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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 ...
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).
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 .
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 hexagonal packing of circles on a 2-dimensional Euclidean plane. These problems are mathematically distinct from the ideas in the circle packing theorem.The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere.
Pascal's pyramid's first five layers. Each face (orange grid) is Pascal's triangle. Arrows show derivation of two example terms. In mathematics, Pascal's pyramid is a three-dimensional arrangement of the trinomial numbers, which are the coefficients of the trinomial expansion and the trinomial distribution. [1]
Every three-dimensional topological manifold which is closed, connected, and has trivial fundamental group is homeomorphic to the three-dimensional sphere. Familiar shapes, such as the surface of a ball (which is known in mathematics as the two-dimensional sphere) or of a torus, are two-dimensional. The surface of a ball has trivial fundamental ...
In general, an n-dimensional CW complex is constructed by taking the disjoint union of a k-dimensional CW complex (for some <) with one or more copies of the n-dimensional ball. For each copy, there is a map that "glues" its boundary (the ( n − 1 ) {\displaystyle (n-1)} -dimensional sphere ) to elements of the k {\displaystyle k} -dimensional ...