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

Used by

The immersion and submersion loci are open Corollary
Immersed submanifolds Definition
Local defining maps for embedded submanifolds Definition
Regular and critical points and values Definition
Smooth embeddings Definition
Coordinate inclusions are immersions and coordinate projections are submersions Example
The constant-rank theorem for manifolds 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. 4 (standard reference, not scraped)
Will J. Merry, Differential Geometry, Definition 6.11 (standard reference, not scraped)