Parallelizable manifold (original) (raw)

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In mathematics, a differentiable manifold of dimension n is called parallelizable if there exist smooth vector fields on the manifold, such that at every point of the tangent vectorsprovide a basis of the tangent space at . Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a global section on A particular choice of such a basis of vector fields on is called a parallelization (or an absolute parallelism) of .

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dbo:abstract In mathematics, a differentiable manifold of dimension n is called parallelizable if there exist smooth vector fields on the manifold, such that at every point of the tangent vectorsprovide a basis of the tangent space at . Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a global section on A particular choice of such a basis of vector fields on is called a parallelization (or an absolute parallelism) of . (en) Une variété différentielle M de classe Ck est dite parallélisablesi son fibré tangent est trivial, c'est-à-dire isomorphe, en tant que fibré vectoriel,à , où est un espace vectoriel de dimension Il revient au même de dire qu'il existe un espace vectoriel E et une forme différentielle telle que pour tout , est un isomorphisme d'espaces vectoriels ; ou encore qu'il existe champs de vecteurs linéairement indépendants en tout point de M, autrement dit un champ de repères. Un isomorphisme de fibrés vectoriels entre et s'appelle un parallèlisme. (fr) 미분위상수학에서 평행화 가능 다양체(平行化可能多樣體, 영어: parallelizable manifold)는 그 접다발이 자명한 매끄러운 다양체이다. (ko) Параллелизуемое многообразие — многообразие размерности , допускающее поле реперов , то есть линейно независимых в каждой точке векторных полей . Поле задает изоморфизм касательного расслоения на , сопоставляющий касательному вектору его координаты относительно репера и его начало. Поэтому параллелизуемое многообразие можно также определить как многообразие, имеющее касательное расслоение. (ru) 数学中，一个 n 维光滑流形 M 为可平行化流形 是指具有向量场 V1, ..., Vn, 使得在 M 中任何一点 P 的切向量 Vi, P 组成 P 点切空间的一组基。等价地说，切丛是平凡丛，所以相伴的线性标架主丛有一个 M 的整体截面。 选取 M 上这样特定的一组向量场的基称为 M 的一个平行化或绝对平行化。 (zh)
dbo:wikiPageExternalLink https://archive.org/details/tensoranalysison00bish https://www.maths.ed.ac.uk/~v1ranick/papers/homsph.pdf
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rdfs:comment In mathematics, a differentiable manifold of dimension n is called parallelizable if there exist smooth vector fields on the manifold, such that at every point of the tangent vectorsprovide a basis of the tangent space at . Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a global section on A particular choice of such a basis of vector fields on is called a parallelization (or an absolute parallelism) of . (en) Une variété différentielle M de classe Ck est dite parallélisablesi son fibré tangent est trivial, c'est-à-dire isomorphe, en tant que fibré vectoriel,à , où est un espace vectoriel de dimension Il revient au même de dire qu'il existe un espace vectoriel E et une forme différentielle telle que pour tout , est un isomorphisme d'espaces vectoriels ; ou encore qu'il existe champs de vecteurs linéairement indépendants en tout point de M, autrement dit un champ de repères. Un isomorphisme de fibrés vectoriels entre et s'appelle un parallèlisme. (fr) 미분위상수학에서 평행화 가능 다양체(平行化可能多樣體, 영어: parallelizable manifold)는 그 접다발이 자명한 매끄러운 다양체이다. (ko) Параллелизуемое многообразие — многообразие размерности , допускающее поле реперов , то есть линейно независимых в каждой точке векторных полей . Поле задает изоморфизм касательного расслоения на , сопоставляющий касательному вектору его координаты относительно репера и его начало. Поэтому параллелизуемое многообразие можно также определить как многообразие, имеющее касательное расслоение. (ru) 数学中，一个 n 维光滑流形 M 为可平行化流形 是指具有向量场 V1, ..., Vn, 使得在 M 中任何一点 P 的切向量 Vi, P 组成 P 点切空间的一组基。等价地说，切丛是平凡丛，所以相伴的线性标架主丛有一个 M 的整体截面。 选取 M 上这样特定的一组向量场的基称为 M 的一个平行化或绝对平行化。 (zh)
rdfs:label Variété parallélisable (fr) 평행화 가능 다양체 (ko) Parallelizable manifold (en) Параллелизуемое многообразие (ru) 可平行化流形 (zh)
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