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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] |
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 mathematical analysis, 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 mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homography
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation. Historically, homographies (and projective spaces) have been introduced to study perspective and projections in Euclidean geometry, and the term ''homography'', which, etymologically, roughly means "similar drawing", dates from this time. At the end of the 19th century, formal definitions of projective spaces were introduced, which differed from extending Euclidean or affine spaces by adding points at infinity. The term "projective transformation" originated in these a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Polygon
In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface has simply connected universal covering surface given by exactly one of the following: *the Riemann sphere, *the complex plane, *the unit disk ''D'' or equivalently the upper half-plane ''H''. In the first case of genus zero, the surface is conformally equivalent to the Riemann sphere. In the second case of genus one, the surface is conformally equivalent to a torus C/Λ for some lattice Λ in C. The fundamental polygon of Λ, if assumed convex, may be taken to be either a period parallelogram or a centrally symmetric hexagon, a result first proved by Fedorov in 1891. In the last case of genus ''g'' > 1, the Riemann surface is conformally equivalent to ''H''/Γ w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-manifolds
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional 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. Introduction Definition A topological space ''X'' is a 3-manifold if it is a second-countable Hausdorff space and if every point in ''X'' has a neighbourhood that is homeomorphic to Euclidean 3-space. Mathematical theory of 3-manifolds The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleinian Model
In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold ''N'' by the quotient space \mathbb^3 / \Gamma where \Gamma is a discrete subgroup of PSL(2,C). Here, the subgroup \Gamma, a Kleinian group, is defined so that it is isomorphic to the fundamental group \pi_1(N) of the surface ''N''. Many authors use the terms ''Kleinian group'' and ''Kleinian model'' interchangeably, letting one stand for the other. The concept is named after Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group .... Many properties of Kleinian models are in direct analogy to those of Fuchsian models; however, overall, the theory is less well developed. A number of unsolved conjectures on Kleinian models are the analogs to theorems on Fuchsian models. See also * Hype ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Teichmüller Space
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller. Each point in a Teichmüller space T(S) may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from S to itself. It can be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for which it is homeomorphic to a ball of dimension 6g-6 for a surface of genus g \ge 2. In this way Teichmüller space can be viewed as the universal covering orbifold of the Riemann moduli space. The Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The study of geometric features of these various structures is an active body of research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasiconformal Map
In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let ''f'' : ''D'' → ''D''′ be an orientation-preserving homeomorphism between open sets in the plane. If ''f'' is continuously differentiable, then it is ''K''-quasiconformal if the derivative of ''f'' at every point maps circles to ellipses with eccentricity bounded by ''K''. Definition Suppose ''f'' : ''D'' → ''D''′ where ''D'' and ''D''′ are two domains in C. There are a variety of equivalent definitions, depending on the required smoothness of ''f''. If ''f'' is assumed to have continuous partial derivatives, then ''f'' is quasiconformal provided it satisfies the Beltrami equation for some complex valued Lebesgue measurable μ satisfying sup , μ, 0. Then ''f'' satisfies () precisely when it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word ''homeomorphism'' comes from the Greek words '' ὅμοιος'' (''homoios'') = similar or same and '' μορφή'' (''morphē'') = shape or form, introduced to mathematics by Henri Poincaré in 1895. Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dehn–Nielsen Theorem
In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topology) deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves. The mapping class group can be defined for arbitrary manifolds (indeed, for arbitrary topological spaces) but the 2-dimensional setting is the most studied in group theory. The mapping class group of surfaces are related to various other groups, in particular braid groups and outer automorphism groups. History The mapping class group appeared in the first half of the twentieth century. Its origins lie in the study of the topology of hyperbolic surfaces, and especially in the study of the intersections of closed curves on these surfaces. The earl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Generated Group
In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of such elements. By definition, every finite group is finitely generated, since ''S'' can be taken to be ''G'' itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated. Examples * Every quotient of a finitely generated group ''G'' is finitely generated; the quotient group is generated by the images of the generators of ''G'' under the canonical projection. * A subgroup of a finitely generated group need not be finitely generated. * A group that is generated by a single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion-free Group
In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module that equals its torsion submodule. A module is torsion-free if its torsion submodule comprises only the zero element. This terminology is more commonly used for modules over a domain, that is, when the regular elements of the ring are all its nonzero elements. This terminology applies to abelian groups (with "module" and "submodule" replaced by "group" and "subgroup"). This is allowed by the fact that the abelian groups are the modules over the ring of integers (in fact, this is the origin of the terminology, that has been introduced for abelian groups before being generalized to modules). In the case of groups that are noncommutative, a ''torsion element'' is an element of finite order. Contrary to the commut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Subset
] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equivalent to saying that the singleton is an open set in the topological space ''S'' (considered as a subspace of ''X''). Another equivalent formulation is: an element ''x'' of ''S'' is an isolated point of ''S'' if and only if it is not a limit point of ''S''. If the space ''X'' is a metric space, for example a Euclidean space, then an element ''x'' of ''S'' is an isolated point of ''S'' if there exists an open ball around ''x'' which contains only finitely many elements of ''S''. Related notions A set that is made up only of isolated points is called a discrete set (see also discrete space). Any discrete subset ''S'' of Euclidean space must be countable, since the isolation of each of its points together with the fact that rational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
