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 into an embedded submanifold is an initial property

Statement

Let $S \subseteq M$ be an embedded submanifold with inclusion $i : S ↪ M$ , and let $G : N \to S$ be a map from a smooth manifold $N$ . Then $G$ is smooth if and only if $i \circ G : N \to M$ is smooth.

Facts & Assumptions

Given: An embedded submanifold $S \subseteq M$ , its inclusion $i$ , and a map $G : N \to S$ .

[F1]

Embedded submanifolds are locally cut out by slice charts (Embedded submanifolds and slice charts).

[L1]

The restricted slice charts define the smooth structure on $S$ (Slice-chart restrictions form a smooth atlas).

[L2]

Ambient charts are diffeomorphisms onto open Euclidean sets (Chart maps are diffeomorphisms onto Euclidean open sets).

Proof

technique · direct

1.1

Assume first that $G$ is smooth. Choose a chart $θ$ on $N$ and a slice chart $φ$ on $M$ around $G (x) \in S$ . In the corresponding restricted chart on $S$ from [L1], the representative of $G$ has values in $R^{k}$ , and the representative of $i \circ G$ is obtained by appending $m - k$ zero coordinates. Hence $i \circ G$ is smooth.

F1L1L2given

1.2

Conversely, assume $i \circ G$ is smooth. In a slice chart on $M$ , the image of $S$ is $R^{k} \times {0}$ by [F1]. Therefore the representative of $i \circ G$ has last $m - k$ coordinates identically zero, and its first $k$ coordinates are exactly the representative of $G$ in the restricted slice chart from [L1]. Those first $k$ coordinates are smooth, so $G$ is smooth.

F1L1L2given

2.1

Steps 1.1 and 1.2 prove both directions.

step 1.1step 1.2∎

Depends on

Used by

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

Dependency tree · two levels

10 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)