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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A constant-rank level set is locally a coordinate slice

Statement

Let $f : U \subseteq R^{m} \to R^{n}$ be $C^{k}$ , $k \geq 1$ , and have constant rank $r$ near $a$ . Put $c = f (a)$ . In the source coordinates of the constant-rank theorem, there is a neighbourhood $W$ of $0$ such that $α (f^{- 1} (c)) \cap W = W \cap ({0}^{r} \times R^{m - r}) .$ Thus a nonempty constant-rank level set is locally a coordinate slice of dimension $m - r$ . If a level set is empty, the pointwise assertion has no instance.

Facts & Assumptions

Given: The map $f$ , point $a$ , value $c = f (a)$ , and constant rank $r$ near $a$ .

[L1]

The level set over $c$ is $f^{- 1} (c) = {x : f (x) = c}$ (Regular and critical points, regular and critical values, and level sets).

[L2]

Local coordinates may be chosen so that $a$ and $c$ become $0$ and the map becomes $(u, v) \mapsto (u, 0)$ (The Euclidean constant-rank normal form).

Proof

technique · direct

1.1

Apply [L2] and restrict to its source coordinate neighbourhood $W$ .

givenL2

2.1

By [L1], a point in $W$ represents a point of $f^{- 1} (c)$ exactly when $(u, 0) = (0, 0)$ , which is exactly the condition $u = 0$ ; the $v \in R^{m - r}$ coordinates are free.

step 1.1L1algebra

3.1

Pulling this slice back by $α^{- 1}$ gives the stated local description. The formula also covers $r = 0$ and $r = m$ through the empty-block convention.

step 2.1∎

Depends on

Used by

A regular level set is locally a Cᵏ graph of dimension m-n Corollary

Dependency tree · two levels

8 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)