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....

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Autores:
Martínez Madrid, Daniela
Tipo de recurso:
Fecha de publicación:
2018
Institución:
Universidad Nacional de Colombia
Repositorio:
Universidad Nacional de Colombia
Idioma:
spa
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oai:repositorio.unal.edu.co:unal/63965
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https://repositorio.unal.edu.co/handle/unal/63965
http://bdigital.unal.edu.co/64628/
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51 Matemáticas / Mathematics
Teorema de Chern-Gauss-Bonnet
Chern-Gauss-Bonnet theorem
Vectores
Vectors
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openAccess
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Atribución-NoComercial 4.0 Internacional
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spelling 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
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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
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dc.rights.license.spa.fl_str_mv Atribución-NoComercial 4.0 Internacional
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dc.rights.accessrights.spa.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv 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
eu_rights_str_mv openAccess
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institution Universidad Nacional de Colombia
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repository.name.fl_str_mv Repositorio Institucional Universidad Nacional de Colombia
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