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A compact C¹ foliation leaf is an embedded hypersurface

Statement

Let $F$ be a $C^{1}$ codimension-one foliation of a smooth Hausdorff manifold $M$ (Smooth manifolds and their smooth charts, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not) and let $L$ be a leaf of $F$ that is compact in its intrinsic leaf-manifold topology (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right). Then the inclusion $L ↪ M$ is a $C^{1}$ embedding, so $L$ is a compact embedded $C^{1}$ hypersurface of $M$ .

Facts & Assumptions

Given: A $C^{1}$ codimension-one foliation $F$ of a smooth Hausdorff manifold $M$ and a compact leaf $L$ .

[F1]

A $C^{1}$ foliation atlas has charts $(x_{α}, t_{α})$ with $C^{1}$ inverse in which plaques are the connected components of the level sets $t_{α} = c$ , and leaves are generated by intersecting plaques; each leaf carries the structure of a one-dimensional-transverse $C^{1}$ manifold of dimension $n - 1$ , with the inclusions of plaques as charts (C¹ codimension-one regular foliations and transverse orientation, C¹ foliation charts preserve plaque equivalence and transverse orientation).

[F2]

A space is compact when every open cover has a finite subcover (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

[F4]

A subspace is compact if and only if every cover by ambient open sets has a finite subcover; the indexed form also holds without a choice axiom (A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it).

[F3]

The ambient manifold is Hausdorff (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not).

[F6]

A subset $S$ of an $m$ -manifold is an embedded $k$ -submanifold when near each of its points there is a smooth chart carrying $S$ onto $R^{k} \times {0}$ (Embedded submanifolds and slice charts).

Proof

technique · direct

1.1F2F3F4given

(Compact-to-Hausdorff without metrization.) The inclusion $j : L \to M$ is continuous and injective. For any closed subset $C$ of compact $L$ , $C$ is compact: add $L ∖ C$ to an open cover of $C$ , take a finite subcover of $L$ , then discard the added set. Pulling back any ambient open cover of $j (C)$ gives a finite subcover by compactness of $C$ ; F4 then makes $j (C)$ compact in its subspace topology. A compact subset $K$ of Hausdorff $M$ is closed: for $z \notin K$ , consider all pairs $(U, V)$ of ambient open sets with $z \in V$ and $U \cap V = \emptyset$ . Hausdorffness makes their first entries cover $K$ ; the indexed form of F4 gives finitely many such pairs covering $K$ . Intersect their second entries, which are neighborhoods of $z$ . This intersection misses $K$ . Thus $j$ takes closed subsets of $L$ to closed subsets of $j (L)$ and has continuous inverse onto its image. No metric or full-AC theorem is invoked.

1.2F1

(Immersion in plaque coordinates.) By F1 and its atlas certificate, the compact intrinsic leaf has a finite $C^{1}$ plaque atlas. In a foliation chart its plaque inclusion is $y \mapsto φ^{- 1} (y, c)$ . Differentiating the identity $φ \circ φ^{- 1} = i d$ shows that $D φ^{- 1}$ is invertible, so this inclusion has rank $n - 1$ . Thus $j$ is an injective $C^{1}$ immersion, including the zero-dimensional case $n = 1$ .

2.1F1F6step 1.1step 1.2∎

(Exclude other branches.) Fix $p \in L$ and an intrinsic plaque-chart neighborhood $V_{L}$ of $p$ . Step 1.1 gives an ambient open neighborhood $O$ with $p \in O$ and $O \cap L \subseteq V_{L}$ . Shrink the foliation chart to a product box inside $O$ . Its intersection with $L$ is the single slice through $p$ , with no other branch of $L$ entering the box. The foliation chart itself is a $C^{1}$ slice chart, and step 1.2 supplies the immersion. Hence $j$ is a $C^{1}$ embedding. F6 is a smooth slice-chart definition; here its explicit analogue in the $C^{1}$ category is used, with no claim that a merely $C^{1}$ leaf is smooth.

Depends on

Used by

Dependency tree · two levels

40 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

Danny Calegari, Foliations and the Geometry of 3-Manifolds (Oxford Mathematical Monographs; complete author-hosted PDF) (standard reference, not scraped)