LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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.

A generic linear projection preserves injectivity and immersion

Statement

Let $f : M^{n} \to R^{N}$ be a smooth embedding with $N > 2 n + 1$ . Then the set of unit vectors $u \in S^{N - 1}$ for which the orthogonal projection

P_{u} : R^{N} \to u^{⊥}

makes $P_{u} \circ f$ an injective immersion is dense in $S^{N - 1}$ .

Facts & Assumptions

Given: A smooth embedding $f : M^{n} \to R^{N}$ with $N > 2 n + 1$ .

[F1]

The secant-direction map $σ_{f}$ is defined on $(M \times M) ∖ Δ_{M}$ , and the tangent-direction map $τ_{f}$ is defined on $T M ∖ 0_{M}$ (Secant and tangent direction maps of a Euclidean embedding).

[L1]

The image of a $C^{1}$ manifold of dimension strictly smaller than the target-manifold dimension is a null set (The image of a lower-dimensional $C^{1}$ manifold is null).

[L2]

A null subset of a positive-dimensional manifold has dense complement (A null set has dense complement in a positive-dimensional manifold).

Proof

technique · direct

1.1

The manifold $(M \times M) ∖ Δ_{M}$ has dimension $2 n$ , and $T M ∖ 0_{M}$ also has dimension $2 n$ . Since $S^{N - 1}$ has dimension $N - 1 > 2 n$ , [L1] shows that both images $σ_{f} ((M \times M) ∖ Δ_{M})$ and $τ_{f} (T M ∖ 0_{M})$ are null subsets of $S^{N - 1}$ .

F1L1given

2.1

By [L2], the complement of the union of those two bad sets is dense in $S^{N - 1}$ . Fix $u$ in that complement.

L2step 1.1choose

3.1

If $P_{u} (f (p)) = P_{u} (f (q))$ , then $f (q) - f (p)$ is parallel to $u$ . Because $f$ is injective, either $p = q$ or $u = \pm σ_{f} (p, q)$ . The second alternative is impossible by step 2.1, so $p = q$ . Thus $P_{u} \circ f$ is injective.

F1step 2.1algebra

3.2

If $d (P_{u} \circ f)_{p} (v) = 0$ for some $v \neq 0$ , then $d f_{p} (v)$ is parallel to $u$ , so $u = \pm τ_{f} (p, v)$ . This again contradicts step 2.1. Hence $d (P_{u} \circ f)_{p}$ is injective for every $p$ , and $P_{u} \circ f$ is an immersion.

F1step 2.1algebra

4.1

Therefore every $u$ outside the secant and tangent images gives an injective immersion after projection, and such $u$ form a dense set.

step 2.1step 3.1step 3.2∎

Depends on

Used by

Projecting a space curve can create a double point Example
FALSE: an arbitrary linear projection of an embedding is an embedding False statement
A generic projection can preserve properness Lemma
The weak Whitney proper embedding theorem Theorem

Dependency tree · two levels

11 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, 2nd ed., Lemma 6.13 (standard reference, not scraped)
Marco Gualtieri, Topology I: Smooth Manifolds, Part 11 (standard reference, not scraped)