ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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The irrational linear foliation of the two-torus

Example

Let $α \in R ∖ Q$ . On $T^{2} = R^{2} / Z^{2}$ , the constant vector field induced by $(1, α)$ defines a regular foliation. Its leaves are the images of $t ⟼ [p + (t, α t)] (p \in R^{2}),$ and each leaf is dense in the torus.

Facts & Assumptions

Given: An irrational number $α$ .

[A1]

Translation by $(1, α)$ on $R^{2}$ descends to the torus.

[A2]

Equip $R^{2} / Z^{2}$ with its standard quotient smooth structure, equivalently the product smooth structure on two circles.

Verification

technique · direct

1.1

On the smooth torus of [A2], the constant line field spanned by $(1, α)$ [A2] is smooth and nowhere zero, so it gives a regular one-dimensional distribution on $T^{2}$ .

A2given

1.2

The integral curve through $[p]$ is the projected affine line [given] $t \mapsto [p + (t, α t)]$ . For $p = 0$ , fix a point $[(u, v)]$ of the torus and a neighborhood of it. Because $α$ is irrational, the set of classes ${[n α] : n \in Z}$ is dense in $R / Z$ . Choose $n$ so that $[n α]$ is arbitrarily close to $[v - α u]$ , and set $t = u + n$ . Then $[(t, α t)] = [(u, α u + n α)]$ has first coordinate $[u]$ and second coordinate arbitrarily close to $[v]$ . Thus the leaf through $[0]$ is dense. Every other leaf is a torus translate of this one, and translations are homeomorphisms, so every leaf is dense.

givenalgebra

2.1

Therefore the torus carries a regular foliation with dense, nonembedded [given] leaves.

given∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Will J. Merry, Differential Geometry (standard reference, not scraped)
Keith Conrad, Local and global Frobenius theorems (standard reference, not scraped)