On Chern's conjecture about the Euler characteristic of affine manifolds
The development the theory of characteristic classes allowed Shiing-Shen Chern to generalize the Gauss Bonnet theorem to Riemannian manifolds of arbitrary dimension. The Chern Gauss Bonnet theorem expresses the Euler characteristic as an integral of a polynomial evaluated on the curvature tensor, i....
- Autores:
-
Martínez Madrid, Daniela
- Tipo de recurso:
- Fecha de publicación:
- 2018
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- spa
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/63965
- Acceso en línea:
- https://repositorio.unal.edu.co/handle/unal/63965
http://bdigital.unal.edu.co/64628/
- Palabra clave:
- 51 Matemáticas / Mathematics
Teorema de Chern-Gauss-Bonnet
Chern-Gauss-Bonnet theorem
Vectores
Vectors
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- openAccess
- License
- Atribución-NoComercial 4.0 Internacional
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Atribución-NoComercial 4.0 InternacionalDerechos reservados - Universidad Nacional de Colombiahttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Arias Abad, CamiloMartínez Madrid, Danieladd8f2a85-7d54-4a7e-af59-b376562b74503002019-07-02T22:20:32Z2019-07-02T22:20:32Z2018https://repositorio.unal.edu.co/handle/unal/63965http://bdigital.unal.edu.co/64628/The development the theory of characteristic classes allowed Shiing-Shen Chern to generalize the Gauss Bonnet theorem to Riemannian manifolds of arbitrary dimension. The Chern Gauss Bonnet theorem expresses the Euler characteristic as an integral of a polynomial evaluated on the curvature tensor, i.e if K is the curvature form of the Levi-Civita connection, the Chern Gauss Bonnet formula is . In particular, the theorem implies that if the Levi Civita connection is _at, the Euler characteristic is zero.An a_ne structure on a manifold is an atlas whose transition functions are a_ne transformations. The existence of such a structure is equivalent to the existence of a _at torsion free connection on the tangent bundle. Around 1955 Chern conjectured the following: Conjecture. The Euler characteristic of a closed affine manifold is zero. Not all fat torsion free connections on TM admit a compatible metric, and therefore, Chern-Weil theory cannot be used in general to write down the Euler class in terms of the curvature. In 1955, Benzécri [1] proved that a closed affine surface has zero Euler characteristic. Later, in 1958, Milnor [11] proved inequalities which completely characterise those oriented rank two bundles over a surface that admit a fiat connection. These inequalities prove that in case of a surface the condition "be torsion free" in Chern's conjecture is not necessary. In 1975, Kostant and Sullivan [9] proved Chern's conjecture in the case where the manifold is complete. In 1977, Smillie [15] proved that the condition that the connection is torsion free matters. For each even dimension greater than 2, Smillie constructed closed manifolds with non-zero Euler characteristic that admit a _at connection on their tangent bundle. In 2015, Klingler [14] proved the conjecture for special affine manifolds. That is, affine manifolds that admit a parallel volume form.Maestríaapplication/pdfspaUniversidad Nacional de Colombia Sede Medellín Facultad de Ciencias Instituto de Matemática Pura y AplicadaInstituto de Matemática Pura y AplicadaMartínez Madrid, Daniela (2018) On Chern's conjecture about the Euler characteristic of affine manifolds. Maestría thesis, Universidad Nacional de Colombia - Sede Medellín.51 Matemáticas / MathematicsTeorema de Chern-Gauss-BonnetChern-Gauss-Bonnet theoremVectoresVectorsOn Chern's conjecture about the Euler characteristic of affine manifoldsTrabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TMORIGINAL1152441431.2018.pdfapplication/pdf897595https://repositorio.unal.edu.co/bitstream/unal/63965/1/1152441431.2018.pdf02ce55b58f136729fbc64db385bbb082MD51THUMBNAIL1152441431.2018.pdf.jpg1152441431.2018.pdf.jpgGenerated Thumbnailimage/jpeg5350https://repositorio.unal.edu.co/bitstream/unal/63965/2/1152441431.2018.pdf.jpg77ea7bfabec3e8948dcafa30dd6804b0MD52unal/63965oai:repositorio.unal.edu.co:unal/639652024-05-01 23:12:08.987Repositorio Institucional Universidad Nacional de Colombiarepositorio_nal@unal.edu.co |
dc.title.spa.fl_str_mv |
On Chern's conjecture about the Euler characteristic of affine manifolds |
title |
On Chern's conjecture about the Euler characteristic of affine manifolds |
spellingShingle |
On Chern's conjecture about the Euler characteristic of affine manifolds 51 Matemáticas / Mathematics Teorema de Chern-Gauss-Bonnet Chern-Gauss-Bonnet theorem Vectores Vectors |
title_short |
On Chern's conjecture about the Euler characteristic of affine manifolds |
title_full |
On Chern's conjecture about the Euler characteristic of affine manifolds |
title_fullStr |
On Chern's conjecture about the Euler characteristic of affine manifolds |
title_full_unstemmed |
On Chern's conjecture about the Euler characteristic of affine manifolds |
title_sort |
On Chern's conjecture about the Euler characteristic of affine manifolds |
dc.creator.fl_str_mv |
Martínez Madrid, Daniela |
dc.contributor.author.spa.fl_str_mv |
Martínez Madrid, Daniela |
dc.contributor.spa.fl_str_mv |
Arias Abad, Camilo |
dc.subject.ddc.spa.fl_str_mv |
51 Matemáticas / Mathematics |
topic |
51 Matemáticas / Mathematics Teorema de Chern-Gauss-Bonnet Chern-Gauss-Bonnet theorem Vectores Vectors |
dc.subject.proposal.spa.fl_str_mv |
Teorema de Chern-Gauss-Bonnet Chern-Gauss-Bonnet theorem Vectores Vectors |
description |
The development the theory of characteristic classes allowed Shiing-Shen Chern to generalize the Gauss Bonnet theorem to Riemannian manifolds of arbitrary dimension. The Chern Gauss Bonnet theorem expresses the Euler characteristic as an integral of a polynomial evaluated on the curvature tensor, i.e if K is the curvature form of the Levi-Civita connection, the Chern Gauss Bonnet formula is . In particular, the theorem implies that if the Levi Civita connection is _at, the Euler characteristic is zero.An a_ne structure on a manifold is an atlas whose transition functions are a_ne transformations. The existence of such a structure is equivalent to the existence of a _at torsion free connection on the tangent bundle. Around 1955 Chern conjectured the following: Conjecture. The Euler characteristic of a closed affine manifold is zero. Not all fat torsion free connections on TM admit a compatible metric, and therefore, Chern-Weil theory cannot be used in general to write down the Euler class in terms of the curvature. In 1955, Benzécri [1] proved that a closed affine surface has zero Euler characteristic. Later, in 1958, Milnor [11] proved inequalities which completely characterise those oriented rank two bundles over a surface that admit a fiat connection. These inequalities prove that in case of a surface the condition "be torsion free" in Chern's conjecture is not necessary. In 1975, Kostant and Sullivan [9] proved Chern's conjecture in the case where the manifold is complete. In 1977, Smillie [15] proved that the condition that the connection is torsion free matters. For each even dimension greater than 2, Smillie constructed closed manifolds with non-zero Euler characteristic that admit a _at connection on their tangent bundle. In 2015, Klingler [14] proved the conjecture for special affine manifolds. That is, affine manifolds that admit a parallel volume form. |
publishDate |
2018 |
dc.date.issued.spa.fl_str_mv |
2018 |
dc.date.accessioned.spa.fl_str_mv |
2019-07-02T22:20:32Z |
dc.date.available.spa.fl_str_mv |
2019-07-02T22:20:32Z |
dc.type.spa.fl_str_mv |
Trabajo de grado - Maestría |
dc.type.driver.spa.fl_str_mv |
info:eu-repo/semantics/masterThesis |
dc.type.version.spa.fl_str_mv |
info:eu-repo/semantics/acceptedVersion |
dc.type.content.spa.fl_str_mv |
Text |
dc.type.redcol.spa.fl_str_mv |
http://purl.org/redcol/resource_type/TM |
status_str |
acceptedVersion |
dc.identifier.uri.none.fl_str_mv |
https://repositorio.unal.edu.co/handle/unal/63965 |
dc.identifier.eprints.spa.fl_str_mv |
http://bdigital.unal.edu.co/64628/ |
url |
https://repositorio.unal.edu.co/handle/unal/63965 http://bdigital.unal.edu.co/64628/ |
dc.language.iso.spa.fl_str_mv |
spa |
language |
spa |
dc.relation.ispartof.spa.fl_str_mv |
Universidad Nacional de Colombia Sede Medellín Facultad de Ciencias Instituto de Matemática Pura y Aplicada Instituto de Matemática Pura y Aplicada |
dc.relation.references.spa.fl_str_mv |
Martínez Madrid, Daniela (2018) On Chern's conjecture about the Euler characteristic of affine manifolds. Maestría thesis, Universidad Nacional de Colombia - Sede Medellín. |
dc.rights.spa.fl_str_mv |
Derechos reservados - Universidad Nacional de Colombia |
dc.rights.coar.fl_str_mv |
http://purl.org/coar/access_right/c_abf2 |
dc.rights.license.spa.fl_str_mv |
Atribución-NoComercial 4.0 Internacional |
dc.rights.uri.spa.fl_str_mv |
http://creativecommons.org/licenses/by-nc/4.0/ |
dc.rights.accessrights.spa.fl_str_mv |
info:eu-repo/semantics/openAccess |
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Atribución-NoComercial 4.0 Internacional Derechos reservados - Universidad Nacional de Colombia http://creativecommons.org/licenses/by-nc/4.0/ http://purl.org/coar/access_right/c_abf2 |
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openAccess |
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