CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 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.

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Every immersion is locally an embedding

Statement

Let $F : M^{m} \to N^{n}$ be a smooth immersion and let $p \in M$ . Then some neighbourhood $U$ of $p$ is sent homeomorphically onto an embedded $m$ -dimensional submanifold of $N$ , and $F ∣_{U}$ is still an immersion.

Facts & Assumptions

Given: A smooth immersion $F : M^{m} \to N^{n}$ and a point $p \in M$ .

[L1]

Near $p$ , suitable coordinates identify $F$ with the coordinate inclusion $u \mapsto (u, 0)$ (Local normal form for immersions).

Proof

technique · direct

1.1

By [L1], after shrinking about $p$ and $F (p)$ there are charts in which $F$ becomes $u \mapsto (u, 0)$ .

L1given

2.1

The coordinate inclusion is injective, is a homeomorphism onto the slice $R^{m} \times {0}$ with the subspace topology, and that slice is an embedded submanifold of the ambient Euclidean space. Pulling this description back through the charts gives the stated neighbourhood $U$ .

step 1.1

3.1

The restricted map remains an immersion because it is locally identified with the coordinate inclusion.

step 1.1step 2.1∎

Depends on

Local normal form for immersions

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Will J. Merry, Differential Geometry, Proposition 6.3 (standard reference, not scraped)
John M. Lee, Introduction to Smooth Manifolds, Immersions (standard reference, not scraped)