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Distribution (differential Geometry)
In differential geometry, a discipline within mathematics, a distribution on a manifold M is an assignment x \mapsto \Delta_x \subseteq T_x M of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle TM. Distributions satisfying a further integrability condition give rise to foliations, i.e. partitions of the manifold into smaller submanifolds. These notions have several applications in many fields of mathematics, e.g. integrable systems, Poisson geometry, non-commutative geometry, sub-Riemannian geometry, differential topology, etc. Even though they share the same name, distributions presented in this article have nothing to do with distributions in the sense of analysis. Definition Let M be a smooth manifold; a (smooth) distribution \Delta assigns to any point x \in M a vector subspace \Delta_x \subset T_xM in a smooth way. More precisely, \Delta consists in a collection \_ of vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Subalgebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contact Structure
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for ' complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem. Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Curve
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. Name Integral curves are known by various other names, depending on the nature and interpretation of the differential equation or vector field. In physics, integral curves for an electric field or magnetic field are known as field lines, and integral curves for the velocity field of a fluid are known as streamlines. In dynamical systems, the integral curves for a differential equation that governs a system are referred to as trajectories or orbits. Definition Suppose that F is a static vector field, that is, a vector-valued function with Cartesian coordinates (''F''1,''F''2,...,''F''''n''), and that x(''t'') is a parametric curve with Cartesian coordinates (''x''1(''t''),''x''2(''t''),...,''x''''n''(''t'')). Then x(''t'') is an integral curve of F if it is a solution of the autonomous system of ordinary differential equations, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chow–Rashevskii Theorem
In sub-Riemannian geometry, the Chow–Rashevskii theorem (also known as Chow's theorem) asserts that any two points of a connected sub-Riemannian manifold, endowed with a bracket generating distribution, are connected by a horizontal path in the manifold. It is named after Wei-Liang Chow who proved it in 1939, and Petr Konstanovich Rashevskii, who proved it independently in 1938. The theorem has a number of equivalent statements, one of which is that the topology induced by the Carnot–Carathéodory metric is equivalent to the intrinsic (locally Euclidean) topology of the manifold. A stronger statement that implies the theorem is the ball–box theorem. See, for instance, and . See also * Orbit (control theory) The notion of orbit of a control system used in mathematical control theory is a particular case of the notion of orbit in group theory. Definition Let \dot q=f(q,u) be a \ ^\infty control system, where belongs to a finite-dimensional manifol ... Refere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Theorem (differential Topology)
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. The theorem generalizes the existence theorem for ordinary differential equations, which guarantees that a single vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of ''r'' vector fields mesh into coordinate grids on ''r''-dimensional integral manifolds. The theorem is foundational in differential topology and calculus on manifolds. Introduction In its most elementary form, the theorem addresses the problem of finding a maximal set of inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Matrices
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned. This notion can be made more precise for an n by m matrix M by partitioning n into a collection \text, and then partitioning m into a collection \text. The original matrix is then considered as the "total" of these groups, in the sense that the (i, j) entry of the original matrix corresponds in a 1-to-1 way with some (s, t) offset entry of some (x,y), where x \in \text and y \in \text. Block matrix algebra arises in general from biproducts in categories of matrices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
