{"id":"https://openalex.org/W7223248677","doi":"https://doi.org/10.48550/arxiv.2610.29360","title":"Curvature, second variation and Jacobi fields in elastic diffeology","display_name":"Curvature, second variation and Jacobi fields in elastic diffeology","publication_year":2026,"publication_date":"2026-10-21","ids":{"openalex":"https://openalex.org/W7223248677","doi":"https://doi.org/10.48550/arxiv.2610.29360"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.29360","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.29360","pdf_url":"https://arxiv.org/pdf/2610.29360","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/2610.29360","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5120300284","display_name":"Yusuke Shiobara","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Shiobara, Yusuke","raw_affiliation_strings":["Department of Science and Technology , Graduate School of Medicine, Science and Technology , Shinshu University , Matsumoto , Nagano 390-8621 , Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Science and Technology , Graduate School of Medicine, Science and Technology , Shinshu University , Matsumoto , Nagano 390-8621 , Japan","institution_ids":["https://openalex.org/I137975476"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5120300284"],"corresponding_institution_ids":["https://openalex.org/I137975476"],"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.8246999979019165,"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":0.8246999979019165,"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":0.17270000278949738,"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":0.0007999999797903001,"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/curvature-tensor","display_name":"curvature tensor","score":0.6859999895095825},{"id":"https://openalex.org/keywords/levi-civita-connection","display_name":"Levi-Civita connection","score":0.8610000014305115},{"id":"https://openalex.org/keywords/sectional-curvature","display_name":"sectional curvature","score":0.8640000224113464},{"id":"https://openalex.org/keywords/curvature-form","display_name":"curvature form","score":0.6620000004768372},{"id":"https://openalex.org/keywords/index-form","display_name":"index form","score":0.9940000176429749},{"id":"https://openalex.org/keywords/jacobi-fields","display_name":"Jacobi fields","score":0.9990000128746033},{"id":"https://openalex.org/keywords/geodesic-flow","display_name":"geodesic flow","score":0.9409999847412109},{"id":"https://openalex.org/keywords/conjugate-points","display_name":"conjugate points","score":0.9549999833106995},{"id":"https://openalex.org/keywords/free-loop-space","display_name":"free loop space","score":0.7699999809265137},{"id":"https://openalex.org/keywords/mapping-spaces","display_name":"mapping spaces","score":0.8320000171661377},{"id":"https://openalex.org/keywords/trace-class-operators","display_name":"trace class operators","score":0.5059999823570251}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8100000023841858},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.652400016784668},{"id":"https://openalex.org/C12520029","wikidata":"https://www.wikidata.org/wiki/Q1147161","display_name":"Scalar curvature","level":3,"score":0.6225000023841858},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.5735999941825867},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5720999836921692},{"id":"https://openalex.org/C183517385","wikidata":"https://www.wikidata.org/wiki/Q855112","display_name":"Riemann curvature tensor","level":3,"score":0.5464000105857849},{"id":"https://openalex.org/C42448751","wikidata":"https://www.wikidata.org/wiki/Q855622","display_name":"Sectional curvature","level":4,"score":0.44530001282691956},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.4104999899864197},{"id":"https://openalex.org/C121854251","wikidata":"https://www.wikidata.org/wiki/Q62932","display_name":"Elasticity (physics)","level":2,"score":0.398499995470047},{"id":"https://openalex.org/C161223928","wikidata":"https://www.wikidata.org/wiki/Q3775868","display_name":"Center of curvature","level":4,"score":0.3756999969482422},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.37549999356269836},{"id":"https://openalex.org/C116858840","wikidata":"https://www.wikidata.org/wiki/Q1340800","display_name":"Endomorphism","level":2,"score":0.37389999628067017},{"id":"https://openalex.org/C12089564","wikidata":"https://www.wikidata.org/wiki/Q1195879","display_name":"Ricci curvature","level":3,"score":0.3499000072479248},{"id":"https://openalex.org/C77791780","wikidata":"https://www.wikidata.org/wiki/Q839832","display_name":"Curvature form","level":5,"score":0.3422999978065491},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.33079999685287476},{"id":"https://openalex.org/C47556283","wikidata":"https://www.wikidata.org/wiki/Q1058314","display_name":"Diffeomorphism","level":2,"score":0.320499986410141},{"id":"https://openalex.org/C136005978","wikidata":"https://www.wikidata.org/wiki/Q1362077","display_name":"First variation","level":2,"score":0.31049999594688416},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.30630001425743103},{"id":"https://openalex.org/C175017881","wikidata":"https://www.wikidata.org/wiki/Q1318998","display_name":"Mean curvature","level":3,"score":0.30140000581741333},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2935999929904938},{"id":"https://openalex.org/C2779668893","wikidata":"https://www.wikidata.org/wiki/Q2250422","display_name":"Constant curvature","level":3,"score":0.2913999855518341},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.2856999933719635},{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.28380000591278076},{"id":"https://openalex.org/C27851653","wikidata":"https://www.wikidata.org/wiki/Q829653","display_name":"Bounded variation","level":2,"score":0.2827000021934509},{"id":"https://openalex.org/C96469262","wikidata":"https://www.wikidata.org/wiki/Q1324364","display_name":"Homogeneous space","level":2,"score":0.2784999907016754},{"id":"https://openalex.org/C47422493","wikidata":"https://www.wikidata.org/wiki/Q2755127","display_name":"Wedge (geometry)","level":2,"score":0.26989999413490295},{"id":"https://openalex.org/C84985374","wikidata":"https://www.wikidata.org/wiki/Q1892644","display_name":"Torsion of a curve","level":5,"score":0.2619999945163727}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.29360","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.29360","pdf_url":"https://arxiv.org/pdf/2610.29360","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.2610.29360","doi":"https://doi.org/10.48550/arxiv.2610.29360","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.29360","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:2610.29360","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.29360","pdf_url":"https://arxiv.org/pdf/2610.29360","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/W7223248677.pdf","grobid_xml":"https://content.openalex.org/works/W7223248677.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W4399511933","https://openalex.org/W4417477854","https://openalex.org/W7172418477"],"related_works":["https://openalex.org/W846433594","https://openalex.org/W2054400520","https://openalex.org/W41032958","https://openalex.org/W39135558","https://openalex.org/W1526896038","https://openalex.org/W2763425874","https://openalex.org/W2066062004","https://openalex.org/W2111764101","https://openalex.org/W7220639948","https://openalex.org/W2567022817"],"abstract_inverted_index":{"We":[0,9,53,93],"develop":[1],"curvature":[2,14,47,56,59,80,150],"theory":[3],"on":[4,25,31,61,76],"Blohmann's":[5],"elastic":[6],"diffeological":[7],"spaces.":[8,33],"introduce":[10],"curvature-tensorial":[11,50],"connections,":[12],"whose":[13],"factors":[15],"uniquely":[16],"through":[17],"$T_3M$,":[18],"and":[19,28,39,57,66,105,171],"show":[20],"that":[21],"this":[22],"property":[23,170],"holds":[24],"finite-dimensional":[26],"manifolds":[27],"is":[29,89,140],"preserved":[30],"mapping":[32,174],"Using":[34],"a":[35,58,118,141,155],"plotwise":[36],"commutator":[37],"identity":[38],"the":[40,45,62,71,78,85,95,99,102,107,111,137,145,162,168],"elasticity":[41],"axioms,":[42],"we":[43,121],"establish":[44],"classical":[46],"symmetries":[48],"for":[49,144],"Levi-Civita":[51],"connections.":[52],"define":[54],"sectional":[55,149],"form":[60,104,113],"algebraic":[63],"exterior":[64],"square,":[65],"demonstrate":[67],"an":[68],"obstruction":[69],"to":[70,84,173],"usual":[72],"Ricci":[73],"trace":[74,91],"construction:":[75],"$C^\\infty(S^1,S^2)$,":[77],"relevant":[79],"endomorphism":[81],"extends":[82],"boundedly":[83],"$L^2$":[86],"completion":[87],"but":[88],"not":[90,166],"class.":[92],"express":[94],"second":[96],"variation":[97],"of":[98,110,181],"energy":[100],"by":[101,114],"index":[103,112,157],"characterize":[106],"null":[108],"space":[109],"Jacobi":[115,123],"fields.":[116],"Given":[117],"geodesic":[119],"flow,":[120],"construct":[122],"fields":[124],"with":[125],"prescribed":[126],"initial":[127],"data":[128],"along":[129],"its":[130],"flow":[131],"lines;":[132],"they":[133],"are":[134],"unique":[135],"when":[136],"real":[138],"line":[139],"curve":[142],"object":[143],"space.":[146],"Finally,":[147],"non-positive":[148],"excludes":[151],"conjugate":[152],"points,":[153],"while":[154],"negative":[156],"direction":[158],"obstructs":[159],"length":[160],"minimization;":[161],"latter":[163],"result":[164],"does":[165],"require":[167],"curve-object":[169],"applies":[172],"spaces":[175,180],"such":[176],"as":[177],"free":[178],"loop":[179],"complete":[182],"Riemannian":[183],"manifolds.":[184]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-10-23T00:00:00"}