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

1.1F1given

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

2.1step 1.1L1algebra

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.

3.1F2step 2.1∎

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 $n$ .

Depends on

Used by

Sphere self-maps are homotopic exactly when their degrees agree Corollary
The kernel distribution of a constant-rank submersion is integrable Corollary
Zero-level symplectic reduction and the dimension formula Corollary
Antipodal points on a round sphere have many minimizing geodesics Counterexample
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
A great sphere is totally geodesic Example
Cartan hadamard for hyperbolic space Example
Coordinate circles give the alternating intersection matrix of a torus Example
Cut locus of a point on a round sphere Example
General and special linear Lie groups Example
Great circles as round-sphere geodesics Example
Hyperbolic space as so zero n one mod so n Example
Killing jacobi fields from rotations Example
Latitude and meridian intersections on the torus Example
Self-intersection of the zero section in an oriented plane bundle Example
SU(2) to SO(3) as a covering homomorphism Example
The catenoid has zero mean curvature but is not totally geodesic Example
The diagonal in the two-sphere has self-intersection two Example
The normal line of an oriented hypersurface is trivial Example
The round sphere has positive constant sectional curvature Example
The special linear group is a codimension-one embedded submanifold Example
Two projective lines have one mod 2 intersection Example
An injective immersion need not be an embedding False statement
Christoffel symbols vanishing at one point implies curvature vanishes there False statement
Naturality alone gives Mayer–Vietoris connector compatibility False statement
Normal coordinates make the metric Euclidean throughout the chart False statement
The intrinsic topology of an immersed submanifold need not be the subspace topology False statement
A compact C¹ leaf has finitely generated fundamental group Lemma
A regular level identifies unparametrized trajectories Lemma
Every integer is realized by a map to the sphere Lemma
A vector bundle section with surjective vertical differential at every zero has a submanifold zero set Proposition
At a conjugate endpoint the index form is degenerate Proposition
Hessian of distance in terms of radial jacobi fields Proposition
Regular common level sets are Lagrangian submanifolds Proposition
Round sphere model geometry Proposition
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
Marsden--Weinstein--Meyer symplectic reduction Theorem

…and 4 more results.

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)