Differentiable Manifold

Differentiable manifold

An n-dimensional (differentiable) manifold M^n is a Hausdorff topological space with a countable (topological) basis, together with a maximal differentiable atlas.

This atlas consists of a family of charts \displaystyle h_\lambda: U_\lambda\to U'_\lambda\subset\mathbb{R}^n, where the domains of the charts, \{U_\lambda\}, form an open cover of M^n, the U'_\lambda are open in \mathbb{R}^n, the charts (local coordinates) h_\lambda are homeomorphisms, and every change of coordinates \displaystyle h_{\lambda\mu}=h_\mu\circ h_\lambda^{-1} is differentiable on its domain of definition h_\lambda(U_\lambda\cap U_\mu).

atlas

Source: Representations of Compact Lie Groups (Graduate Texts in Mathematics)

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  1. Pingback: Differentiable Manifold — Singapore Maths Tuition | Mathpresso

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