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# Mathematics

Questions in category: 子流形 (Submanifolds).

1

## [Thm]常秩映射定理

Posted by haifeng on 2015-07-24 21:19:42 last update 2015-07-24 21:19:42 | Answers (0) | 收藏

$f^{-1}(q)=\{p\in M\mid f(p)=q\}$

References:

2

## 一类与 $S^2$ 微分同胚的二维曲面

Posted by haifeng on 2015-07-24 21:10:36 last update 2015-07-24 21:30:25 | Answers (1) | 收藏

$F:\ \mathbb{R}^3\rightarrow\mathbb{R},\quad F(x,y,z)=(x^2+y^2)^2+z^2.$

3

## 图像流形(graph manifold)

Posted by haifeng on 2015-07-24 13:05:36 last update 2015-07-26 16:39:11 | Answers (1) | 收藏

$\Gamma_f=\{(p,q)\in M\times N\mid f(p)=q\}$

$\Gamma_f=\text{graph}(f)=\{(x,f(x))\in\mathbb{R}^{n+m}\mid x\in\mathbb{R}^n\}$

4

## 证明: $C^k$ 映射限制到正则子流形上仍为 $C^k$ 映射.

Posted by haifeng on 2015-07-24 11:27:56 last update 2015-07-24 11:27:56 | Answers (1) | 收藏

5

## 证明 $n$ 维环面均可嵌入到 $\mathbb{R}^{n+1}$ 中

Posted by haifeng on 2015-07-24 11:19:54 last update 2015-07-24 11:26:52 | Answers (0) | 收藏

Hint: 使用归纳法.

$T^2\times S^1\rightarrow\mathbb{R}^3\times\mathbb{R}$.

6

## $S^n$ 能否嵌入到 $\mathbb{R}^n$ 中去?

Posted by haifeng on 2015-07-22 21:55:30 last update 2015-07-22 21:55:30 | Answers (1) | 收藏

$S^n$ 能否嵌入到 $\mathbb{R}^n$ 中去?

7

## $f(t)=(e^{i2\pi t},e^{i2\pi\alpha t})$ 当 $\alpha$ 是正无理数时, 其像在 $S^1\times S^1$ 中稠密.

Posted by haifeng on 2015-07-22 10:25:25 last update 2015-07-28 13:44:46 | Answers (3) | 收藏

$f:\ \mathbb{R}\rightarrow S^1\times S^1,\quad f(t)=(e^{i2\pi t},e^{i2\pi\alpha t}).$

8

## 说明 $\{(x,y)\in\mathbb{R}^2\mid y^2=x^2(x+1)\}$ 是 $\mathbb{R}^2$ 的浸入子流形

Posted by haifeng on 2015-07-21 10:45:27 last update 2015-07-21 10:45:27 | Answers (0) | 收藏

$\{(x,y)\in\mathbb{R}^2\mid y^2=x^2(x+1)\}$

9

## 双纽线

Posted by haifeng on 2015-07-21 10:43:43 last update 2015-07-21 11:35:50 | Answers (1) | 收藏

$f(t)=(\frac{t^3+t}{t^4+1},\frac{t^3-t}{t^4+1}),\quad t\in\mathbb{R}.$

$(x^2+y^2)^2=x^2-y^2,$

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