plus":[112],"normal":[114],"direction":[115,309,387],"$K$:":[116],"exactly":[117,171,242,390],"linearised":[122,137,244],"immersions,":[123],"at":[124,157,203,213,219,286,314],"covector":[126],"both":[129],"branches.":[130],"So":[131,166,381],"injective":[134,153],"modulo":[135,144],"immersions":[138],"everywhere,":[139],"nowhere":[142,263],"elliptic":[143],"diffeomorphisms":[145],"alone.":[146],"**Transverse-traceless":[148],"tensors.**":[149],"The":[150,190,303],"transverse-traceless":[155,168,274],"tensors":[156,275],"$\\xi$":[158],"iff":[159],"$\\xi^{\\mathsf":[160],"T}(\\sigma_1g":[161],"K)\\xi":[163],"\\neq":[164],"slice":[169,181],"serves":[170,185],"where":[172],"first":[174],"Newton":[175],"tensor":[176],"definite,":[178],"adapted":[182],"to":[183,465],"$K$":[184],"everywhere.":[186],"count.**":[189],"nine":[191],"Codazzi":[192],"components":[193],"satisfy":[194],"four":[195],"identities,":[196],"which":[197,257],"leaves":[198,310],"five":[199],"conditions":[200],"per":[201],"point,":[202],"shape":[206,221],"operator.":[207],"Cartan's":[208],"test":[209],"holds":[210],"equality":[212],"order":[214],"three,":[215],"in":[216,265,322,334],"exact":[217],"arithmetic,":[218],"sixteen":[220],"operators,":[222],"umbilic":[224],"included,":[226],"so":[228],"off":[229],"proper":[231],"real-algebraic":[232],"set.":[233],"There":[234],"formal":[236],"solutions":[237],"are":[241,290],"immersions.":[245],"**Meagreness.**":[247],"closed":[250,335],"$F$,":[256],"contains":[258],"realizable":[260,312],"metrics,":[261],"dense":[264],"domain.":[268],"round":[271,330,360],"sphere.**":[272],"linearisation":[277],"$\\mathrm{Cod}\\circ\\tfrac12(-\\Delta":[279],"+":[280,367,369],"2)$,":[281],"singular":[283],"values":[284],"$n(n^2-1)/\\sqrt2$":[285],"level":[287],"$n$.":[288],"These":[289],"Branson's":[291],"multipliers,":[292],"they":[294],"give":[295,439],"bound":[297],"$\\lVert":[298],"DF\\,h\\rVert^2":[299],"\\geq":[300],"288\\lVert":[301],"h\\rVert^2$.":[302],"straight":[304],"path":[305],"along":[306],"order,":[316],"while":[317],"radial":[319,344,363],"lift":[320],"stays":[321],"it.":[323],"**Shells.**":[325],"metrics":[327,354],"foliated":[328],"2-spheres":[331],"we":[332],"solve":[333],"first-order":[338,386],"problem":[339],"for":[340],"deformations":[341],"coefficient":[345,364],"shell":[348,412],"harmonic.":[349],"write":[351],"down":[352],"all":[356],"hypersurfaces":[357],"swept":[358],"2-spheres:":[361],"$H^2":[366],"(\\alpha":[368],"\\Phi\\,b\\cdot\\hat":[370],"n)^2$,":[371],"$b$":[373],"free":[375],"tilt":[376],"spheres'":[379],"3-planes.":[380],"$\\ell":[383],"1$":[385],"realized":[389],"tilt,":[393],"no":[395],"in-plane":[396,402],"drift":[397,403],"centres.":[400],"An":[401],"realizes":[404],"together":[406],"shear":[409],"labels.":[413],"rest.**":[416],"treat":[418],"degenerate":[420],"locus":[421],"through":[422],"two":[423,445],"polynomial":[424],"forms":[425],"them":[431],"bidegree":[434],"Rivertz's":[436],"equalities.":[437],"Gauss\u2013Codazzi":[440],"flatness":[443],"real":[446],"$\\mathfrak{su}(2)$":[447],"connections,":[448],"connection":[450],"Lawn":[453],"Roth's":[455],"two-spinor":[456],"characterization;":[457],"domain,":[461],"realizability":[462],"equivalent":[464],"single":[467],"parallel":[468],"spinor.":[469],"Every":[470],"displayed":[471],"result":[472],"reproduced":[474],"script;":[477],"Appendix":[478],"V":[479],"indexes":[480],"them.":[481]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-02T09:45:03.842371","created_date":"2026-10-30T00:00:00"}