Dynamical Systems: An Introduction by Luis Barreira & Claudia Valls

Author:Luis Barreira & Claudia Valls [Barreira, Luis & Valls, Claudia]
Language: eng
Format: epub
Tags: Differential Equations, Numerical Analysis, Non-Euclidean, Geometry, General, Mathematics, Linear & Nonlinear Programming, Mathematical Analysis
ISBN: 9781447148357
Google: 1XGRcE0OaksC
Publisher: Springer Science & Business Media
Published: 2013-01-02T00:00:00+00:00

5.1 Smooth Manifolds

In this section we recall some basic notions of the theory of smooth manifolds.

Definition 5.1

A set M is said to admit a differentiable structure of dimension n∈ℕ if there exist injective maps φ i :U i →M in open sets U i ⊂ℝ n for i∈I such that: 1.⋃ i∈I φ i (U i )=M;

2.for any i,j∈I such that

the preimages and are open and the map is of class C 1.

Each map φ i :U i →M is called a chart or a coordinate system. Given a differentiable structure on M, we consider the topology on M formed by the sets A⊂M such that is open for every i∈I.

Definition 5.2

A set M is said to be a (smooth) manifold of dimension n if it admits a differentiable structure of dimension n and is a Hausdorff topological space with countable basis.

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