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 submersion is locally a coordinate projection

Statement

Let $k \geq 1$ and let $f : U \subseteq R^{m} \to R^{n}$ be $C^{k}$ . Near a submersion point there are $C^{k}$ coordinates in which the map is $(u, v) \mapsto u$ . If $m = n$ , it is a local $C^{k}$ diffeomorphism.

Facts & Assumptions

Given: A submersion point $a$ of $f$ .

[L1]

At a submersion point $D f (a)$ is surjective and has rank $n$ (Submersions and immersions between Euclidean open sets); the rank-at-least- $n$ locus is open (Differential rank is lower semicontinuous).

[L2]

A constant-rank- $n$ map has local normal form $(u, v) \mapsto (u, 0)$ with the target zero block in $R^{0}$ (The Euclidean constant-rank normal form).

Proof

technique · direct

1.1

By [L1], $D f$ has rank at least $n$ on a neighbourhood of $a$ ; it cannot have larger rank, so its rank is constantly $n$ there.

givenL1

2.1

Apply [L2]. Because $n - r = 0$ , its normal form is exactly the projection $(u, v) \mapsto u$ .

step 1.1L2

3.1

If $m = n$ , the $v$ block is also empty, so the normal form is the identity and $f$ is a local $C^{k}$ diffeomorphism.

step 2.1∎

Depends on

Used by

Euclidean submersions are open maps Corollary

Dependency tree · two levels

15 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, Submersion Theorem (standard reference, not scraped)
L. W. Tu, An Introduction to Manifolds, Section 11.2 (standard reference, not scraped)