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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 .
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.
The distribution/foliation is regular If and only if the Poisson manifold is regular. More generally, the image of the anchor map ρ : A → T M {\displaystyle \rho :A\to TM} of any Lie algebroid A → M {\displaystyle A\to M} defines a singular distribution which is automatically integrable, and the leaves of the associated singular foliation ...
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 ...
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 ...
(Note that the foliation need not be regular; i.e. the leaves of the foliation might not be embedded submanifolds.) An integrability condition is a condition on the α i {\displaystyle \alpha _{i}} to guarantee that there will be integral submanifolds of sufficiently high dimension.
Intersection lineations are linear structures formed by the intersection of any two surfaces in a three-dimensional space. The trace of bedding on an intersecting foliation plane commonly appears as colour stripes generally parallel to local fold's hinges. Intersection lineations can also be due to the intersection of two foliations.