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On Chern's conjecture about the Euler characteristic of affine manifolds

Martínez Madrid, Daniela (2018) On Chern's conjecture about the Euler characteristic of affine manifolds. Maestría thesis, Universidad Nacional de Colombia - Sede Medellín.

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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.

Tipo de documento:Tesis/trabajos de grado - Thesis (Maestría)
Colaborador / Asesor:Arias Abad, Camilo
Información adicional:Línea de Investigación: Geometría Diferencial
Palabras clave:Teorema de Chern-Gauss-Bonnet, Chern-Gauss-Bonnet theorem, Vectores, Vectors
Temática:5 Ciencias naturales y matemáticas / Science > 51 Matemáticas / Mathematics
Unidad administrativa:Sede Medellín > Facultad de Ciencias > Instituto de Matemática Pura y Aplicada
Código ID:64628
Enviado por : Daniela Martínez Madrid
Enviado el día :22 Junio 2018 17:19
Ultima modificación:27 Febrero 2019 20:26
Ultima modificación:27 Febrero 2019 20:26
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