CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-24

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A regular level set is locally a $C^{k}$ graph of dimension $m - n$

Statement

Let $f : U \subseteq R^{m} \to R^{n}$ be $C^{k}$ , $k \geq 1$ , and let $c$ be a regular value. Near each point, a regular level set is a $C^{k}$ graph over $\ker D f (a)$ of dimension $m - n$ .

More precisely, for $a \in f^{- 1} (c)$ put $K = \ker D f (a)$ and $E = K^{⊥}$ , so $R^{m} = K \oplus E$ . There are neighbourhoods $P \subseteq K$ of $0$ and $Q \subseteq E$ of $0$ and a $C^{k}$ map $g : P \to Q$ with $g (0) = 0$ and $D g (0) = 0$ such that, near $a$ , $f^{- 1} (c) = {a + u + g (u) : u \in P} .$ The empty fibre satisfies the regular-value convention vacuously, and when $m = n$ the local graph has zero-dimensional domain and is the isolated point $a$ .

Facts & Assumptions

Given: The stated map, regular value $c$ , and a point $a \in f^{- 1} (c)$ .

[L1]

Surjectivity of $D f (a)$ persists nearby, and a constant-rank level is locally a coordinate slice (Regular and critical points, regular and critical values, and level sets, Differential rank is lower semicontinuous, A constant-rank level set is locally a coordinate slice).

[L2]

If $K$ is a subspace of the finite-dimensional Euclidean inner-product space $R^{m}$ , then $R^{m} = K \oplus K^{⊥}$ ; rank-nullity gives $\dim K = m - n$ , and the inverse function theorem turns an invertible derivative into a local $C^{1}$ diffeomorphism whose inverse is $C^{k}$ when the original map is $C^{k}$ (For a subspace $W$ of a finite-dimensional inner product space, $V = W \oplus W^{⊥}$ , Rank-nullity: $\dim_{F} V = nullity T + rank T$ , The Euclidean inverse function theorem, A local inverse of a $C^{k}$ regular map is $C^{k}$ ).

Proof

technique · direct

1.1

By [L1], $f$ has constant rank $n$ near $a$ , and its fibre is a $C^{k}$ coordinate slice. By [L2], put $E = K^{⊥}$ , so $R^{m} = K \oplus E$ and $\dim K = m - n$ .

givenL1L2construct

2.1

The projection of that slice to $K$ along $E$ has derivative equal to the identity at $a$ : its tangent there is $K$ , because differentiating the normal-form slice and undoing the source coordinates gives $\ker D f (a)$ . By [L2], this projection is a local $C^{k}$ diffeomorphism.

step 1.1L2algebra

3.1

Inverting the projection writes the slice uniquely as $a + u + g (u)$ . Its derivative at $0$ takes values both in $E$ and in the tangent $K$ , so $D g (0) = 0$ ; the zero-dimensional case is the same statement with $P = {0}$ .

step 2.1algebra

4.1

This gives the asserted graph and dimension at every point of a nonempty regular fibre, while the empty-fibre case is vacuous.

step 3.1∎

Depends on

Used by

The cone x²+y²=z² has a rank drop at its apex Counterexample
The cusp y²=x³ has a rank drop at the origin Counterexample
A Euclidean sphere is a regular level set with tangent hyperplanes Example
A positive-definite quadratic ellipsoid is a regular level set Example
The graph of a Cᵏ Euclidean map is a regular level set Example
The one-sheeted hyperboloid is a regular surface of revolution Example
The orthogonal group is a regular level set of dimension n(n-1)/2 Example
FALSE: every level set of a smooth map is locally a graph False statement
Regular level surfaces have local regular parametrizations with the same tangent plane Theorem
Tangent vectors to a regular level set are exactly its curve velocities Theorem

Dependency tree · two levels

36 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, Regular Level Set Theorem (standard reference, not scraped)
L. W. Tu, An Introduction to Manifolds, Section 11.2 (standard reference, not scraped)

A regular level set is locally a CkCk graph of dimension m−nm−n