TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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A regular level set is an embedded submanifold

Statement

Let $F : M^{m} \to N^{n}$ be smooth, let $q \in N$ be a regular value, and assume $F^{- 1} (q)$ is nonempty. Then $F^{- 1} (q)$ is an embedded submanifold of codimension $n$ . Equivalently, it has dimension $m - n$ .

Facts & Assumptions

Given: A smooth map $F : M^{m} \to N^{n}$ and a regular value $q \in N$ with nonempty fibre.

[F1]

A regular value is one whose fibre points are all submersion points; the fibre may be empty (Regular and critical points and values).

[F2]

Codimension means ambient dimension minus submanifold dimension, and embedded submanifolds are defined by slice charts (Codimension and hypersurfaces, Embedded submanifolds and slice charts).

[L1]

Near any submersion point, suitable coordinates put $F$ into the form $(u, v) \mapsto u$ (Local normal form for submersions).

Proof

technique · direct

1.1

Let $p \in F^{- 1} (q)$ be arbitrary. By [F1], $F$ is a submersion at $p$ .

F1given

2.1

Apply [L1] at $p$ . In suitable charts around $p$ and $q$ , the map becomes $(u, v) \mapsto u$ on $R^{n} \times R^{m - n}$ . After centering $q$ , the fibre is ${0} \times R^{m - n}$ . Permuting the two source-coordinate blocks sends it to the standard slice $R^{m - n} \times {0}$ . Thus $F^{- 1} (q)$ is locally an embedded submanifold.

step 1.1L1algebra

3.1

Since every point of the fibre has such a slice neighbourhood, $F^{- 1} (q)$ is an embedded submanifold. Its local model has dimension $m - n$ , so by [F2] the codimension is $n$ .

F2step 2.1∎

Depends on

Used by

Countably many concentric circles give an injective immersion that is not an embedding Counterexample
A cylinder is the preimage of a circle under a projection Example
The special linear group is a codimension-one embedded submanifold Example
An injective immersion need not be an embedding False statement
The intrinsic topology of an immersed submanifold need not be the subspace topology False statement
The graph of a smooth map is an embedded submanifold Proposition
The tangent space of a regular level set is the kernel Proposition
Transverse intersections of coordinate slices have the expected local form Proposition
The preimage theorem for submanifolds under submersions Theorem

Dependency tree · two levels

9 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

John M. Lee, Introduction to Smooth Manifolds, Level Sets (standard reference, not scraped)
Will J. Merry, Differential Geometry, Theorem 6.10 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, Theorem 3.3 (standard reference, not scraped)