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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Local normal form for submersions

Statement

Let $F : M^{m} \to N^{n}$ be a smooth submersion at $p \in M$ . Then there are charts near $p$ and $F (p)$ in which the coordinate representative of $F$ is

$(u, v) ⟼ u$

near the distinguished point.

Facts & Assumptions

Given: A smooth map $F : M^{m} \to N^{n}$ that is a submersion at $p$ .

[L1]

The submersion locus is open (The immersion and submersion loci are open).

[L2]

A constant-rank- $n$ map has local normal form $(u, v) \mapsto (u, 0)$ in adapted coordinates (The constant-rank theorem for manifolds).

Proof

technique · direct

1.1L1given

Since $F$ is a submersion at $p$ , the linear map $d F_{p} : T_{p} M \to T_{F (p)} N$ is surjective. Its target has dimension $n$ , so its rank is $n$ . Thus [L1] gives a neighbourhood on which $F$ has constant rank $n$ .

2.1step 1.1L2

Apply [L2] with $r = n$ . The target normal factor has dimension $0$ , so the normal form reads $(u, v) \mapsto u$ .

3.1step 2.1∎

This is the claimed submersion normal form.

Depends on

Used by

Every submersion is an open map Corollary
Maximal compact subgroups exist and are conjugate in a connected finite center semisimple Lie group Corollary
Cotangent reduction for a principal bundle at zero Example
Disk bundles over compact bases are compact manifolds with boundary Lemma
Open manifolds admit exhaustions with no caps Lemma
Regular exhaustion and Dirichlet solutions on relatively compact surface domains Lemma
Regular sublevels are compact manifolds with boundary Lemma
Cartan decomposition gives the invariant metric and curvature of G mod K Proposition
Invariant Hamiltonians descend to reduced Hamiltonians Proposition
Reduction commutes with products Proposition
The pullback foliation under a transverse map Proposition
The tangent space of a regular level set is the kernel Proposition
A regular level set is an embedded submanifold Theorem
Cartan decomposition identifies p with the noncompact symmetric space Theorem
Conjugacy of Cartan subalgebras Theorem
Liouville–Arnold action–angle theorem Theorem
Reduction in stages for free proper regular actions Theorem

Dependency tree · two levels

10 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, Submersions (standard reference, not scraped)
Will J. Merry, Differential Geometry, Proposition 6.13 (standard reference, not scraped)