PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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The inclusion of an embedded submanifold is a smooth embedding

Statement

If $S \subseteq M$ is an embedded submanifold equipped with the smooth structure from its restricted slice charts, then the inclusion $i : S ↪ M$ is a smooth embedding.

Facts & Assumptions

Given: An embedded submanifold $S \subseteq M$ with its slice-chart smooth structure.

[F1]

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

[L1]

The restricted slice charts form a smooth atlas on $S$ with the subspace topology (Slice-chart restrictions form a smooth atlas).

Proof

technique · direct

1.1L1given

The inclusion $i : S ↪ M$ is injective and, by [L1], a homeomorphism of $S$ onto its image because the domain topology of $S$ is exactly the subspace topology.

1.2L1

In a restricted slice chart on $S$ and the ambient slice chart on $M$ , the coordinate representative of $i$ is $u \mapsto (u, 0)$ . Its differential is the coordinate inclusion, hence injective, so $i$ is an immersion.

2.1F1step 1.1step 1.2∎

Steps 1.1 and 1.2 verify the clauses in [F1], so $i$ is a smooth embedding.

Depends on

Used by

Closed embedded submanifolds of complete Riemannian manifolds are complete Corollary
Normal and conormal bundles of an embedded submanifold Definition
Transverse embedded submanifolds Definition
Assuming countable choice, an ambient metric identifies the two normal bundles Proposition
The smooth structure of an embedded submanifold is unique Proposition
The tubular neighbourhood theorem in a smooth ambient manifold Theorem

Dependency tree · two levels

7 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, Embedded Submanifolds (standard reference, not scraped)