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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 ...
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 ...
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 .
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.
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 ...
In other words, every point admits a foliation chart, i.e. the distribution is tangent to the leaves of a foliation. Moreover, this local characterisation coincides with the definition of integrability for a G {\displaystyle G} -structures , when G {\displaystyle G} is the group of real invertible upper-triangular block matrices (with ( n × n ...
If the heat is too intense, foliation will be weakened due to the nucleation and growth of new randomly oriented crystals and the rock will become a hornfels. [1] If minimal heat is applied to a rock with a preexisting foliation and without a change in mineral assemblage, the cleavage will be strengthened by growth of micas parallel to foliation.
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 ...