V2EX trivial bundle

Trivial Bundle

释义 Definition(中文)

trivial bundle(平凡丛/平凡纤维丛):在数学(尤其是微分几何与代数拓扑)中,指与底空间 \(B\) 和纤维 \(F\) 的直积 \(B \times F\) 同构(丛同构)的纤维丛。直观地说,它“没有扭结”,整体结构就是把同一个纤维 \(F\) 在每个点上平铺到 \(B\) 上。
(注:在不同语境下也可指平凡向量丛 trivial vector bundle。)

发音 Pronunciation(IPA)

/trvil bndl/

例句 Examples

A trivial bundle looks like a product space \(B \times F\).
平凡丛看起来就像直积空间 \(B \times F\)。

Over a contractible base space, every vector bundle is isomorphic to a trivial bundle.
在可缩的底空间上,每个向量丛都同构于一个平凡丛。

词源 Etymology(中文)

trivial 源自拉丁语 trivialis(“三岔路口的、常见的”),后来引申为“普通的、不复杂的”。在数学里常用来表示“没有额外结构/没有扭曲、可视为最简单情形”。bundle 原意为“捆、束”,在数学中译作“丛”,强调“把许多纤维像一束一样组织在底空间之上”的结构。

相关词 Related Words

文献与作品 Literary / Notable Works

  • Allen Hatcher, Algebraic Topology(讨论纤维丛与相关构造时常用到“trivial bundle/平凡丛”作为基本对照例)
  • John W. Milnor & James D. Stasheff, Characteristic Classes(特征类与向量丛中频繁出现“trivial bundle”)
  • Raoul Bott & Loring W. Tu, Differential Forms in Algebraic Topology(涉及丛、截面与平凡化时出现该术语)
  • Loring W. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes(讲连接与特征类时以平凡丛作基础例)
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