PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-29

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Chart maps are diffeomorphisms onto Euclidean open sets

Statement

Let $(M, S)$ be a smooth manifold and let $(U, φ)$ be a smooth chart. Give $U$ the restricted smooth structure of the open submanifold and $φ (U)$ its standard smooth structure. Then the corestriction $φ : U \to φ (U)$ is a diffeomorphism.

Facts & Assumptions

Given: A smooth manifold $(M, S)$ and a smooth chart $(U, φ)$ .

[F1]

A smooth chart is a chart of the maximal atlas, hence a homeomorphism of the open set $U$ onto the open subset $φ (U) \subseteq R^{n}$ (Smooth manifolds and their smooth charts, Manifold charts, coordinate domains, and coordinate functions).

[F2]

The open subset $φ (U)$ carries the standard smooth structure generated by the identity chart (Open subsets of Euclidean space have the standard smooth structure), and the open subset $U$ carries the restricted structure of $M$ (An open subset of a smooth manifold has a canonical restricted smooth structure).

[F3]

A map between smooth manifolds is smooth when its representative with respect to suitable smooth charts is smooth ( $C^{r}$ and smooth maps between smooth manifolds).

[F4]

A diffeomorphism is a bijective smooth map with smooth inverse (Diffeomorphisms and local diffeomorphisms of manifolds).

[A1]

The identity map of an open subset of $R^{n}$ is smooth, each iterated coordinate derivative being a constant function.

Proof

technique · direct

1.1givenF1

By [F1] the corestriction $φ : U \to φ (U)$ is a bijective [given, F1] homeomorphism.

1.2givenF2F3A1

$φ$ is smooth: with the smooth chart $(U, φ)$ of $U$ from [F2] and the identity chart of $φ (U)$ from [F2], the representative is $i d \circ φ \circ φ^{- 1} = {i d}_{φ (U)}$ , smooth by [A1], so [F3] applies.

1.3givenF2F3A1

$φ^{- 1}$ is smooth: with the identity chart of $φ (U)$ and the [given, F2, F3, A1] chart $(U, φ)$ of $U$ , the representative is $φ \circ φ^{- 1} \circ i d = {i d}_{φ (U)}$ , again smooth by [A1], so [F3] applies.

2.1F4step 1.1step 1.2step 1.3∎

Steps 1.1-1.3 give a bijective smooth map with smooth inverse, which is [F4, step 1.1, step 1.2, step 1.3] exactly a diffeomorphism by [F4].

Depends on

Used by

The immersion and submersion loci are open Corollary
Polar coordinates do not give a chart at the origin Counterexample
Coordinate derivations at a point Definition
A chart bump at a point with prescribed support Lemma
Slice-chart restrictions form a smooth atlas Lemma
The orientation double cover is canonically oriented and preserves closedness Lemma
A smooth function with zero differential is constant on each connected component Proposition
A smooth map of locally maximal rank has locally constant rank Proposition
A vector bundle projection is a surjective submersion Proposition
Smoothness into an embedded submanifold is an initial property Proposition
The diagonal is an embedded submanifold Proposition
The tangent space of a regular level set is the kernel Proposition
The total space of a rank-r bundle has dimension dim M + r Proposition
The zero section is a smooth embedding Proposition
Embedded submanifolds admit local defining submersions Theorem
Local existence, uniqueness, and smooth dependence for manifold integral curves Theorem
The constant-rank theorem for manifolds Theorem
The smooth inverse function theorem on manifolds Theorem
Time-dependent vector fields have local smooth evolution operators Theorem

Dependency tree · two levels

20 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, §2.4 (standard reference, not scraped)