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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 Morse foliation F on a manifold M is a singular transversely oriented codimension one foliation of class with isolated singularities such that: each singularity of F is of Morse type, each singular leaf L contains a unique singularity p ; in addition, if ind p = 1 {\displaystyle \operatorname {ind} p=1} then L ∖ p {\displaystyle L ...
which is a product of the square root of the determinant of the four-dimensional metric tensor for the full spacetime and its Ricci scalar. This is the Lagrangian from the Einstein–Hilbert action. The desired outcome of the derivation is to define an embedding of three-dimensional spatial slices in the four-dimensional spacetime.