CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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 immersion and submersion loci are open

Statement

Let $F : M^{m} \to N^{n}$ be a smooth map. The set of points where $F$ is an immersion is open in $M$ , and the set of points where $F$ is a submersion is open in $M$ .

Facts & Assumptions

Given: A smooth map $F : M^{m} \to N^{n}$ .

[F1]

$F$ is an immersion at $p$ exactly when ${rank}_{p} F = m$ , and it is a submersion at $p$ exactly when ${rank}_{p} F = n$ (Immersions, submersions, and constant-rank maps).

[L1]

For a smooth Euclidean map, the locus where the differential has rank at least a fixed integer is open (Differential rank is lower semicontinuous).

[L2]

Every manifold chart is a diffeomorphism onto an open Euclidean set (Chart maps are diffeomorphisms onto Euclidean open sets).

Proof

technique · direct

1.1F1L1L2given

Fix an immersion point $p$ . Choose charts around $p$ and $F (p)$ as in [L2], and let $f$ be the coordinate representative of $F$ . Then $rank D f (φ (p)) = m$ by [F1]. Since the differential of a map $R^{m} \to R^{n}$ cannot have rank above $m$ , [L1] gives a Euclidean neighbourhood on which the rank stays at least $m$ , hence exactly $m$ .

1.2F1L1L2given

The same argument with $n$ in place of $m$ shows that near any submersion point the rank stays equal to $n$ , so the submersion locus is open. Here the upper bound $rank D f \leq n$ is the relevant maximal-rank bound.

2.1step 1.1L2

Translating step 1.1 back through the source chart, every point of a smaller neighbourhood of $p$ is again an immersion point. Hence the immersion locus is open.

The immersion and submersion loci are open

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources