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.

Immersions, submersions, and constant-rank maps

Definition

Let $F : M^{m} \to N^{n}$ be a smooth map.

$F$ is an immersion at $p \in M$ when $d F_{p}$ is injective.
$F$ is a submersion at $p \in M$ when $d F_{p}$ is surjective.
$F$ has constant rank $r$ on $A \subseteq M$ when ${rank}_{p} F = r$ for every $p \in A$ (The rank of a smooth map at a point).

The map is an immersion or submersion without qualification when the corresponding pointwise condition holds at every point of $M$ .

Depends on

The rank of a smooth map at a point

Dependency tree · 2 levels

Sources