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foliations

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Definitions of foliations in various dictionaries:

noun - (botany) the process of forming leaves

noun - (geology) the arrangement of leaflike layers in a rock

noun - (architecture) leaf-like architectural ornament

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Plural form of foliation.
The process of splitting into thin sheets or laminae.

Foliations might refer to

Foliations might be related to
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 Rn into the cosets x + Rp of the standardly embedded subspace Rp. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable (of class Cr), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1 (otherwise, C0 is a topological foliation). The number p (the dimension of the leaves) is called the dimension of the foliation and q = n - p is called its codimension. * In physics (relativity), by foliation (or slicing) it is meant that the manifold (spacetime) is decomposed into hypersurfaces of dimension p and there exists a smooth scalar field which is regular in the sense that its gradient never vanishes, such that each hypersurface is a level surface of this scalar field. Since the scalar field is regular, the hypersurfaces are non-intersecting. It is usually assumed that the manifold is globally hyperbolic, all hypersurfaces are spacelike, and the foliation covers the whole manifold. Each hypersurface is called a leaf or a slice of the foliation.

Foliations might be related to

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 Rn into the cosets x + Rp of the standardly embedded subspace Rp. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable (of class Cr), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1 (otherwise, C0 is a topological foliation). The number p (the dimension of the leaves) is called the dimension of the foliation and q = n - p is called its codimension.
* In physics (relativity), by foliation (or slicing) it is meant that the manifold (spacetime) is decomposed into hypersurfaces of dimension p and there exists a smooth scalar field which is regular in the sense that its gradient never vanishes, such that each hypersurface is a level surface of this scalar field. Since the scalar field is regular, the hypersurfaces are non-intersecting. It is usually assumed that the manifold is globally hyperbolic, all hypersurfaces are spacelike, and the foliation covers the whole manifold. Each hypersurface is called a leaf or a slice of the foliation.

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