DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Regular and critical points and values

Definition

Let $F : M \to N$ be a smooth map.

A point $p \in M$ is a regular point of $F$ when $F$ is a submersion at $p$ , and a critical point otherwise (Immersions, submersions, and constant-rank maps).
A point $q \in N$ is a regular value of $F$ when every point of $F^{- 1} (q)$ is a regular point. This includes the empty-fibre case.
A point $q \in N$ that is not a regular value is a critical value.

Depends on

Immersions, submersions, and constant-rank maps

Used by

A value outside the image can still be regular Example
A regular value need not belong to the image False statement
The tangent space of a regular level set is the kernel Proposition
A regular level set is an embedded submanifold Theorem

Dependency tree · two levels

2 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, Ch. 5 (standard reference, not scraped)
Will J. Merry, Differential Geometry, Definition 6.9 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, Definition 12 (standard reference, not scraped)