流形边界 (boundary of a manifold)

带边流形的边界：半空间坐标下最后一坐标为0的点；不同于任意子集的闭包减内部，也不同于链群中∂的像。

别名：boundary；manifold boundary；边界

词性：名词

已登记的原文定义

Nakahara · second-edition-local · §5.1.2 · PDF 第192页

Before we give examples, we briefly comment on manifolds with boundaries. So far, we have assumed that the coordinate neighbourhood  $U_{i}$  is homeomorphic to an open set of  $\mathbb{R}^m$ . In some applications, however, this turns out to be too restrictive and we need to relax this condition. If a topological space  $M$  is covered by a family of open sets  $\{U_i\}$  each of which is homeomorphic to an open set of  $H^{m} \equiv \{(x^{1},\ldots ,x^{m}) \in \mathbb{R}^{m} | x^{m} \geq 0\}$ ,  $M$  is said to be a manifold with a boundary, see figure 5.3. The set of points which are mapped to points with  $x^{m} = 0$  is called the boundary of  $M$ , denoted by  $\partial M$ . The coordinates of  $\partial M$  may be given by  $m - 1$  numbers  $(x^{1},\dots,x^{m - 1},0)$ . Now we have to be careful when we define the smoothness. The map  $\psi_{ij} : \varphi_j(U_i \cap U_j) \to \varphi_i(U_i \cap U_j)$  is defined on an open set of  $H^{m}$  in general, and  $\psi_{ij}$  is said to be smooth if it is  $C^\infty$  in an open set of  $\mathbb{R}^m$  which contains  $\varphi_j(U_i \cap U_j)$ . Readers are encouraged to use their imagination since our definition is in harmony with our intuitive notions about boundaries. For example, the boundary of the solid ball  $D^3$  is the sphere  $S^2$  and the boundary of the sphere is an empty set.

带边流形的边界：半空间坐标下最后一坐标为0的点；不同于任意子集的闭包减内部，也不同于链群中∂的像。

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阅读原文位置 · zh-CN

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以下按已登记位置排列。出现顺序、原文引用与已登记依赖分别展示；最早位置不代表全书首次定义。

Nakahara · second-edition-local

已登记出处中的最早位置：§5.1 · zh-CN。

位置	语言	登记角色
§5.1	zh-CN · 英文上下文	定义入口
§11.1	zh-CN · 英文上下文	讨论
§11.5	zh-CN · 英文上下文	讨论
§12.3	zh-CN · 英文上下文	讨论
§12.4	zh-CN · 英文上下文	讨论
§12.8	zh-CN · 英文上下文	讨论
§13.4	zh-CN · 英文上下文	讨论
§13.5	zh-CN · 英文上下文	讨论
§13.6	zh-CN · 英文上下文	讨论

来源短摘录与定位

§5.1 · zh-CN · machine_draft · unreviewed