DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-24

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.

Submersions and immersions between Euclidean open sets

Definition

Let $m, n \geq 1$ , let $U \subseteq R^{m}$ and $V \subseteq R^{n}$ be open, and let $f : U \to V$ be $C^{1}$ .

$f$ is an immersion at $a \in U$ when $D f (a) : R^{m} \to R^{n}$ is injective (The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial), equivalently when ${rank}_{a} f = m$ (The rank of a derivative and constant-rank Euclidean maps). Thus an immersion point can exist only when $m \leq n$ .
$f$ is a submersion at $a \in U$ when $D f (a)$ is surjective (Injection, surjection, bijection), equivalently when ${rank}_{a} f = n$ . Thus a submersion point can exist only when $n \leq m$ .

The map is an immersion or submersion when the corresponding condition holds at every point of $U$ .

Depends on

Used by

Dependency tree · two levels

12 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

J. M. Lee, Introduction to Smooth Manifolds, Theorems 8.8-8.11 (standard reference, not scraped)
L. W. Tu, An Introduction to Manifolds, Sections 11.1-11.2 (standard reference, not scraped)