CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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A Euclidean immersion is locally the canonical inclusion and is locally an embedding

Statement

Let $k \geq 1$ and let $f : U \subseteq R^{m} \to R^{n}$ be $C^{k}$ . Near an immersion point there are $C^{k}$ coordinates in which the map is the canonical inclusion $u \mapsto (u, 0)$ . After restricting its domain, $f$ is an embedding onto its local image. If $m = n$ , it is a local $C^{k}$ diffeomorphism.

Facts & Assumptions

Given: An immersion point $a$ of $f$ .

[L1]

At an immersion point $D f (a)$ is injective and has rank $m$ , and the rank-at-least- $m$ locus is open (Submersions and immersions between Euclidean open sets, Differential rank is lower semicontinuous).

[L2]

An embedding is an injective map whose corestriction is a homeomorphism onto its image (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological); the constant-rank normal form at rank $m$ is $u \mapsto (u, 0)$ (The Euclidean constant-rank normal form).

Proof

technique · direct

1.1

By [L1], the derivative has constant rank $m$ near $a$ .

givenL1

2.1

By [L2], coordinate diffeomorphisms turn the restriction of $f$ into $i (u) = (u, 0)$ . This map is injective and its inverse on $i [U]$ is the continuous projection onto the first $m$ coordinates.

step 1.1L2

3.1

Conjugating by the coordinate diffeomorphisms shows that the restricted $f$ is an embedding. If $m = n$ , the zero block is empty and the normal form is a local diffeomorphism.

step 2.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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

J. M. Lee, Introduction to Smooth Manifolds, Immersion Theorem (standard reference, not scraped)
L. W. Tu, An Introduction to Manifolds, Section 11.2 (standard reference, not scraped)