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Open subsets of Euclidean space have the standard smooth structure

Statement

Let $n \geq 1$ and let $W \subseteq R^{n}$ be open. Then $W$ is a smooth $n$ -manifold: the one-chart atlas ${(W, {i d}_{W})}$ is a smooth atlas, and the smooth structure it generates is the one induced on the open subset $W$ of $R^{n}$ . A chart $(U, φ)$ on $W$ belongs to this structure exactly when both transition maps between $φ$ and the identity are smooth, that is, exactly when $φ : U \to φ (U)$ and its inverse are smooth as maps between Euclidean open sets.

Facts & Assumptions

Given: An integer $n \geq 1$ and an open subset $W \subseteq R^{n}$ .

[F1]

$R^{n}$ is Hausdorff, being metrizable, and its rational open boxes form a countable basis, so it is second countable (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not, Second countability: an at most countable basis for the topology, $Q^{n}$ is a countable dense subset of $R^{n}$ , and rational open boxes form a countable basis).

[F2]

A chart is a homeomorphism from an open set of the manifold onto an open subset of $R^{n}$ (Manifold charts, coordinate domains, and coordinate functions).

[F3]

Two charts are smoothly compatible when their domains are disjoint or both transition maps are smooth (Smoothly compatible charts and the smoothness of Euclidean transition maps).

[F4]

A smooth atlas is a family of pairwise smoothly compatible charts whose domains cover the manifold (Smooth atlases).

[L1]

If $(M, S)$ is a smooth $n$ -manifold and $U \subseteq M$ is open with the subspace topology, then $U$ is a topological $n$ -manifold, and the restricted charts $(V \cap U, φ ∣_{V \cap U})$ form a smooth atlas whose generated maximal atlas is independent of the presenting atlas (An open subset of a smooth manifold has a canonical restricted smooth structure).

[A1]

The identity map ${i d}_{W}$ on an open subset of $R^{n}$ is smooth: each coordinate function has every iterated coordinate derivative equal to a constant function, hence existing and continuous.

Proof

technique · direct

1.1

The one-chart family ${(R^{n}, {i d}_{R^{n}})}$ is a smooth atlas on $R^{n}$ : [F1] gives the topological-manifold hypotheses, [F2] makes the identity a chart, and [A1] makes the identity transition smooth, so [F3] and [F4] apply.

givenF1F2F3F4A1

2.1

Since $W \subseteq R^{n}$ is open, [L1] applied to the smooth manifold $R^{n}$ from step 1.1 gives the restricted smooth structure on $W$ .

L1step 1.1

3.1

If a chart $(U, φ)$ on $W$ belongs to that restricted structure, then it is smoothly compatible with the identity chart $(W, {i d}_{W})$ ; hence the two transition maps are exactly $φ$ and $φ^{- 1}$ , and [F3] makes both smooth as maps between Euclidean open sets.

F3step 2.1

4.1

Conversely, if $φ : U \to φ (U)$ and $φ^{- 1}$ are smooth as maps between Euclidean open sets, then $(U, φ)$ is smoothly compatible with $(W, {i d}_{W})$ by [F3], so it belongs to the maximal atlas generated by the identity chart on $W$ . Together with steps 2.1 and 3.1 this proves the stated characterization and identifies it with the restricted smooth structure.

F3L1step 2.1step 3.1∎

Depends on

Used by

Euclidean spaces and Euclidean open subsets as smooth manifolds Example
Chart maps are diffeomorphisms onto Euclidean open sets Proposition

Dependency tree · two levels

29 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Rob van der Vorst, Introduction to differentiable manifolds, §1, Example 1.5 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, §2.3 (standard reference, not scraped)