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2-dimensional section of Reeb foliation 3-dimensional model of Reeb foliation. In mathematics (differential geometry), a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space R n into the cosets x + R p of the standardly embedded ...
Given any Poisson bivector field on a 3-dimensional manifold, the bivector field , for any (), is ... the symplectic foliation is the usual (singular) foliation ...
In mathematics, the Reeb foliation is a particular foliation of the 3-sphere, introduced by the French mathematician Georges Reeb (1920–1993). It is based on dividing the sphere into two solid tori , along a 2- torus : see Clifford torus .
The definition of singular foliation can be made precise in several equivalent ways. Actually, in the literature there is a plethora of variations, reformulations and generalisations of the Stefan-Sussman theorem, using different notion of singular foliations according to which applications one has in mind, e.g. Poisson geometry [ 10 ] [ 11 ...
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (a tangent plane) to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer ...
Hyperbolic geometry is the most rich and least understood of the eight geometries in dimension 3 (for example, for all other geometries it is not hard to give an explicit enumeration of the finite-volume manifolds with this geometry, while this is far from being the case for hyperbolic manifolds).
Definition. A p-dimensional, class C r foliation of an n-dimensional manifold M is a decomposition of M into a union of disjoint connected submanifolds {L α} α∈A, called the leaves of the foliation, with the following property: Every point in M has a neighborhood U and a system of local, class C r coordinates x=(x 1, ⋅⋅⋅, x n) : U→R ...
An advantage of Haefliger structures over foliations is that they are closed under pullbacks.More precisely, given a Haefliger structure on , defined by a Haefliger cocycle , and a continuous map :, the pullback Haefliger structure on is defined by the open cover () and the cocycle .