Non-squeezing theorem (original) (raw)
En géométrie symplectique, le théorème de non-plongement de Gromov, ou théorème de non-tassement de Gromov, affirme l'impossibilité de plonger de manière symplectique une boule de rayon R dans un cylindre symplectique de rayon r<R. * Portail de la géométrie
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| dbo:abstract | The non-squeezing theorem, also called Gromov's non-squeezing theorem, is one of the most important theorems in symplectic geometry. It was first proven in 1985 by Mikhail Gromov. The theorem states that one cannot embed a ball into a cylinder via a symplectic map unless the radius of the ball is less than or equal to the radius of the cylinder. The theorem is important because formerly very little was known about the geometry behind symplectic maps. One easy consequence of a transformation being symplectic is that it preserves volume. One can easily embed a ball of any radius into a cylinder of any other radius by a volume-preserving transformation: just picture squeezing the ball into the cylinder (hence, the name non-squeezing theorem). Thus, the non-squeezing theorem tells us that, although symplectic transformations are volume-preserving, it is much more restrictive for a transformation to be symplectic than it is to be volume-preserving. (en) En géométrie symplectique, le théorème de non-plongement de Gromov, ou théorème de non-tassement de Gromov, affirme l'impossibilité de plonger de manière symplectique une boule de rayon R dans un cylindre symplectique de rayon r<R. * Portail de la géométrie (fr) 심플렉틱 기하학에서 심플렉틱 용량(symplectic容量, 영어: symplectic capacity)은 심플렉틱 다양체의 2차원 "넓이"를 측정하는 방법이다. 그로모프 조임 불가능성 정리(Громов조임不可能性定理, 영어: Gromov non-squeezing theorem)에 따라, 심플렉틱 용량이 존재한다. (ko) Теорема про симплектичного верблюда — одна з основних теорем в симплектичній геометрії. Теорема говорить, що кулю можливо вкласти в циліндр, зберігаючи природну симплектичну форму, тільки якщо радіус кулі не перевищує радіуса циліндра. (uk) Теорема о симплектическом верблюде — одна из основных теорем в симплектической геометрии.Теорема гласит, что шар возможно вложить в цилиндр сохраняя естественную симплектическую форму,только если радиус шара не превосходит радиуса цилиндра. (ru) |
| dbo:wikiPageExternalLink | https://arxiv.org/abs/1208.5969v1 http://www.math.sunysb.edu/~dusa/ewmcambrevjn23.pdf |
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| rdfs:comment | En géométrie symplectique, le théorème de non-plongement de Gromov, ou théorème de non-tassement de Gromov, affirme l'impossibilité de plonger de manière symplectique une boule de rayon R dans un cylindre symplectique de rayon r<R. * Portail de la géométrie (fr) 심플렉틱 기하학에서 심플렉틱 용량(symplectic容量, 영어: symplectic capacity)은 심플렉틱 다양체의 2차원 "넓이"를 측정하는 방법이다. 그로모프 조임 불가능성 정리(Громов조임不可能性定理, 영어: Gromov non-squeezing theorem)에 따라, 심플렉틱 용량이 존재한다. (ko) Теорема про симплектичного верблюда — одна з основних теорем в симплектичній геометрії. Теорема говорить, що кулю можливо вкласти в циліндр, зберігаючи природну симплектичну форму, тільки якщо радіус кулі не перевищує радіуса циліндра. (uk) Теорема о симплектическом верблюде — одна из основных теорем в симплектической геометрии.Теорема гласит, что шар возможно вложить в цилиндр сохраняя естественную симплектическую форму,только если радиус шара не превосходит радиуса цилиндра. (ru) The non-squeezing theorem, also called Gromov's non-squeezing theorem, is one of the most important theorems in symplectic geometry. It was first proven in 1985 by Mikhail Gromov. The theorem states that one cannot embed a ball into a cylinder via a symplectic map unless the radius of the ball is less than or equal to the radius of the cylinder. The theorem is important because formerly very little was known about the geometry behind symplectic maps. (en) |
| rdfs:label | Théorème de non-plongement de Gromov (fr) 심플렉틱 용량 (ko) Non-squeezing theorem (en) Теорема о симплектическом верблюде (ru) Теорема про симплектичного верблюда (uk) |
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