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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 ...
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 leaf is a torus T 2 bounding a solid torus with the Reeb foliation. The theorem was proved by Sergei Novikov in 1964. Earlier, Charles Ehresmann had conjectured that every smooth codimension-one foliation on S 3 had a compact leaf, which was known to be true for all known examples; in particular, the Reeb foliation has a compact leaf that ...
The distribution/foliation is regular if and only if the action is free. Given a Poisson manifold ( M , π ) {\displaystyle (M,\pi )} , the image of π ♯ = ι π : T ∗ M → T M {\displaystyle \pi ^{\sharp }=\iota _{\pi }:T^{*}M\to TM} is a singular distribution which is always integrable; the leaves of the associated singular foliation are ...
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 n such that ...
A 3-torus in this sense is an example of a 3-dimensional compact manifold. It is also an example of a compact abelian Lie group . This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication).