Publication: Teorema de Poncaré-Hopf
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2019
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Abstract
En este trabajo estudiamos las relaciones existentes entre funciones —campos tangentes y funciones reales— definidas sobre variedades diferenciables y la topología de dichas variedades. Para ello usamos diversas técnicas de Topología Diferencial. Los resultados principales son el Teorema del Indice de Poincaré-Hopf y la fórmula de Gauss-Bonnet para hipersuperficies de dimensión par. Básicamente ambos resultados muestran que ciertas cantidades geométricas —el índice total de un campo tangente y la curvatura íntegra— son invariantes topológicos de las variedades donde están definidas. Para la obtención de estos teoremas nuestra principal herramienta será el grado topológico de Brouwer-Kronecker; con su ayuda podremos definir la noción clave de este artículo: el índice de un campo tangente en una singularidad aislada. En el trascurso de este escrito también desarrollamos los principios de la Teoría de Morse, los cuales nos permiten demostrar el Teorema de Reeb. Finalmente, también estudiamos bajo que condiciones se puede garantizar la existencia de campos tangentes a variedades nunca nulos.
In this work we study the relationships between functions —vector fields and real functions— defined on smooth manifolds and the topology of the manifolds themselves. We will mostly use tools from Differential Topology. Main results are the Poincar´e-Hopf Index Theorem and the Gauss-Bonnet formula for hypersurfaces of even dimension. Basically both results show that certain geometrical quantities —the total index of a vector field and the integral curvature— are invariants of the manifolds where they are defined. In order to obtain these theorems our main tool will be the Brouwer-Kronecker topological degree; with it we will be able to define the key notion of this article: the index of a vector field at an isolated singularity. Along the way we will also give a short introduction to Morse Theory, which in turn allows us to prove the Reeb Theorem. Finally, we study under which hypothesis we can be certain that non-zero vector fields defined on manifolds exist.
In this work we study the relationships between functions —vector fields and real functions— defined on smooth manifolds and the topology of the manifolds themselves. We will mostly use tools from Differential Topology. Main results are the Poincar´e-Hopf Index Theorem and the Gauss-Bonnet formula for hypersurfaces of even dimension. Basically both results show that certain geometrical quantities —the total index of a vector field and the integral curvature— are invariants of the manifolds where they are defined. In order to obtain these theorems our main tool will be the Brouwer-Kronecker topological degree; with it we will be able to define the key notion of this article: the index of a vector field at an isolated singularity. Along the way we will also give a short introduction to Morse Theory, which in turn allows us to prove the Reeb Theorem. Finally, we study under which hypothesis we can be certain that non-zero vector fields defined on manifolds exist.
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