Nakahara Definition 5.3：子流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
词性：名词
已登记的原文定义
Nakahara · second-edition-local · §5.2.7 · PDF 第207页
Definition 5.3. (Immersion, submanifold, embedding) Let $f: M \to N$ be a smooth map and let $\dim M \leq \dim N$ .Nakahara Definition 5.3：子流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §5.2.7 · PDF 第207页
(a) The map $f$ is called an immersion of $M$ into $N$ if $f_{*}: T_{p}M \to T_{f(p)}N$ is an injection (one to one), that is $\operatorname{rank} f_{*} = \dim M$ .
(b) The map $f$ is called an **embedding** if $f$ is an injection and an immersion. The image $f(M)$ is called a **submanifold** of $N$ . [In practice, $f(M)$ thus defined is diffeomorphic to $M$ .]Nakahara Definition 5.3：子流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §5.2.7 · PDF 第207页 · 定义续文（环境外解释或示例）
If $f$ is an immersion, $f^{*}$ maps $T_{p}M$ isomorphically to an $m$ -dimensional vector subspace of $T_{f(p)}N$ since $\mathrm{rank}f_{*} = \dim M$ . From theorem 2.1, we also find $\ker f_{*} = \{0\}$ . If $f$ is an embedding, $M$ is diffeomorphic to $f(M)$ . Examples will clarify these rather technical points. Consider a map $f: S^1 \to \mathbb{R}^2$ in figure 5.11(a). It is an immersion since a one-dimensional tangent space of $S^1$ is mapped by $f_{*}$ to a subspace of $T_{f(p)}\mathbb{R}^2$ . The image $f(S^1)$ is not a submanifold of $\mathbb{R}^2$ since $f$ is not an injection. The map $g: S^1 \to \mathbb{R}^2$ in figure 5.11(b) is an embedding and $g(S^1)$ is a submanifold of $\mathbb{R}^2$ . Clearly, an embedding is an immersion although the converse is not necessarily true. In the previous section, we occasionally mentioned the embedding of $S^n$ into $\mathbb{R}^{n + 1}$ . Now this meaning is clear; if $S^n$ is embedded by $f: S^n \to \mathbb{R}^{n + 1}$ then $S^n$ is diffeomorphic to $f(S^n)$ .Nakahara Definition 5.3：子流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
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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：§5.2 · zh-CN。
来源短摘录与定位
§5.2 · zh-CN · machine_draft · unreviewed
在结束本节之前，我们给出流形的子流形(submanifold)的定义，并在此澄清嵌入的含义。
§5.2 · zh-CN · machine_draft · unreviewed
正文请在原位置阅读；长段落、数学边界或未定位引用不截断嵌入，精确范围与指纹保存在结构化索引。
§5.2 · zh-CN · machine_draft · unreviewed
正文请在原位置阅读；长段落、数学边界或未定位引用不截断嵌入，精确范围与指纹保存在结构化索引。
§7.1 · zh-CN · machine_draft · unreviewed
正文请在原位置阅读；长段落、数学边界或未定位引用不截断嵌入，精确范围与指纹保存在结构化索引。
原文中已定位的引用（不自动视为依赖）
- href=“02-section-5-2.md#definition-5-3”
- href=”../chapter-02/01-section-2-1.md#bound-definition-dc76bb871b2cda8079b57014”
- href=“02-section-5-2.md#bound-definition-e491da5cce081c4491486e21”
- href=“02-section-5-2.md#bound-definition-2421240c344eac78ae5e2266”
- href=”../chapter-05/02-section-5-2.md#bound-definition-2421240c344eac78ae5e2266”
- href=”../chapter-05/02-section-5-2.md#bound-definition-e491da5cce081c4491486e21”
- href=”../chapter-05/02-section-5-2.md#section-5-2”