Angel "Java" Lopez en Blog

Publicado el 29 de Diciembre, 2013, 15:01

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Quiero hoy comenzar esta serie de notas, sobre un tema que me persigue: las variedades (los "manifold" en inglés) y en especial, las variedades suaves.

En mis tiempos, no se estudiaba variedades o "manifolds". Se estudiaba geometría diferencial. Pero en este siglo, me encontré con las variedades, las variedades suaves, y otros temas relacionados, en "el Penrose". Esta serie de posts nace de mi necesidad de publicar notas, aisladas, arbitrarias, pero que quiero que perduren, sobre el gran tema de las variedades suaves.

Nota 1

En estos días, leo al comienzo del prefacio de "Introduction to Smooth Manifolds", de Lee, Springer:

Manifolds are everywhere. These generalizations of curves and surfaces to arbitrarily many dimensions provide the mathematical context for  understanding "space" in all of its manifestations. Today, the tools of manifold theory are indispensable in most major subfields of pure mathematics, and outside of pure mathematics they are becoming increasingly important to scientists in such diverse fields as genetics, robotics, econometrics,  computer graphics, biomedical imaging, and, of course, the undisputed leader among consumers (and inspirers) of mathematics—theoretical physics. No longer a specialized subject that is studied only by differential geometers, manifold theory is now one of the basic skills that all mathematics students should acquire as early as possible.

Over the past few centuries, mathematicians have developed a wondrous collection of conceptual machines designed to enable us to peer ever more deeply into the invisible world of geometry in higher dimensions. Once their operation is mastered, these powerful machines enable us to think geometrically about the 6-dimensional zero set of a polynomial in four complex variables, or the 10-dimensional manifold of 5 x 5 orthogonal  matrices, as easily as we think about the familiar 2-dimensional sphere in R3. The price we pay for this power, however, is that the machines are built out of layer upon layer of abstract structure. Starting with the familiar raw materials of Euclidean spaces, linear algebra, and multivariable  calculus, one must progress through topological spaces, smooth atlases, tangent bundles, cotangent bundles, immersed and embedded submanifolds,  tensors, Riemannian metrics, differential forms, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more—just to get to the point where one can even think about studying specialized applications of manifeld theory such as gauge theory or symplectic topology.

Como dice el texto, hoy las variedades están en todos lados. Fue "el Penrose" el que me puso sobreaviso sobre los "manifolds". Las variedades están relacionadas con los espacios topológicos, imposición de estructura adicional, mapas de coordenadas, diferenciación independiente del mapa de coordenados elegido, etc.


In mathematics, a manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhoodthat is homeomorphic to the Euclidean space of dimension n. Lines and circles, but notfigure eights, are one-dimensional manifolds. Two-dimensional manifolds are also calledsurfaces. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot.

Ese es el tema. Por una parte, un espacio topológico. Por otra, una estructura adicional, que permite considerar temas como la diferenciación bien definida (independiente de la carta de coordenadas elegida).

Nos leemos!

Angel "Java" Lopez