PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-31

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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.

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A smooth map of locally maximal rank has locally constant rank

Statement

Let $F : M \to N$ be smooth and let $p \in M$ . Suppose there is a neighbourhood $U$ of $p$ such that ${rank}_{x} F \leq {rank}_{p} F$ for every $x \in U$ . Then $F$ has constant rank on some neighbourhood of $p$ .

Facts & Assumptions

Given: A smooth map $F : M \to N$ , a point $p \in M$ , and a neighbourhood $U$ on which the rank never exceeds ${rank}_{p} F$ .

[F1]

${rank}_{x} F$ is the rank of the differential at $x$ (The rank of a smooth map at a point).

[L1]

For Euclidean smooth maps, the set where the differential has rank at least a fixed value is open (Differential rank is lower semicontinuous).

[L2]

Charts identify neighbourhoods in manifolds with open Euclidean sets (Chart maps are diffeomorphisms onto Euclidean open sets).

Proof

technique · direct

1.1

Choose charts around $p$ and $F (p)$ as in [L2], and let $f$ be the coordinate representative of $F$ . By [F1], $D f (φ (p))$ has rank $r : = {rank}_{p} F$ , and the hypothesis says nearby ranks are at most $r$ .

F1L2given

2.1

By [L1], the locus where $D f$ has rank at least $r$ is open. Since $φ (p)$ lies in that locus and nearby ranks are never above $r$ , there is a smaller Euclidean neighbourhood on which the rank is both at least $r$ and at most $r$ , hence exactly $r$ .

step 1.1L1

3.1

Pulling that neighbourhood back through the source chart, $F$ has constant rank $r$ on a neighbourhood of $p$ .

step 2.1L2∎

Depends on

Used by

Rank at one point need not determine nearby rank False statement

Dependency tree · two levels

17 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, Maps of Constant Rank (standard reference, not scraped)