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 some papers on general relativity by mathematical physicists, the term foliation (or slicing) is used to describe a situation where the relevant Lorentz manifold (a (p+1)-dimensional spacetime) has been decomposed into hypersurfaces of dimension p, specified as the level sets of a real-valued smooth function (scalar field) whose gradient is everywhere non-zero; this smooth function is moreover usually assumed to be a time function, meaning that its gradient is everywhere time-like, so that its level-sets are all space-like hypersurfaces. In deference to standard mathematical terminology, these hypersurface are often called the leaves (or sometimes slices) of the foliation. Note that while this situation does constitute a codimension-1 foliation in the standard mathematical sense, examples of this type are actually globally trivial; while the leaves of a (mathematical) codimension-1 foliation are always locally the level sets of a function, they generally cannot be expressed this way globally, as a leaf may pass through a local-trivializing chart infinitely many times, and the holonomy around a leaf may also obstruct the existence of a globally-consistent defining functions for the leaves.
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In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M. The notion of G-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields. For example, for the orthogonal group, an O(n)-structure defines a Riemannian metric, and for the special linear group an SL(n,R)-structure is the same as a volume form.
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields.
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below. A topological space is a 3-manifold if it is a second-countable Hausdorff space and if every point in has a neighbourhood that is homeomorphic to Euclidean 3-space.
The goal of this course/seminar is to introduce the students to some contemporary aspects of geometric group theory. Emphasis will be put on Artin's Braid groups and Thompson's groups.
We consider the singular set in the thin obstacle problem with weight vertical bar x(n +1)vertical bar(a) for a epsilon (-1, 1), which arises as the local extension of the obstacle problem for the fractional Laplacian (a nonlocal problem). We develop a ref ...
Ghys and Sergiescu proved in the 1980s that Thompson's group T, and hence F, admits actions by C-infinity diffeomorphisms of the circle. They proved that the standard actions of these groups are topologically conjugate to a group of C-infinity diffeomorphi ...
In central Europe, many geo-energy reservoirs have revealed to be hosted in transverse isotropic crystalline rock, where the rock's mechanical and hydraulic transport properties are poorly constrained. Here, we performed triaxial experiments on Cresciano G ...