{"id":"https://openalex.org/W7211895045","doi":"https://doi.org/10.48550/arxiv.2609.04299","title":"Harmonic morphisms of sub-Riemannian Lie groups","display_name":"Harmonic morphisms of sub-Riemannian Lie groups","publication_year":2026,"publication_date":"2026-09-03","ids":{"openalex":"https://openalex.org/W7211895045","doi":"https://doi.org/10.48550/arxiv.2609.04299"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.04299","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.04299","pdf_url":"https://arxiv.org/pdf/2609.04299","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","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.04299","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5028748648","display_name":"Sebastiano Nicolussi Golo","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Golo, Sebastiano Nicolussi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5044990686","display_name":"Andrea Pinamonti","orcid":"https://orcid.org/0000-0003-1000-8607","observed_orcids":["https://orcid.org/0000-0003-1000-8607"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Pinamonti, Andrea","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5007367086","display_name":"Ben Warhurst","orcid":"https://orcid.org/0000-0002-2770-4871","observed_orcids":["https://orcid.org/0000-0002-2770-4871"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Warhurst, Ben","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":0.7128999829292297,"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":0.7128999829292297,"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":0.26759999990463257,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"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/harmonic-morphisms","display_name":"harmonic morphisms","score":0.996999979019165},{"id":"https://openalex.org/keywords/harmonic-coordinates","display_name":"harmonic coordinates","score":0.9629999995231628},{"id":"https://openalex.org/keywords/carnot-groups","display_name":"Carnot groups","score":0.6930000185966492},{"id":"https://openalex.org/keywords/sub-riemannian-manifolds","display_name":"sub-Riemannian manifolds","score":0.8479999899864197}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8396999835968018},{"id":"https://openalex.org/C137212723","wikidata":"https://www.wikidata.org/wiki/Q1948412","display_name":"Morphism","level":2,"score":0.6564000248908997},{"id":"https://openalex.org/C96469262","wikidata":"https://www.wikidata.org/wiki/Q1324364","display_name":"Homogeneous space","level":2,"score":0.646399974822998},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5910999774932861},{"id":"https://openalex.org/C51568863","wikidata":"https://www.wikidata.org/wiki/Q664495","display_name":"Lie algebra","level":2,"score":0.5214999914169312},{"id":"https://openalex.org/C187915474","wikidata":"https://www.wikidata.org/wiki/Q622679","display_name":"Lie group","level":2,"score":0.5127000212669373},{"id":"https://openalex.org/C127934551","wikidata":"https://www.wikidata.org/wiki/Q1148098","display_name":"Harmonic","level":2,"score":0.5038999915122986},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.3930000066757202},{"id":"https://openalex.org/C86108249","wikidata":"https://www.wikidata.org/wiki/Q2996688","display_name":"Harmonic coordinates","level":4,"score":0.3808000087738037},{"id":"https://openalex.org/C58910974","wikidata":"https://www.wikidata.org/wiki/Q1017106","display_name":"Adjoint representation","level":3,"score":0.3702000081539154},{"id":"https://openalex.org/C22365015","wikidata":"https://www.wikidata.org/wiki/Q1003162","display_name":"Simple Lie group","level":3,"score":0.36719998717308044},{"id":"https://openalex.org/C78804095","wikidata":"https://www.wikidata.org/wiki/Q1133024","display_name":"Lie superalgebra","level":5,"score":0.35199999809265137},{"id":"https://openalex.org/C98214594","wikidata":"https://www.wikidata.org/wiki/Q850275","display_name":"Conformal map","level":2,"score":0.35010001063346863},{"id":"https://openalex.org/C73648015","wikidata":"https://www.wikidata.org/wiki/Q6543817","display_name":"Lie conformal algebra","level":3,"score":0.34540000557899475},{"id":"https://openalex.org/C186489112","wikidata":"https://www.wikidata.org/wiki/Q763735","display_name":"Representation of a Lie group","level":3,"score":0.3197000026702881},{"id":"https://openalex.org/C197273675","wikidata":"https://www.wikidata.org/wiki/Q13220368","display_name":"Representation theory","level":2,"score":0.3050000071525574},{"id":"https://openalex.org/C5106717","wikidata":"https://www.wikidata.org/wiki/Q3427416","display_name":"Fundamental representation","level":4,"score":0.304500013589859},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2971000075340271},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.2953999936580658},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.28630000352859497},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.2736999988555908},{"id":"https://openalex.org/C143135876","wikidata":"https://www.wikidata.org/wiki/Q357300","display_name":"Lie theory","level":5,"score":0.2660999894142151},{"id":"https://openalex.org/C6270764","wikidata":"https://www.wikidata.org/wiki/Q5659257","display_name":"Harmonic map","level":2,"score":0.2653000056743622},{"id":"https://openalex.org/C144091092","wikidata":"https://www.wikidata.org/wiki/Q3611543","display_name":"Graded Lie algebra","level":3,"score":0.26429998874664307},{"id":"https://openalex.org/C203946495","wikidata":"https://www.wikidata.org/wiki/Q1017106","display_name":"Adjoint representation of a Lie algebra","level":4,"score":0.26420000195503235},{"id":"https://openalex.org/C627467","wikidata":"https://www.wikidata.org/wiki/Q599027","display_name":"Harmonic function","level":2,"score":0.26409998536109924},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.25429999828338623}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.04299","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.04299","pdf_url":"https://arxiv.org/pdf/2609.04299","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.2609.04299","doi":"https://doi.org/10.48550/arxiv.2609.04299","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.04299","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:2609.04299","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.04299","pdf_url":"https://arxiv.org/pdf/2609.04299","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7211895045.pdf","grobid_xml":"https://content.openalex.org/works/W7211895045.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4386107353","https://openalex.org/W1603911705","https://openalex.org/W1963618346","https://openalex.org/W7133352257","https://openalex.org/W4402837612","https://openalex.org/W2075419256","https://openalex.org/W162831094","https://openalex.org/W2476357857","https://openalex.org/W4406031842","https://openalex.org/W2003286344"],"abstract_inverted_index":{"We":[0,38],"prove":[1],"that":[2,21,50],"harmonic":[3,33,43,52,75,116,125],"morphisms":[4,34,53,117],"between":[5],"sub-Riemannian":[6,15,61,78,119],"Lie":[7,36,79,90,120],"groups":[8,121],"are":[9,11,18,54],"smooth,":[10],"symmetries":[12,55],"of":[13,42,56,69,77,92,118],"the":[14,40,57,66,70,89,93,95,113],"Laplacian":[16,58],"and":[17,49,102],"conformal":[19],"submersions":[20],"satisfy":[22],"a":[23,74,85],"particular":[24],"PDE.":[25],"Moreover,":[26],"we":[27,64],"give":[28],"some":[29],"partial":[30],"results":[31],"for":[32,45,100],"beyond":[35],"groups.":[37],"show":[39],"existence":[41],"coordinates":[44],"harmonically":[46],"homogeneous":[47],"spaces,":[48],"smooth":[51],"in":[59,108,112],"all":[60],"manifolds.":[62],"Finally,":[63],"compute":[65],"first":[67],"variation":[68],"horizontal":[71],"energy":[72],"along":[73],"morphism":[76],"groups:":[80],"it":[81],"is":[82],"given":[83],"by":[84],"linear":[86],"form":[87],"on":[88],"algebra":[91],"target,":[94],"modular":[96],"mismatch,":[97],"which":[98],"vanishes":[99],"Riemannian":[101,114],"Carnot":[103],"targets,":[104],"but":[105],"surprisingly":[106],"not":[107,123],"general.":[109],"Therefore,":[110],"unlike":[111],"case,":[115],"need":[122],"be":[124],"maps.":[126]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-08T00:00:00"}
