DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Integral manifolds of a distribution

Definition

Let $D$ be a smooth distribution on $M$ . A connected injectively immersed submanifold $i : N \to M$ is an integral manifold of $D$ when

$d i_{q} (T_{q} N) = D_{i (q)}$

for every $q \in N$ .

This definition uses the intrinsic manifold structure on $N$ ; the image $i (N) \subseteq M$ need not carry the subspace topology.

Depends on

Used by

Integrable distributions Definition
The coordinate-plane distribution and its affine leaves Example
The subspace topology on a leaf is always its manifold topology False statement
Integral manifolds are locally contained in plaques Lemma
Integrable distributions are involutive Proposition
Integral manifolds have the distribution dimension Proposition
Local diffeomorphisms carry distributions and integral manifolds Proposition
Existence and uniqueness of maximal connected integral manifolds Theorem

Dependency tree · two levels

8 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

Will J. Merry, Differential Geometry (standard reference, not scraped)
Keith Conrad, Local and global Frobenius theorems (standard reference, not scraped)