CorollaryStatement: 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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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.

3.1step 2.1step 1.2∎

Steps 2.1 and 1.2 prove the two openness claims.

Depends on

Used by

Local normal form for immersions Corollary
Local normal form for submersions Corollary
Refuted: a homotopy that is immersive at every earlier time is a regular homotopy Counterexample
The critical value set of a smooth map is sigma-compact Proposition
Transversality is stable on a compact source Proposition
Regular homotopy allows self-intersections but never rank drop Remark

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, Ch. 4 (standard reference, not scraped)
Will J. Merry, Differential Geometry, Definition 6.11 (standard reference, not scraped)