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Anosov Flow
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contraction". Anosov systems are a special case of Axiom A systems. Anosov diffeomorphisms were introduced by Dmitri Victorovich Anosov, who proved that their behaviour was in an appropriate sense ''generic'' (when they exist at all). Dmitri V. Anosov, ''Geodesic flows on closed Riemannian manifolds with negative curvature'', (1967) Proc. Steklov Inst. Mathematics. 90. Overview Three closely related definitions must be distinguished: * If a differentiable map ''f'' on ''M'' has a hyperbolic structure on the tangent bundle, then it is called an Anosov map. Examples include the Bernoulli map,_and_Arnold's_cat_map.html" ;"title=", 1)^\infty : x \mapsto (x_0, x_1, x_2, ..., and Arnold's cat map">, 1)^\infty : x \mapsto (x_0, x_1, x_2, ..., ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infranil Manifold
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or provide more detailed expositions of the definitions given below. * Connection * Curvature * Metric space * Riemannian manifold See also: * Glossary of general topology * Glossary of differential geometry and topology * List of differential geometry topics Unless stated otherwise, letters ''X'', ''Y'', ''Z'' below denote metric spaces, ''M'', ''N'' denote Riemannian manifolds, , ''xy'', or , xy, _X denotes the distance between points ''x'' and ''y'' in ''X''. Italic ''word'' denotes a self-reference to this glossary. ''A caveat'': many terms in Riemannian and metric geometry, such as ''convex function'', ''convex set'' and others, do not have exactly the same meaning as in general mathematical usage. __NOTOC__ A Alexandrov space a g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Map (Lie Theory)
In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups. The ordinary exponential function of mathematical analysis is a special case of the exponential map when G is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects. Definitions Let G be a Lie group and \mathfrak g be its Lie algebra (thought of as the tangent space to the identity element of G). The exponential map is a map :\exp\colon \mathfrak g \to G which can be defined in several different ways. The typical modern definition is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, 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 [x,y]. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative property, associative. Lie algebras are closely related to Lie groups, which are group (mathematics), 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 space, connected Lie group unique up to finite coverings (Lie's third theorem). This Lie group–Lie algebra correspondence, correspondence allows one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isometries
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' meaning "equal", and μέτρον ''metron'' meaning "measure". Introduction Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry; the isometry that relates them is either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection. Isometries are often used in constructions where one space i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PSL2(R)
In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: : \mbox(2,\mathbf) = \left\. It is a connected non-compact simple real Lie group of dimension 3 with applications in geometry, topology, representation theory, and physics. SL(2, R) acts on the complex upper half-plane by fractional linear transformations. The group action factors through the quotient PSL(2, R) (the 2 × 2 projective special linear group over R). More specifically, :PSL(2, R) = SL(2, R) / , where ''I'' denotes the 2 × 2 identity matrix. It contains the modular group PSL(2, Z). Also closely related is the 2-fold covering group, Mp(2, R), a metaplectic group (thinking of SL(2, R) as a symplectic group). Another related group is SL±(2, R), the group of real 2 × 2 matrices with determinant ±1; this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a ''vector bundle'' of rank 1. Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner. In topological applications, this vector space is usually real or complex. The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the positive and negative reals each to a point; whereas removing the origin from the complex plane ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with # An action of G on P, analogous to (x, g)h = (x, gh) for a product space. # A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) \mapsto x. Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of (x,e). Likewise, there is not generally a projection onto G generalizing the projection onto the second factor, X \times G \to G that exists for the Cartesian product. They may also have a complicated topology that prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space. A common example of a principal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuchsian Group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces. There are some variations of the definition: sometimes the Fuchsian group is assumed to be finitely generated, sometimes it is allowed to be a subgroup of PGL(2,R) (so that it contains orientation-reversing elements), and sometimes it is allowed to be a Kleinian group (a discrete subgroup of PSL(2,C)) which is conjugate to a subgroup of PSL(2,R). Fuchsian groups are used to create Fuchsian models of Riemann surfaces. In this case, the group may be called the Fuchsian group of the surface. In some sense, Fuchsian groups do for non-Euclidean geometry what crystallographic groups do for Euclidean geometry. Some Escher graphics are based on t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to the set of complex numbers with positive imaginary part: :\mathcal \equiv \ ~. The term arises from a common visualization of the complex number as the point in the plane endowed with Cartesian coordinates. When the axis is oriented vertically, the "upper half-plane" corresponds to the region above the axis and thus complex numbers for which > 0. It is the domain of many functions of interest in complex analysis, especially modular forms. The lower half-plane, defined by 0. Proposition: Let ''A'' and ''B'' be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes ''A'' to ''B''. :Proof: First shift the center of ''A'' to (0,0). Then take λ = (diame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuchsian Model
In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface ''R'' as a quotient of the upper half-plane H by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation. The concept is named after Lazarus Fuchs. A more precise definition By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. More precisely this theorem states that a Riemann surface R which is not isomorphic to either the Riemann sphere (the elliptic case) or a quotient of the complex plane by a discrete subgroup (the parabolic case) must be a quotient of the hyperbolic plane \mathbb H by a subgroup \Gamma acting properly discontinuously and freely. In the Poincaré half-plane model for the hyperbolic plane the group of biholomorphic transformations is the group \mathrm_2(\mathbb R) acting by homographies, and the uniformization theorem means that there exists a discrete, torsion-free subgroup \Gamma \subset \mathrm_2(\ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
