{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/19590"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/19590","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Symplectic and complex foliations","abstract":"\"Symplectic (not necessarily Riemannian) foliations have a transversely symplectic structure for which many standard results of symplectic geometry have their transverse analogues: the dual bundle to the transverse bundle of a foliation is a manifold with a canonical symplectic foliation, the Darboux theorem may be established, basic (holonomy invariant) functions may be regarded as transverse Hamiltonian functions for which Hamiltonian \"\"vector fields\"\" are holonomy invariant classes of vector fields in the transverse bundle, and this structure induces a holonomy invariant Poisson bracket structure on the space of basic functions.\"","abstract_html":""Symplectic (not necessarily Riemannian) foliations have a transversely symplectic structure for which many standard results of symplectic geometry have their transverse analogues: the dual bundle to the transverse bundle of a foliation is a manifold with a canonical symplectic foliation, the Darboux theorem may be established, basic (holonomy invariant) functions may be regarded as transverse Hamiltonian functions for which Hamiltonian ""vector fields"" are holonomy invariant classes of vector fields in the transverse bundle, and this structure induces a holonomy invariant Poisson bracket structure on the space of basic functions."","abstract_has_math":false,"creators":["Scofield, Paul David"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Bishop, Richard L."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:12:21Z","date_published":"2011-05-07T12:12:21Z","updated_at":"2026-07-22T22:25:14Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1990 Scofield, Paul David"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9114405","(UMI)AAI9114405"],"render_values":[{"text":"AAI9114405","href":null,"code":true},{"text":"(UMI)AAI9114405","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/19590","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Bishop, Richard L."]},{"key":"dc:creator","label":"Author","values":["Scofield, Paul David"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:12:21Z","10000-01-01","1990"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1990 Scofield, Paul David"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9114405","(UMI)AAI9114405","http://hdl.handle.net/2142/19590"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"Symplectic (not necessarily Riemannian) foliations have a transversely symplectic structure for which many standard results of symplectic geometry have their transverse analogues: the dual bundle to the transverse bundle of a foliation is a manifold with a canonical symplectic foliation, the Darboux theorem may be established, basic (holonomy invariant) functions may be regarded as transverse Hamiltonian functions for which Hamiltonian \"\"vector fields\"\" are holonomy invariant classes of vector fields in the transverse bundle, and this structure induces a holonomy invariant Poisson bracket structure on the space of basic functions.\"","If a foliation is both almost symplectic and Riemannian, then it can be given a compatible holonomy invariant complex structure, making it a Hermitian foliation. This leads somewhat naturally to a discussion of complex, almost complex, and Kahler foliations. In particular, we obtain a transverse Newlander-Nirenberg Theorem characterizing the integrability of an almost complex foliation by the vanishing of a transverse Nijenhuis torsion tensor.","As an example we consider a family of foliations of dimension one, codimension $2n$ on the ($2n$ + 1)-dimensional sphere. For $n$ = 1 we exhibit their transverse symplectic structure and investigate their transverse Hamiltonian systems. For all $n$ we exhibit the transverse Kahler structure of these foliations.","Made available in DSpace on 2011-05-07T12:12:21Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114405.pdf: 1961721 bytes, checksum: 4daff3eda2cd0b27fc9ff546ac090940 (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:38:05Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:15:47-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Symplectic and complex foliations"]}]}],"canonical_facts":{"dc:contributor":["Bishop, Richard L."],"dc:creator":["Scofield, Paul David"],"dc:date":["2011-05-07T12:12:21Z","10000-01-01","1990"],"dc:description":["\"Symplectic (not necessarily Riemannian) foliations have a transversely symplectic structure for which many standard results of symplectic geometry have their transverse analogues: the dual bundle to the transverse bundle of a foliation is a manifold with a canonical symplectic foliation, the Darboux theorem may be established, basic (holonomy invariant) functions may be regarded as transverse Hamiltonian functions for which Hamiltonian \"\"vector fields\"\" are holonomy invariant classes of vector fields in the transverse bundle, and this structure induces a holonomy invariant Poisson bracket structure on the space of basic functions.\"","If a foliation is both almost symplectic and Riemannian, then it can be given a compatible holonomy invariant complex structure, making it a Hermitian foliation. This leads somewhat naturally to a discussion of complex, almost complex, and Kahler foliations. In particular, we obtain a transverse Newlander-Nirenberg Theorem characterizing the integrability of an almost complex foliation by the vanishing of a transverse Nijenhuis torsion tensor.","As an example we consider a family of foliations of dimension one, codimension $2n$ on the ($2n$ + 1)-dimensional sphere. For $n$ = 1 we exhibit their transverse symplectic structure and investigate their transverse Hamiltonian systems. For all $n$ we exhibit the transverse Kahler structure of these foliations.","Made available in DSpace on 2011-05-07T12:12:21Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114405.pdf: 1961721 bytes, checksum: 4daff3eda2cd0b27fc9ff546ac090940 (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:38:05Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:15:47-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9114405","(UMI)AAI9114405","http://hdl.handle.net/2142/19590"],"dc:language":["eng"],"dc:rights":["Copyright 1990 Scofield, Paul David"],"dc:subject":["Mathematics"],"dc:title":["Symplectic and complex foliations"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:14Z"}