Orbifold (original) (raw)
In der Topologie ist eine Orbifaltigkeit (englisch: Orbifold) eine Verallgemeinerung einer Mannigfaltigkeit.
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| dbo:abstract | In der Topologie ist eine Orbifaltigkeit (englisch: Orbifold) eine Verallgemeinerung einer Mannigfaltigkeit. (de) En topología, orbifold (Orbidad u orbivariedad) es la generalización de una variedad diferenciable, consistente en un espacio topológico (llamado espacio subyacente) con una estructura de orbifold (véase abajo). El espacio subyacente localmente aparece como un cociente de un espacio euclídeo bajo la acción de un grupo finito de isometrías. El ejemplo principal del espacio subyacente es un espacio cociente de una variedad bajo la acción de un grupo finito de difeomorfismos. En particular, una variedad con borde lleva una estructura natural de orbifold, puesto que es Z2-factor de su . Un espacio factor de una variedad a lo largo de una S1-acción diferenciable sin puntos fijos lleva estructura de orbifold (este no es un caso particular del ejemplo principal). La estructura de orbifold da una natural para las variedades abiertas en su espacio subyacente, donde cada estrato corresponde a un conjunto de puntos singulares del mismo tipo. Debe ser observado que un espacio topológico puede llevar muchas estructuras de orbifold diversas. Por ejemplo, considere O el orbifold asociado a un espacio factor de la 2-esfera a lo largo de una rotación de π, es homeomorfo a la 2-esfera, pero la estructura natural de orbifold es diferente. Es posible adoptar la mayoría de las propiedades de variedades a los orbifolds y estas propiedades son generalmente diferentes de las propiedades correspondientes del espacio subyacente. En el ejemplo antedicho, su grupo fundamental de orbifold es Z2 y su característica euleriana de orbifold es 1. (es) In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. Definitions of orbifold have been given several times: by Ichirô Satake in the context of automorphic forms in the 1950s under the name V-manifold; by William Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the name orbifold, after a vote by his students; and by André Haefliger in the 1980s in the context of Mikhail Gromov's programme on CAT(k) spaces under the name orbihedron. Historically, orbifolds arose first as surfaces with singular points long before they were formally defined. One of the first classical examples arose in the theory of modular forms with the action of the modular group on the upper half-plane: a version of the Riemann–Roch theorem holds after the quotient is compactified by the addition of two orbifold cusp points. In 3-manifold theory, the theory of Seifert fiber spaces, initiated by Herbert Seifert, can be phrased in terms of 2-dimensional orbifolds. In geometric group theory, post-Gromov, discrete groups have been studied in terms of the local curvature properties of orbihedra and their covering spaces. In string theory, the word "orbifold" has a slightly different meaning, discussed in detail below. In two-dimensional conformal field theory, it refers to the theory attached to the fixed point subalgebra of a vertex algebra under the action of a finite group of automorphisms. The main example of underlying space is a quotient space of a manifold under the properly discontinuous action of a possibly infinite group of diffeomorphisms with finite isotropy subgroups. In particular this applies to any action of a finite group; thus a manifold with boundary carries a natural orbifold structure, since it is the quotient of its double by an action of . One topological space can carry different orbifold structures. For example, consider the orbifold O associated with a quotient space of the 2-sphere along a rotation by ; it is homeomorphic to the 2-sphere, but the natural orbifold structure is different. It is possible to adopt most of the characteristics of manifolds to orbifolds and these characteristics are usually different from correspondent characteristics of underlying space. In the above example, the orbifold fundamental group of O is and its orbifold Euler characteristic is 1. (en) En mathématiques, un orbifold (parfois appelé aussi orbivariété) est une généralisation de la notion de variété contenant de possibles singularités. Ces espaces ont été introduits explicitement pour la première fois par Ichirō Satake en 1956 sous le nom de V-manifolds. Pour passer de la notion de variété (différentiable) à celle d'orbifold, on ajoute comme modèles locaux tous les quotients d'ouverts de par l'action de groupes finis. L'intérêt pour ces objets a été ravivé considérablement à la fin des années 70 par William Thurston en relation avec sa conjecture de géométrisation. En physique, ces espaces ont été considérés initialement comme espaces de compactification en théorie des cordes car malgré la présence de singularités la théorie y est bien définie.Lorsqu'ils sont utilisés dans le cadre plus particulier de la théorie des supercordes, les orbifolds autorisés doivent avoir la propriété supplémentaire d'être des variétés de Calabi-Yau afin de préserver une quantité minimale de supersymétrie. Mais dans le cas où des singularités sont présentes, il s'agit là d'une extension de la définition originale des espaces de Calabi-Yau car ceux-ci sont en principe des espaces sans singularité. (fr) 기하학에서, 오비폴드(영어: orbifold)는 국소적으로 유한군의 선형작용에 대한 유클리드 공간의 몫공간과 동형인 위상 공간이다. 매끄러운 다양체의 개념의 일반화이며, 다양체와 달리 특정한 형태의 특이점을 가질 수 있다. (ko) Nelle discipline matematiche della topologia, della geometria e della teoria dei gruppi, un orbifold (contrazione dell'inglese orbit-manifold, "varietà orbitale", tradotto talvolta in italiano con orbivarietà) è una generalizzazione del concetto di varietà. È uno spazio topologico (chiamato lo spazio sottostante o spazio soggiacente) con una struttura di orbifold (vedi sotto). Lo spazio sottostante somiglia localmente allo spazio quoziente di uno spazio euclideo sotto l'azione lineare di un gruppo finito. (it) Em topologia, orbivariedade (orbifold em inglês) é uma generalização do conceito de variedade diferenciável, consistindo em um espaço topológico com uma estrutura orbital definida. (pt) Орбифолд, или орбиобра́зие, — неформально говоря, это многообразие с особенностями, которые выглядят как фактор евклидова пространства по конечной группе. Один из объектов исследования в алгебраической топологии, алгебраической и дифференциальной геометрии, теории особенностей. (ru) Орбівиди (англ. Orbifold)— неформально кажучи, це многовид з особливостями, які виглядають як фактор евклідового простору за скінченною групою. Один з об'єктів дослідження в алгебричній топології, алгебричній і диференціальній геометрії, теорії особливостей. (uk) |
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| dbp:quote | This terminology should not be blamed on me. It was obtained by a democratic process in my course of 1976–77. An orbifold is something with many folds; unfortunately, the word "manifold" already has a different definition. I tried "foldamani", which was quickly displaced by the suggestion of "manifolded". After two months of patiently saying "no, not a manifold, a manifoldead," we held a vote, and "orbifold" won. (en) |
| dbp:source | explaining the origin of the word "orbifold" (en) |
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| rdfs:comment | In der Topologie ist eine Orbifaltigkeit (englisch: Orbifold) eine Verallgemeinerung einer Mannigfaltigkeit. (de) 기하학에서, 오비폴드(영어: orbifold)는 국소적으로 유한군의 선형작용에 대한 유클리드 공간의 몫공간과 동형인 위상 공간이다. 매끄러운 다양체의 개념의 일반화이며, 다양체와 달리 특정한 형태의 특이점을 가질 수 있다. (ko) Nelle discipline matematiche della topologia, della geometria e della teoria dei gruppi, un orbifold (contrazione dell'inglese orbit-manifold, "varietà orbitale", tradotto talvolta in italiano con orbivarietà) è una generalizzazione del concetto di varietà. È uno spazio topologico (chiamato lo spazio sottostante o spazio soggiacente) con una struttura di orbifold (vedi sotto). Lo spazio sottostante somiglia localmente allo spazio quoziente di uno spazio euclideo sotto l'azione lineare di un gruppo finito. (it) Em topologia, orbivariedade (orbifold em inglês) é uma generalização do conceito de variedade diferenciável, consistindo em um espaço topológico com uma estrutura orbital definida. (pt) Орбифолд, или орбиобра́зие, — неформально говоря, это многообразие с особенностями, которые выглядят как фактор евклидова пространства по конечной группе. Один из объектов исследования в алгебраической топологии, алгебраической и дифференциальной геометрии, теории особенностей. (ru) Орбівиди (англ. Orbifold)— неформально кажучи, це многовид з особливостями, які виглядають як фактор евклідового простору за скінченною групою. Один з об'єктів дослідження в алгебричній топології, алгебричній і диференціальній геометрії, теорії особливостей. (uk) En topología, orbifold (Orbidad u orbivariedad) es la generalización de una variedad diferenciable, consistente en un espacio topológico (llamado espacio subyacente) con una estructura de orbifold (véase abajo). El espacio subyacente localmente aparece como un cociente de un espacio euclídeo bajo la acción de un grupo finito de isometrías. La estructura de orbifold da una natural para las variedades abiertas en su espacio subyacente, donde cada estrato corresponde a un conjunto de puntos singulares del mismo tipo. (es) En mathématiques, un orbifold (parfois appelé aussi orbivariété) est une généralisation de la notion de variété contenant de possibles singularités. Ces espaces ont été introduits explicitement pour la première fois par Ichirō Satake en 1956 sous le nom de V-manifolds. Pour passer de la notion de variété (différentiable) à celle d'orbifold, on ajoute comme modèles locaux tous les quotients d'ouverts de par l'action de groupes finis. L'intérêt pour ces objets a été ravivé considérablement à la fin des années 70 par William Thurston en relation avec sa conjecture de géométrisation. (fr) In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. In string theory, the word "orbifold" has a slightly different meaning, discussed in detail below. In two-dimensional conformal field theory, it refers to the theory attached to the fixed point subalgebra of a vertex algebra under the action of a finite group of automorphisms. (en) |
| rdfs:label | Orbifaltigkeit (de) Orbifold (es) Orbifold (fr) Orbifold (it) 오비폴드 (ko) Orbifold (en) Orbivariedade (pt) Орбифолд (ru) Орбівид (uk) |
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