Regular foliation atlases

Depends on

• Plaques of a flat chart

Used by

• A noncompact foliation violating Novikov's compactness conclusions Counterexample
• Dead-end components Definition
• Leaves of a regular foliation Definition
• Local transversals to a regular foliation Definition
• Saturated neighbourhoods of a leaf Definition
• Smooth foliations tangent to the boundary Definition
• Stable leaves Definition
• The accessible manifold of a leaf Definition
• The Bott partial connection on the normal bundle of a foliation Definition
• Transversely oriented codimension-one foliations Definition
• The Mobius-band line foliation Example
• The product foliation Example
• The product foliation near a compact leaf with trivial holonomy Example
• A characteristic disk with essential boundary data produces a vanishing cycle Lemma
• A compact leafwise nullhomotopy persists under a transverse deformation Lemma
• A finite characteristic circuit has C² regular port traces 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 nonzero limitwise-nullhomotopy class forces a compact boundary leaf Lemma
• A nonzero limitwise-nullhomotopy class yields a compact boundary leaf Lemma
• A saddle polycycle has a smooth transverse family on either adjacent annulus Lemma
• A vanishing cycle determines a nonzero limitwise-nullhomotopy class Lemma
• Gluing manifolds with boundary along a boundary diffeomorphism Lemma
• Holonomy respects path concatenation and reversal Lemma
• Limitwise-nullhomotopy predicate descends to a normal subgroup Lemma
• Relative generic position for characteristic disk maps Lemma
• Restriction of a foliation transverse to the boundary Lemma
• The holonomy germ is independent of the foliation chart chain Lemma
• The normal model map restricts to a diffeomorphism onto a saturated neighbourhood Lemma
• Transverse holonomy transport is well defined and equivariant on the model Lemma
• Mapping torus foliations realize global Reeb stable examples Proposition
• The pullback foliation under a transverse map Proposition
• The quotient foliation under a free and properly discontinuous foliated action Proposition
• Holonomy depends only on leafwise homotopy relative to endpoints Theorem
• Local Reeb stability for compact leaves with finite holonomy Theorem
• 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

• John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)
• Will J. Merry, Differential Geometry (standard reference, not scraped)