Alphabeta MathHow statement and proof provenance work
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Leaves of a regular foliation
Definition
Given a regular foliation atlas on , call each connected component of a slice
an atlas plaque. Declare two points equivalent when they can be joined by a finite chain of atlas plaques in which consecutive plaques intersect. A leaf is an equivalence class for this plaque-chain relation. Thus a leaf is the underlying subset obtained by continuing one local plaque through overlapping foliation charts, not an arbitrary connected union of distinct leaves. This item defines only that underlying subset; it does not yet assign the subset an intrinsic smooth manifold structure.
Depends on
Used by
- Dead-end components Definition
- Leafwise paths and leafwise homotopy relative to endpoints Definition
- Limit cycles of a leaf Definition
- Limitwise-nullhomotopy predicate on based loops Definition
- Saturated neighbourhoods of a leaf Definition
- Taut codimension-one foliations Definition
- The accessible manifold of a leaf Definition
- The holonomy representation and the holonomy group of a leaf Definition
- The monodromy groupoid of a foliation Definition
- Vanishing cycles of a codimension-one foliation Definition
- The Kronecker foliation of the torus has dense leaves and trivial leaf holonomy Example
- The Möbius band's central leaf has reflection holonomy Example
- The Mobius-band line foliation Example
- A compact leafwise nullhomotopy persists under a transverse deformation Lemma
- A fixed leafwise cap gives a joint transverse product with exact collar data Lemma
- A leafwise path determines a germ of a transverse diffeomorphism Lemma
- A no-transversal leaf bounds a positive accessibility region with finite inward boundary Lemma
- A non-closed leaf of a codimension-one foliation meets a closed transversal Lemma
- A noncompact leaf of a compact C2 foliation meets a positive closed transversal Lemma
- A nonzero limitwise-nullhomotopy class forces a compact boundary leaf Lemma
- A nonzero limitwise-nullhomotopy class yields a compact boundary leaf Lemma
- Closedness of compact leaves diffeomorphic to a finite-fundamental-group leaf Lemma
- Compact leaves near a compact reference leaf are one-sheeted collar graphs Lemma
- Finite general position for a leafwise loop Lemma
- Fixed transverse fences and their finite crossing words Lemma
- Restriction of a foliation transverse to the boundary Lemma
- The covering of a leaf associated with the holonomy kernel exists Lemma
- Suspension holonomy is the germ of the represented monodromy action Proposition
- The isotropy of the holonomy groupoid is the leaf holonomy group Proposition
- The pullback foliation under a transverse map Proposition
- The quotient foliation under a free and properly discontinuous foliated action Proposition
- Regular foliations and integrable distributions correspond Theorem
Dependency tree · two levels
2 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
- Will J. Merry, Differential Geometry (standard reference, not scraped)