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拓扑流形

拓樸空間 来自维基百科,自由的百科全书

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数学中,拓扑流形( topological manifold )是一个“局部上看起来像是 ”的拓朴空间,是微分几何的主要研究对象。所有其他类型的流形( manifolds )都是带有额结构的拓扑流形。例如可微流形是一个带有额外的“微分结构”的拓扑流形;而光滑流形则要求这个“微分结构”要是无穷可微的。

形式定义

一个 拓扑流形(或简称流形)是一个拓扑空间 ,满足以下性质[1]

  1. 豪斯多夫空间
  2. 第二可数空间
  3. 对于每个 中的点,找的到一个该点的邻域 ,使得 同胚
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范例

  • 连续函数的图形。
  • 维的球体。
  • 射影空间
  • 环面

范例

维流形

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Projective manifolds

  • Projective spaces over the reals, complexes, or quaternions are compact manifolds.
    • Real projective space RPn is a n-dimensional manifold.
    • Complex projective space CPn is a 2n-dimensional manifold.
    • Quaternionic projective space HPn is a 4n-dimensional manifold.
  • Manifolds related to projective space include Grassmannians, flag manifolds, and Stiefel manifolds.

Other manifolds

  • Differentiable manifolds are a class of topological manifolds equipped with a differential structure.
  • Lens spaces are a class of differentiable manifolds that are quotients of odd-dimensional spheres.
  • Lie groups are a class of differentiable manifolds equipped with a compatible group structure.
  • The E8 manifold is a topological manifold which cannot be given a differentiable structure.
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参考文献

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