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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Smoothness of a map on an embedded submanifold is local in the ambient space

Statement

Let $S \subseteq M$ be an embedded submanifold and let $f : S \to N$ be a map to a smooth manifold $N$ . Then $f$ is smooth if and only if every point $p \in S$ has an open neighbourhood $U \subseteq M$ and a smooth map $\tilde{f} : U \to N$ such that $\tilde{f} ∣_{U \cap S} = f ∣_{U \cap S}$ .

Facts & Assumptions

Given: An embedded submanifold $S \subseteq M$ and a map $f : S \to N$ .

[F1]

Embedded submanifolds have slice charts (Embedded submanifolds and slice charts).

[L1]

A map into an embedded submanifold is smooth exactly when its ambient composite is smooth (Smoothness into an embedded submanifold is an initial property).

[L2]

Smooth maps are continuous (Smooth maps are continuous).

Proof

technique · direct

1.1

Assume $f$ is smooth. Fix $p \in S$ and choose a slice chart $φ : U \to Ω \subseteq R^{m}$ with $φ (S \cap U) = Ω \cap (R^{k} \times {0})$ . In these coordinates, the representative of $f ∣_{S \cap U}$ depends only on the first $k$ variables. Extend it to all of $Ω$ by ignoring the last $m - k$ coordinates. Transporting this extension back gives a smooth $\tilde{f} : U \to N$ with $\tilde{f} ∣_{U \cap S} = f ∣_{U \cap S}$ .

F1L2givenconstruct

2.1

Conversely, suppose every point has such an ambient extension. On each $U \cap S$ , the restriction of $\tilde{f}$ to the embedded submanifold $U \cap S$ is smooth by [L1], so $f$ is smooth near every point of $S$ . Smooth maps being local on the source, $f$ is smooth on all of $S$ .

L1step 1.1given

3.1

Therefore the ambient-extension criterion is equivalent to smoothness of $f$ .

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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, Restricting Maps to Submanifolds (standard reference, not scraped)