DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

The rank of a smooth map at a point

Definition

Let $F : M \to N$ be a smooth map and let $p \in M$ . The rank of $F$ at $p$ is

${rank}_{p} F : = rank (d F_{p}),$

the rank of the linear map $d F_{p} : T_{p} M \to T_{F (p)} N$ (The differential of a smooth map).

This is chart-independent because the differential changes under coordinates by composition with linear isomorphisms, so its rank is unchanged.

Depends on

The differential of a smooth map

Used by

Immersions, submersions, and constant-rank maps Definition
A smooth map of locally maximal rank has locally constant rank Proposition
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)