V2EX diffeomorphism

Diffeomorphism

定义 Definition

diffeomorphism(微分同胚):在微分几何与拓扑中,指两个光滑流形之间的一个双射映射,它本身是光滑(可无限次微分)的,并且它的逆映射也同样光滑。直观上,它表示一种“可平滑拉伸/弯曲但不撕裂、不粘合”的一一对应变形。(该词也有更一般的“可微同胚”用法,但最常见指光滑情形。)

发音 Pronunciation (IPA)

/dfimrfzm/

词源 Etymology

来自 Greek 词根组合:**di-**(表示“两个/双重”)+ morphē(“形状”)+ -ism(名词后缀);并与数学中的 differential(微分的)概念结合,形成表示“在微分意义下保持结构的一一对应变形”的术语。

例句 Examples

A circle is diffeomorphic to an ellipse.
圆与椭圆在微分意义下是同胚的(它们之间存在微分同胚)。

If two manifolds are related by a diffeomorphism, then many smooth geometric properties can be transferred from one to the other.
如果两个流形通过一个微分同胚联系起来,那么许多光滑的几何性质就可以从一个“搬运”到另一个。

相关词 Related Words

文学与典籍 Literary Works

  • John M. Lee, Introduction to Smooth Manifolds(多处用于定义与讨论流形之间的微分同胚)
  • Victor Guillemin & Alan Pollack, Differential Topology(以 diffeomorphism 为核心概念之一)
  • Morris W. Hirsch, Differential Topology(系统使用 diffeomorphism 讨论分类与结构)
  • John Milnor, Topology from the Differentiable Viewpoint(以可微/光滑观点讨论拓扑主题,频繁出现该词)
  • Michael Spivak, Calculus on Manifolds(在光滑映射与坐标变换语境中出现)
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