TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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Every smooth manifold embeds in some finite-dimensional Euclidean space

Statement

Every smooth manifold embeds smoothly in some finite-dimensional Euclidean space. If the manifold is noncompact, one can choose such an embedding in the form

M ↪ R^{N} \times R, p \mapsto (G (p), ρ (p)),

where $G$ is bounded and $ρ$ is a smooth proper exhaustion function.

Facts & Assumptions

Given: A smooth manifold $M$ .

[L1]

A compact smooth manifold admits a finite coordinate-bump embedding into some Euclidean space (A finite coordinate-bump map embeds a compact manifold in some Euclidean space).

[F1]

The noncompact Whitney construction in the cited Chapter 6 source produces a finite-dimensional embedding in the form $p ⟼ (G (p), ρ (p)),$ where the coordinate-bump component $G$ is bounded and the final coordinate $ρ$ is a smooth proper exhaustion function.

Proof

technique · direct

1.1

If $M$ is compact, [L1] already gives a smooth embedding of $M$ into some finite-dimensional Euclidean space.

L1given

1.2

If $M$ is noncompact, [F1] gives an embedding $(G, ρ)$ with $G$ bounded and $ρ$ proper.

F1given

2.1

Combining the compact case from step 1.1 and the noncompact case from step 1.2 proves the theorem.

step 1.1step 1.2∎

Depends on

A finite coordinate-bump map embeds a compact manifold in some Euclidean space

Used by

The weak Whitney proper embedding theorem Theorem

Dependency tree · two levels

4 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., Chapter 6 (standard reference, not scraped)
Marco Gualtieri, Topology I: Smooth Manifolds, Part 11 (standard reference, not scraped)