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An open subset of a smooth manifold has a canonical restricted smooth structure

Statement

Let $(M, S)$ be a smooth $n$ -manifold and let $U \subseteq M$ be open, carrying the subspace topology. Then $U$ is a topological $n$ -manifold. For every smooth atlas $A$ with $[A] = S$ , the family of restricted charts

$A ∣_{U} : = {(V \cap U, φ ∣_{V \cap U}) : (V, φ) \in A} .$

is a smooth atlas on $U$ , and the maximal atlas it generates is independent of the presenting atlas $A$ : it depends only on the structure $S$ . This maximal atlas is the restricted smooth structure of $U$ .

Facts & Assumptions

Given: A smooth $n$ -manifold $(M, S)$ , an open subset $U \subseteq M$ , and a smooth atlas $A$ with $[A] = S$ .

[F1]

The open sets of the subspace $U$ are exactly the traces $O \cap U$ of open sets of $M$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

[F2]

A chart $(V, φ)$ has $V$ open in $M$ and $φ : V \to φ (V)$ a homeomorphism onto an open subset of $R^{n}$ (Manifold charts, coordinate domains, and coordinate functions).

[F3]

Hausdorffness and second countability are hereditary ( $T_{0}$ , $T_{1}$ , and Hausdorffness are hereditary, Second countability is hereditary).

[F4]

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).

[F5]

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

[L1]

Two smooth atlases generate the same maximal atlas exactly when their union is a smooth atlas (Each smooth atlas is contained in a unique maximal smooth atlas).

[A1]

If $V$ is open in $M$ and $U$ is open in $M$ , then $V \cap U$ is open in the subspace $U$ , its image $φ (V \cap U)$ is open in $R^{n}$ , and the restriction $φ ∣_{V \cap U} : V \cap U \to φ (V \cap U)$ is a homeomorphism.

Proof

technique · direct

1.1

$U$ is Hausdorff and second countable by [F3]. For $p \in U$ choose a chart $(V, φ)$ of a smooth atlas of $M$ with $p \in V$ , which exists because atlases cover $M$ by [F5]; by [F1] and [A1] the set $V \cap U$ is open in $U$ , its image $φ (V \cap U)$ is open in $R^{n}$ , and $φ ∣_{V \cap U} : V \cap U \to φ (V \cap U)$ is a homeomorphism. Hence $U$ is locally Euclidean of dimension $n$ and is a topological $n$ -manifold.

givenF3F5F1A1

1.2

Each $(V \cap U, φ ∣_{V \cap U})$ is a chart on $U$ by [A1] and [F2], and [given, F2, F5, A1] the domains $V \cap U$ cover $U$ because the domains $V$ of $A$ cover $M$ by [F5].

givenF2F5A1

2.1

For $(V, φ), (W, ψ) \in A$ , the transition of the two [given, F4, F5, step 1.2] restricted charts on $φ (V \cap W \cap U)$ is the restriction of $ψ \circ φ^{- 1}$ , which is smooth on $φ (V \cap W)$ by [F4] whenever the overlap is nonempty; restricting to the open subset $φ (V \cap W \cap U)$ keeps every iterated coordinate derivative existing and continuous, so the restricted transition is smooth. The disjoint-domain clause of [F4] covers the case $V \cap W \cap U = \emptyset$ . Hence the members of $A ∣_{U}$ are pairwise smoothly compatible, and [F5] makes $A ∣_{U}$ a smooth atlas.

givenF4F5step 1.2

3.1

If $B$ is another smooth atlas with [given, F4, L1, step 2.1] $[B] = S = [A]$ , then $A \cup B$ is a smooth atlas by [L1]. Its restrictions give $A ∣_{U} \cup B ∣_{U} = (A \cup B) ∣_{U}$ , and the cross-pair transitions are restrictions of smooth transitions exactly as in step 2.1, so $A ∣_{U} \cup B ∣_{U}$ is a smooth atlas on $U$ ; applying [L1] to the two restricted atlases yields $[A ∣_{U}] = [B ∣_{U}]$ . The restricted structure therefore depends only on $S$ .

givenF4L1step 2.1∎

Depends on

Used by

Diffeomorphisms and local diffeomorphisms of manifolds Definition
Chart maps are diffeomorphisms onto Euclidean open sets Proposition
Open subsets of Euclidean space have the standard smooth structure Proposition
Restrictions, corestrictions, and products of smooth maps are smooth Proposition
Smoothness is local on the source Proposition

Dependency tree · two levels

22 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, §2 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, §2.2 (standard reference, not scraped)