复流形

本书暂定概念：尚未进入正式概念库。下面是本书给出的定义原文。

定义 1 · 8 复流形

可微流形是容许可微结构的拓扑空间。这里我们再介绍一种在物理学中有意义的结构。在初等复分析中，偏导数须满足柯西-黎曼关系。此时我们不仅谈论函数的可微性，还谈论它的解析性。复流形(complex manifold)容许一种复结构，在其中每个坐标邻域都同胚于 ${\mathbb{C}}^n$，并且从一个坐标系到另一个坐标系的转换是解析的。

英文原文

A differentiable manifold is a topological space which admits differentiable structures. Here we introduce another structure which has relevance in physics. In elementary complex analysis, the partial derivatives are required to satisfy the Cauchy-Riemann relations. We talk not only of the differentiability but also of the analyticity of a function in this case. A complex manifold admits a complex structure in which each coordinate neighbourhood is homeomorphic to ${\mathbb{C}}^n$ and the transition from one coordinate system to the other is analytic.

出处：8 复流形 · node a-18e2d1a6f347c6613f51293d · 源节点 SHA-256 130359270a2fde62…

定义 2 · 定义 8.1

若下列公理成立，则 $M$ 称为复流形(complex manifold)：

(i) $M$ 是拓扑空间。
(ii) $M$ 上给定了若干对 $\{(U_i,{\phi }_i)\}$ 构成的一族。
(iii) $\{U_i\}$ 是一族覆盖 $M$ 的开集。映射 ${\phi }_i$ 是从 $U_i$ 到 ${\mathbb{C}}^m$ 的某个开子集 $U$ 的同胚。[因此，$M$ 是偶数维的。]
(iv) 若 $U_i$ 与 $U_j$ 满足 $U_i\cap U_j\ne \varnothing$，则从 ${\phi }_i(U_i\cap U_j)$ 到 ${\phi }_j(U_i\cap U_j)$ 的映射 ${\psi }_{ji}={\phi }_j\circ {\phi }_i^{-1}$ 是全纯的。

8

数 $m$ 称为 $M$ 的复维数，记作 ${\dim }_{\mathbb{C}}M=m$。实维数 $2m$ 记作 ${\dim }_{\mathbb{R}}M$，或简记为 $\dim M$。设 $z^{\mu }={\phi }_i(p)$ 与 $w^{\nu }={\phi }_j(p)$ 分别是点 $p\in U_i\cap U_j$ 在坐标卡 $(U_i,{\phi }_i)$ 与 $(U_j,{\phi }_j)$ 中的（复）坐标。公理 (iv) 断言函数 $w^{\nu }=u^{\nu }+\mathrm{i}{v}^{\nu }$（$1\le \nu \le m$）关于 $z^{\mu }=x^{\mu }+\mathrm{i}{y}^{\mu }$ 全纯，即

$$\frac{\partial u^v}{\partial x^v}=\frac{\partial v^v}{\partial y^v}\frac{\partial u^v}{\partial y^v}=-\frac{\partial v^v}{\partial x^v}1\le \mu ,v\le m.$$

这些公理保证复流形上的微积分运算可以独立于所选取的特殊坐标来进行。例如，${\mathbb{C}}^n$ 是最简单的复流形：单个坐标卡就覆盖了整个空间，且 $\phi$ 是恒等映射。

设 $\{(U_i,{\phi }_i)\}$ 与 $\{(V_j,{\psi }_j)\}$ 是 $M$ 的两个图册。若两个图册的并仍是一个满足定义 8.1各公理的图册，则称它们定义了同一个复结构。一个复流形可以带有若干个复结构（见例 8.2）。

英文原文

$M$ is a complex manifold if the following axioms hold,

(i) $M$ is a topological space.
(ii) $M$ is provided with a family of pairs $\{(U_i,{\phi }_i)\}$ .
(iii) $\{U_i\}$ is a family of open sets which covers $M$ . The map ${\phi }_i$ is a homeomorphism from $U_i$ to an open subset $U$ of ${\mathbb{C}}^m$ . [Hence, $M$ is even dimensional.]
(iv) Given $U_i$ and $U_j$ such that $U_i\cap U_j\ne \varnothing$ , the map ${\psi }_{ji}={\phi }_j\circ {\phi }_i^{-1}$ from ${\phi }_i(U_i\cap U_j)$ to ${\phi }_j(U_i\cap U_j)$ is holomorphic.

8

The number $m$ is called the complex dimension of $M$ and is denoted as ${\dim }_{\mathbb{C}}M=m$ . The real dimension $2m$ is denoted either by ${\dim }_{\mathbb{R}}M$ or simply by $\dim M$ . Let $z^{\mu }={\phi }_i(p)$ and $w^{\nu }={\phi }_j(p)$ be the (complex) coordinates of a point $p\in U_i\cap U_j$ in the charts $(U_i,{\phi }_i)$ and $(U_j,{\phi }_j)$ , respectively. Axiom (iv) asserts that the function $w^{\nu }=u^{\nu }+\mathrm{i}{v}^{\nu }$ ( $1\le \nu \le m$ ) is holomorphic in $z^{\mu }=x^{\mu }+\mathrm{i}{y}^{\mu }$ , namely

$$\frac{\partial u^v}{\partial x^v}=\frac{\partial v^v}{\partial y^v}\frac{\partial u^v}{\partial y^v}=-\frac{\partial v^v}{\partial x^v}1\le \mu ,v\le m.$$

These axioms ensure that calculus on complex manifolds can be carried out independently of the special coordinates chosen. For example, ${\mathbb{C}}^n$ is the simplest complex manifold. A single chart covers the whole space and $\phi$ is the identity map.

Let $\{(U_i,{\phi }_i)\}$ and $\{(V_j,{\psi }_j)\}$ be atlases of $M$ . If the union of two atlases is again an atlas which satisfies the axioms of definition 8.1, they are said to define the same complex structure. A complex manifold may carry a number of complex structures (see example 8.2).

出处：8.1 复流形 · node a-adba1a1991a4cfbb4410981a · 源节点 SHA-256 0b70cb5ddd5d133d…

定义 3 · 14.1 黎曼面上的微分几何

这就验证了 ${\Sigma }_g$ 是一个复流形(complex manifold)。此外还有

$${\mathrm{e}}^{2\sigma (z,\bar{z})}={\mathrm{e}}^{2{\sigma }^{\prime }(w,\bar{w})}|\partial w/\partial z{|}^2.\text{\hspace{1em}(14.9)}$$

英文原文

which verifies that ${\Sigma }_g$ is a complex manifold. We also have

$${\mathrm{e}}^{2\sigma (z,\bar{z})}={\mathrm{e}}^{2{\sigma }^{\prime }(w,\bar{w})}|\partial w/\partial z{|}^2.\text{\hspace{1em}(14.9)}$$

出处：14.1 黎曼面上的微分几何 · node a-e5dcaf9b1fd8efc564b6fe05 · 源节点 SHA-256 3a4215ce1bea4da5…