Base natural

Base natural

Base natural

En geometría diferencial, una base natural es un tipo base vectorial del espacio tangente a una variedad diferenciable que puede ser asociada naturalmente a un sistema de coordenadas curvilíneo.

Definición de base natural

Dada una variedad diferenciable \mathcal{M} de dimensión n y un conjunto de n campos vectoriales V_1,...,V_n\;\in T\mathcal{M} definidos sobre un conjunto abierto A esa variedad tales que en cada punto forman una base vectorial del espacio tangente a la variedad, es una base natural si y sólo si existe una carta local o sistema de coordenadas local \varphi:A \subset \mathcal{M} \to \R^n \land \varphi(P) \mapsto (x_1(P),...,x_n(P)) que cumpla alguna de las siguientes condiciones:

  1. Los campos vectoriales coinciden con las derivadas a lo largo de las curvas coordenadas asociadas a las coordenadas locales: {V_i|}_P = \left(\part_{x_i}\right)_P.
  2. La base dual asociada a los n campos vectoriales en cada punto, está formada por 1-formas exactas, es decir, existen \theta^1,...,\theta^n\ \in T^*\mathcal{M} y una carta local, como la descrita anteriormente, tal que \theta^i(V_j) =\delta_j^i y \theta^i = dx^i\;.
Obtenido de "Base natural"

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