TheoremStatement: 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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The Euclidean constant-rank normal form

Statement

Let $k \geq 1$ , let $f : U \subseteq R^{m} \to R^{n}$ be $C^{k}$ , and suppose $f$ has constant rank $r$ on a neighbourhood of $a$ . There are local $C^{k}$ coordinate diffeomorphisms $α$ at $a$ and $β$ at $f (a)$ , both sending the distinguished point to $0$ , such that $β \circ f \circ α^{- 1} (u, v) = (u, 0)$ for $(u, v) \in R^{r} \times R^{m - r}$ near $0$ . The final zero lies in $R^{n - r}$ ; every zero-dimensional block is omitted.

Facts & Assumptions

Given: The stated $C^{k}$ map, point, and constant rank $r$ .

[L1]

Source rank coordinates make $f \circ Φ^{- 1}$ equal to $(u, h (u))$ after shrinking, and a differentiable map with zero derivative on a connected open set is constant (A nonzero rank minor supplies the source coordinates for the constant-rank theorem, In source rank coordinates, the remaining components depend only on the rank coordinates, A differentiable map on a connected open Euclidean set has zero derivative exactly when it is constant).

[L2]

Finite sums, differences, componentwise maps, and composites of $C^{k}$ Euclidean maps are $C^{k}$ ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition).

Proof

technique · direct

1.1

Translate the source and target distinguished points to $0$ and apply [L1], obtaining $g (u, v) = (u, h (u))$ on a product neighbourhood.

givenL1

2.1

Define the target shear $β (z, w) = (z, w - h (z))$ . It is a $C^{k}$ diffeomorphism with explicit inverse $(z, w) \mapsto (z, w + h (z))$ by [L2].

step 1.1L2construct

3.1

The composite satisfies $β (g (u, v)) = (u, 0)$ . When $r = 0$ , [L1] makes $f$ locally constant before the target translation; when $r = m$ or $r = n$ , the empty blocks make the same displayed formula literal.

step 1.1step 2.1L1

4.1

Taking $α$ to be the translated source coordinate map gives the asserted local normal form.

step 3.1∎

Depends on

Used by

Dependency tree · two levels

24 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, Theorem 7.13 (standard reference, not scraped)
L. W. Tu, An Introduction to Manifolds, Theorem 11.1 (standard reference, not scraped)