PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-31

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The image of a smooth embedding is an embedded submanifold

Statement

Let $F : M^{m} \to N^{n}$ be a smooth embedding. Then $F (M) \subseteq N$ is an embedded $m$ -submanifold, and the corestriction $F : M \to F (M)$ is a diffeomorphism.

Facts & Assumptions

Given: A smooth embedding $F : M^{m} \to N^{n}$ .

[F1]

A smooth embedding is an injective immersion and a homeomorphism onto its image with the subspace topology (Smooth embeddings).

[F2]

Embedded submanifolds are described by slice charts (Embedded submanifolds and slice charts).

[L1]

Near any point, an immersion has coordinates of the form $u \mapsto (u, 0)$ (Local normal form for immersions).

Proof

technique · direct

1.1F1L1given

Fix $p \in M$ . By [F1] the map $F$ is an immersion, so [L1] gives charts near $p$ and $F (p)$ in which $F$ becomes $u \mapsto (u, 0)$ .

2.1F1F2step 1.1

In those target coordinates, the image of a small neighbourhood of $p$ is exactly the coordinate slice $R^{m} \times {0}$ . Because [F1] also says that $F$ is a homeomorphism onto its image, we may shrink the neighbourhood in the target so that no other points of $M$ map into that same slice patch. Hence the image is locally a slice chart in the sense of [F2].

3.1F1step 2.1∎

Since every point of $F (M)$ has such a slice neighbourhood, $F (M)$ is an embedded $m$ -submanifold. The corestriction $F : M \to F (M)$ is already a homeomorphism by [F1], and in the local coordinates of step 1.1 it is the identity on the $R^{m}$ factor, so it is a diffeomorphism.

Depends on

Used by

Dependency tree · two levels

9 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

John M. Lee, Introduction to Smooth Manifolds, Embeddings (standard reference, not scraped)
Will J. Merry, Differential Geometry, Definition 6.1 (standard reference, not scraped)