Differential topology (original) (raw)
الطوبولوجيا التفاضلية (بالإنجليزية: Differential topology) هي المجال الذي يتعامل مع الوظائف المختلفة على المشعبات المختلفة. يرتبط ارتباطا وثيقا بالهندسة التفاضلية ويكوّن معاً النظرية الهندسية للمشعبات المختلفة.
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| dbo:abstract | Dins l'entorn de la matemàtica, la topologia diferencial és una branca de coneixements que considera les varietats diferenciables i les funcions entre elles. Estudia les possibles que les varietats poden portar. És un ciència adjacent a la geometria diferencial. La topologia diferencial fa servir una de les principals eines de la teoria d'intersecció: la , per establir els seus principals resultats. Algunes de les qüestions que aquesta ciència tracta de respondre són: * Quantes estructures diferenciables té una 2-varietat? ¿I una 3-varietat? * Pot una certa varietat diferenciable ser embedded (de l'anglès: embedded ) en una altra? * Si dos varietats diferenciables són homeomorfes són difeomorfes? * Quines varietats diferenciables són frontera de varietats compactes? (ca) الطوبولوجيا التفاضلية (بالإنجليزية: Differential topology) هي المجال الذي يتعامل مع الوظائف المختلفة على المشعبات المختلفة. يرتبط ارتباطا وثيقا بالهندسة التفاضلية ويكوّن معاً النظرية الهندسية للمشعبات المختلفة. (ar) Στα μαθηματικά, η διαφορική τοπολογία είναι το πεδίο που ασχολείται με σε . Είναι στενά συνδεδεμένη με τη διαφορική γεωμετρία και μαζί συνθέτουν τη γεωμετρική θεωρία των διαφοροποιήσιμων πολλαπλοτήτων. (el) In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the geometric properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its homotopy type, or the structure of its diffeomorphism group. Because many of these coarser properties may be captured algebraically, differential topology has strong links to algebraic topology. The central goal of the field of differential topology is the classification of all smooth manifolds up to diffeomorphism. Since dimension is an invariant of smooth manifolds up to diffeomorphism type, this classification is often studied by classifying the (connected) manifolds in each dimension separately: * In dimension 1, the only smooth manifolds up to diffeomorphism are the circle, the real number line, and allowing a boundary, the half-closed interval and fully closed interval . * In dimension 2, every closed surface is classified up to diffeomorphism by its genus, the number of holes (or equivalently its Euler characteristic), and whether or not it is orientable. This is the famous classification of closed surfaces. Already in dimension two the classification of non-compact surfaces becomes difficult, due to the existence of exotic spaces such as Jacob's ladder. * In dimension 3, William Thurston's geometrization conjecture, proven by Grigori Perelman, gives a partial classification of compact three-manifolds. Included in this theorem is the Poincaré conjecture, which states that any closed, simply connected three-manifold is homeomorphic (and in fact diffeomorphic) to the 3-sphere. Beginning in dimension 4, the classification becomes much more difficult for two reasons. Firstly, every finitely presented group appears as the fundamental group of some 4-manifold, and since the fundamental group is a diffeomorphism invariant, this makes the classification of 4-manifolds at least as difficult as the classification of finitely presented groups. By the word problem for groups, which is equivalent to the halting problem, it is impossible to classify such groups, so a full topological classification is impossible. Secondly, beginning in dimension four it is possible to have smooth manifolds that are homeomorphic, but with distinct, non-diffeomorphic smooth structures. This is true even for the Euclidean space , which admits many exotic structures. This means that the study of differential topology in dimensions 4 and higher must use tools genuinely outside the realm of the regular continuous topology of topological manifolds. One of the central open problems in differential topology is the four-dimensional smooth Poincaré conjecture, which asks if every smooth 4-manifold that is homeomorphic to the 4-sphere, is also diffeomorphic to it. That is, does the 4-sphere admit only one smooth structure? This conjecture is true in dimensions 1, 2, and 3, by the above classification results, but is known to be false in dimension 7 due to the Milnor spheres. Important tools in studying the differential topology of smooth manifolds include the construction of smooth topological invariants of such manifolds, such as de Rham cohomology or the intersection form, as well as smoothable topological constructions, such as smooth surgery theory or the construction of cobordisms. Morse theory is an important tool which studies smooth manifolds by considering the critical points of differentiable functions on the manifold, demonstrating how the smooth structure of the manifold enters into the set of tools available. Often times more geometric or analytical techniques may be used, by equipping a smooth manifold with a Riemannian metric or by studying a differential equation on it. Care must be taken to ensure that the resulting information is insensitive to this choice of extra structure, and so genuinely reflects only the topological properties of the underlying smooth manifold. For example, the Hodge theorem provides a geometric and analytical interpretation of the de Rham cohomology, and gauge theory was used by Simon Donaldson to prove facts about the intersection form of simply connected 4-manifolds. In some cases techniques from contemporary physics may appear, such as topological quantum field theory, which can be used to compute topological invariants of smooth spaces. Famous theorems in differential topology include the Whitney embedding theorem, the hairy ball theorem, the Hopf theorem, the Poincaré–Hopf theorem, Donaldson's theorem, and the Poincaré conjecture. (en) Die Differentialtopologie ist ein Teilgebiet der Mathematik, das globale geometrische Invarianten untersucht, die nicht durch eine Metrik oder eine symplektische Form definiert werden. Die untersuchten Invarianten sind meist Invarianten topologischer Räume, die zusätzlich eine differenzierbare Struktur tragen, also von differenzierbaren Mannigfaltigkeiten. Beispielsweise stellt die De-Rham-Kohomologie eine Verbindung zwischen analytischen Eigenschaften und topologischen Invarianten der Mannigfaltigkeit her. Oft werden Mittel der Analysis und der Theorie der Differentialgleichungen benutzt, um über die Topologie des Raumes Auskunft zu bekommen. Dies geschieht beispielsweise in der Morse-Theorie oder der aus der Physik kommenden Yang-Mills-Theorie. Letztere führt zu sogenannten exotischen R4s, d. h. vierdimensionalen euklidischen Räumen, die zwar homöomorph, aber nicht diffeomorph zum Standard-R4 sind. Solche exotischen Räume kommen erst ab Dimension vier vor. Ein anderes bekanntes Beispiel sind Milnors exotische 7-Sphären. Ihre Entdeckung 1956 stellte einen entscheidenden Wendepunkt in der Differentialtopologie dar. Wegbereiter der modernen Differentialtopologie sind Bernhard Riemann und Henri Poincaré. Wichtige Vertreter im 20. Jahrhundert sind Hassler Whitney, John Willard Milnor und Simon Donaldson. Jüngere Entwicklungen haben Verbindungen zur Physik aufgezeigt, für die vor allem der String-Theoretiker und Fields-Medaille-Träger Edward Witten steht. (de) En matemáticas, la topología diferencial es una rama de conocimientos que considera las variedades diferenciables y a las funciones diferenciables entre ellas. Estudia las posibles estructuras diferenciables que las variedades pueden portar. Es una ciencia adyacente a la geometría diferencial. La topología diferencial usa una de las principales herramientas de la teoría de intersección: transversalidad, para establecer sus principales resultados. Algunas de las cuestiones que esta ciencia trata de responder son: * ¿Cuántas estructuras diferenciables tiene una 2-variedad? ¿Y una 3-variedad? * ¿Puede una cierta variedad diferenciable ser encajada (del inglés: embedded) en otra? * Si dos variedades diferenciables son homeomorfas ¿son difeomorfas? * ¿Qué variedades diferenciables son frontera de variedades compactas? (es) La topologie différentielle est une branche des mathématiques qui étudie les fonctions différentiables définies sur des variétés différentielles, ainsi que les applications différentiables entre variétés différentielles. Elle est reliée à la géométrie différentielle, discipline avec laquelle elle se conjugue pour construire une théorie géométrique des variétés différentiables. (fr) 微分位相幾何学もしくは微分トポロジー（英語：differential topology）は、多様体のに注目する幾何学の一分野。という位相のみでは決まらないものを扱うため純粋な位相幾何学として扱うのは難しい部分もあるが,位相が与えられている多様体のつまり微積分ができるような構造を調べるということで位相多様体を調べるもので,まで込めた多様体に距離や曲率を定めて研究を行う微分幾何学に比べ自由度は高いことから位相幾何学であるとされている。解析学や微分幾何学と位相幾何学の学際研究が非常に有益なことは初期から知られており、局所的な性質を扱う微分幾何学と大域的な性質を扱う位相幾何学の対照的な2分野による多様体の研究は双方の発展を促した。古くはフェリックス・クラインやアンリ・ポアンカレまで遡れ、現在微分位相幾何学と呼ばれているものはルネ・トムやジョン・ミルナーといった数学者によって創り出された。 (ja) In matematica, la topologia differenziale è una parte della topologia che usa gli strumenti del calcolo infinitesimale. L'oggetto principalmente studiato è la varietà differenziabile, una generalizzazione a più dimensioni delle curve e delle superfici. La geometria differenziale è un settore contiguo ed in parte sovrapposto, che studia le varietà da un punto di vista più "rigido": in geometria differenziale si introducono e studiano concetti geometrici come quello di angolo, distanza, geodetica, curvatura, che non sono presenti in topologia. Parallelamente, la topologia algebrica e la geometria algebrica applicano gli strumenti dell'algebra alla topologia e alla geometria. In molti casi, l'uso dell'algebra e del calcolo infinitesimale danno risultati analoghi, benché espressi con formalismi completamente diversi. (it) 미분위상수학(微分位相數學, 영어: differential topology)은 매끄러운 다양체의 위상수학적 성질을 연구하는 위상수학의 한 분과이다. 미분기하학과 밀접한 관계를 다루지만, 미분기하학과 달리 미분 동형에 대하여 불변인 성질들을 주로 다룬다. (ko) Differentiaaltopologie onderzoekt eigenschappen van "gladde" ruimten die ongewijzigd blijven bij "gladde" (dat wil zeggen onbeperkt differentieerbare) vervormingen. Ze onderscheidt zich daarmee enerzijds van de differentiaalmeetkunde, die slechts afstandbewarende transformaties (isometrieën) toelaat, en anderzijds de topologie, die willekeurige vervormingen (mits continu en bijectief) toelaat. Het basisobject is een -dimensionale differentieerbare variëteit, ruwweg gezegd: een puntenverzameling die plaatselijk in kaart wordt gebracht door reële coördinaten, waarbij de kaartentransformaties in overlappingsgebieden onbeperkt differentieerbaar zijn. Een afbeelding tussen twee differentieerbare variëteiten wordt altijd stilzwijgend onbeperkt differentieerbaar ondersteld (net zoals een afbeelding tussen twee topologische ruimten geacht wordt continu te zijn). Zo'n afbeelding heet een diffeomorfisme als ze bijectief is en haar inverse ook differentieerbaar is; de twee differentieerbare variëteiten zijn dan voor de differentiaaltopologie in wezen gelijkwaardig en men zegt dat zij diffeomorf zijn. Zo is het oppervlak van een ellipsoïde diffeomorf met een boloppervlak (een diffeomorfisme kan gegeven worden door centrale projectie vanuit het middelpunt), maar is niet daaraan isometrisch, tenzij het al een boloppervlak is. Met deze begrippen laat de eerste alinea zich ook wat preciseren:differentiaaltopologie onderzoekt alleen die eigenschappen van differentieerbare variëteiten die invariant zijn onder diffeomorfie, wat betekent dat zodra een differentieerbare variëteit de eigenschap in kwestie heeft, iedere daarmee diffeomorfe variëteit hem ook heeft. Veel stellingen uit de differentiaaltopologie luiden analoog met stellingen uit de topologie, maar zijn vaak eenvoudiger te bewijzen. Dat geldt onder meer voor de : Iedere compacte tweedimensionale differentieerbare variëteit is diffeomorf met een oppervlak van (boloppervlak met handvatten) of een projectief vlak met handvatten (hier mag ). Niettemin springen de in het oog, dit zijn differentieerbare structuren op klassieke topologische variëteiten (bv. de 7-sfeer of ) die topologisch niet te onderscheiden zijn van hun "klassieke" tegenhanger, maar waarvan de differentieerbare structuur essentieel verschillend is. Dus een homeomorfisme tussen die twee zal nooit een diffeomorfisme kunnen zijn. (nl) Topologia różniczkowa – dział topologii korzystający z pojęć i metod analizy matematycznej; zajmuje się zwłaszcza rozmaitościami różniczkowymi i różniczkowymi odwzorowaniami, w szczególności dyfeomorfizmami, zanurzeniami różniczkowymi i wiązkami wektorowymi. (pl) Em matemática, a topologia diferencial é a área dedicada ao estudo das funções diferenciáveis sobre variedades diferenciáveis. Ela é bastante relacionada à geometria diferencial e em conjunto elas constituem a teoria geométrica das variedades diferenciáveis. (pt) Диференціальна топологія є розділом математики, в якому досліджуються диференційовані функції на диференційованих многовидах. Вона тісно пов'язана з диференціальною геометрією і разом вони складають геометричну теорію диференційованих многовидів. (uk) 微分拓撲是一個处理在微分流形上的可微函数的数学领域。很自然地，它是在研究微分方程理論的过程中被提出來的。微分幾何是用微積分來研究幾何的学问。这些领域非常接近，在物理学，特别在相对论方面有许多的应用。它们合在一起还建立了可从动力系统观点直接研究的、可微流形的几何理论。 (zh) |
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| rdfs:comment | الطوبولوجيا التفاضلية (بالإنجليزية: Differential topology) هي المجال الذي يتعامل مع الوظائف المختلفة على المشعبات المختلفة. يرتبط ارتباطا وثيقا بالهندسة التفاضلية ويكوّن معاً النظرية الهندسية للمشعبات المختلفة. (ar) Στα μαθηματικά, η διαφορική τοπολογία είναι το πεδίο που ασχολείται με σε . Είναι στενά συνδεδεμένη με τη διαφορική γεωμετρία και μαζί συνθέτουν τη γεωμετρική θεωρία των διαφοροποιήσιμων πολλαπλοτήτων. (el) La topologie différentielle est une branche des mathématiques qui étudie les fonctions différentiables définies sur des variétés différentielles, ainsi que les applications différentiables entre variétés différentielles. Elle est reliée à la géométrie différentielle, discipline avec laquelle elle se conjugue pour construire une théorie géométrique des variétés différentiables. (fr) 微分位相幾何学もしくは微分トポロジー（英語：differential topology）は、多様体のに注目する幾何学の一分野。という位相のみでは決まらないものを扱うため純粋な位相幾何学として扱うのは難しい部分もあるが,位相が与えられている多様体のつまり微積分ができるような構造を調べるということで位相多様体を調べるもので,まで込めた多様体に距離や曲率を定めて研究を行う微分幾何学に比べ自由度は高いことから位相幾何学であるとされている。解析学や微分幾何学と位相幾何学の学際研究が非常に有益なことは初期から知られており、局所的な性質を扱う微分幾何学と大域的な性質を扱う位相幾何学の対照的な2分野による多様体の研究は双方の発展を促した。古くはフェリックス・クラインやアンリ・ポアンカレまで遡れ、現在微分位相幾何学と呼ばれているものはルネ・トムやジョン・ミルナーといった数学者によって創り出された。 (ja) 미분위상수학(微分位相數學, 영어: differential topology)은 매끄러운 다양체의 위상수학적 성질을 연구하는 위상수학의 한 분과이다. 미분기하학과 밀접한 관계를 다루지만, 미분기하학과 달리 미분 동형에 대하여 불변인 성질들을 주로 다룬다. (ko) Topologia różniczkowa – dział topologii korzystający z pojęć i metod analizy matematycznej; zajmuje się zwłaszcza rozmaitościami różniczkowymi i różniczkowymi odwzorowaniami, w szczególności dyfeomorfizmami, zanurzeniami różniczkowymi i wiązkami wektorowymi. (pl) Em matemática, a topologia diferencial é a área dedicada ao estudo das funções diferenciáveis sobre variedades diferenciáveis. Ela é bastante relacionada à geometria diferencial e em conjunto elas constituem a teoria geométrica das variedades diferenciáveis. (pt) Диференціальна топологія є розділом математики, в якому досліджуються диференційовані функції на диференційованих многовидах. Вона тісно пов'язана з диференціальною геометрією і разом вони складають геометричну теорію диференційованих многовидів. (uk) 微分拓撲是一個处理在微分流形上的可微函数的数学领域。很自然地，它是在研究微分方程理論的过程中被提出來的。微分幾何是用微積分來研究幾何的学问。这些领域非常接近，在物理学，特别在相对论方面有许多的应用。它们合在一起还建立了可从动力系统观点直接研究的、可微流形的几何理论。 (zh) Dins l'entorn de la matemàtica, la topologia diferencial és una branca de coneixements que considera les varietats diferenciables i les funcions entre elles. Estudia les possibles que les varietats poden portar. És un ciència adjacent a la geometria diferencial. La topologia diferencial fa servir una de les principals eines de la teoria d'intersecció: la , per establir els seus principals resultats. Algunes de les qüestions que aquesta ciència tracta de respondre són: (ca) In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the geometric properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its homotopy type, or the structure of its diffeomorphism group. Because many of these coarser properties may be captured algebraically, differential topology has strong links to algebraic topology. (en) En matemáticas, la topología diferencial es una rama de conocimientos que considera las variedades diferenciables y a las funciones diferenciables entre ellas. Estudia las posibles estructuras diferenciables que las variedades pueden portar. Es una ciencia adyacente a la geometría diferencial. La topología diferencial usa una de las principales herramientas de la teoría de intersección: transversalidad, para establecer sus principales resultados. Algunas de las cuestiones que esta ciencia trata de responder son: (es) Die Differentialtopologie ist ein Teilgebiet der Mathematik, das globale geometrische Invarianten untersucht, die nicht durch eine Metrik oder eine symplektische Form definiert werden. Die untersuchten Invarianten sind meist Invarianten topologischer Räume, die zusätzlich eine differenzierbare Struktur tragen, also von differenzierbaren Mannigfaltigkeiten. (de) In matematica, la topologia differenziale è una parte della topologia che usa gli strumenti del calcolo infinitesimale. L'oggetto principalmente studiato è la varietà differenziabile, una generalizzazione a più dimensioni delle curve e delle superfici. La geometria differenziale è un settore contiguo ed in parte sovrapposto, che studia le varietà da un punto di vista più "rigido": in geometria differenziale si introducono e studiano concetti geometrici come quello di angolo, distanza, geodetica, curvatura, che non sono presenti in topologia. (it) Differentiaaltopologie onderzoekt eigenschappen van "gladde" ruimten die ongewijzigd blijven bij "gladde" (dat wil zeggen onbeperkt differentieerbare) vervormingen. Ze onderscheidt zich daarmee enerzijds van de differentiaalmeetkunde, die slechts afstandbewarende transformaties (isometrieën) toelaat, en anderzijds de topologie, die willekeurige vervormingen (mits continu en bijectief) toelaat. Veel stellingen uit de differentiaaltopologie luiden analoog met stellingen uit de topologie, maar zijn vaak eenvoudiger te bewijzen. Dat geldt onder meer voor de : (nl) |
| rdfs:label | Differential topology (en) طوبولوجيا تفاضلية (ar) Topologia diferencial (ca) Differentialtopologie (de) Διαφορική τοπολογία (el) Topología diferencial (es) Topologie différentielle (fr) Topologia differenziale (it) 微分位相幾何学 (ja) 미분위상수학 (ko) Differentiaaltopologie (nl) Topologia różniczkowa (pl) Topologia diferencial (pt) Дифференциальная топология (ru) Диференціальна топологія (uk) 微分拓扑 (zh) |
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