Smooth Manifold

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Smooth Manifold

Definition (Smooth Manifold)

A Smooth or $C^{\infty}$ manifold is a Topological Manifold $M$ together with a maximal atlas $\mathfrak{M}$ , i.e. it is the pair $(M,\mathfrak{M})$ .

Remark

A manifold is said to have dimension $n$ if all of its connected components have dimension $n$ .

Remark

A 1-dimensional manifold is also called a curve, a 2-dimensional manifold a surface, and an n-dimensional manifold an n-manifold.

In practice, to check that a Topological Manifold $M$ is a , it is not necessary to exhibit a maximal atlas. The existence of any Atlas on $M$ will do, because of the following proposition.

Proposition (5.10)

Any Atlas $\mathfrak{U} = \{(\mathcal{U}_{\alpha}, \phi_{\alpha} )\}$ on a Locally Euclidean Space is contained in a unique maximal atlas.

Remark

In summary, to show that a Topological Space $M$ is a $C^{\infty}$ manifold, it suffices to check that:

$M$ is Hausdorff and Second-countable,
$M$ has a $C^{\infty}$ Atlas (not necessarily maximal).

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