文章与日志 - 流形
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en流形的基本概念|Basic Notions of Manifold
http://krsna.lamost.org/engine/node/729
<p><p>Basic Notions of Manifold</p>
<p>When we mention Euclid space, maybe it induct excess meanings. So,<br />
we can try to construct an object that its meaning is less than<br />
Euclid space but however, is meaningful and analyzable.</p>
<p>At least, we need some points, then, if we want a point is<br />
analyzable, it must has its neighborhood, moreover, to make it<br />
computable or realizable, let its neighborhood homeomorphism with<br />
an open set of Euclid space.</p>
<p>Such points constitutes a set, then make this set is a separated<br />
set, a Hausdorff set, we definite such a set a <b>manifold</b>.</p>
<p>The point `x\in M`, M is a manifold. U is a neighborhood of x, and<br />
there is a homeomorphism `\phi:U\mapsto \phi(U)`. then, Any such a<br />
$(U,\phi(U))$ means a realization of any point of U: $\phi(y)$ is<br />
the coordination of $y\in U$.</p>
<p>So, we definite $(U, \phi(U))$ a <b>map chart</b> of M.</p>
<p>Because any point of a manifold has its neighborhood as well as an<br />
homeomorphism, then for any pair of points (x, y), their<br />
neighborhood (U, V) have intersection or have not:</p>
<p>i) if $U\cap V\neq\theta$, then $\phi(U)\circ\varphi(V)^{-1}$ and<br />
$\varphi(V)\circ\phi(U)^{-1}$ are two real functions from an open<br />
set to another of Euclid space. we can treat them as coordinations<br />
transformation, we can let them be $C^{r}$.</p>
<p>ii) if $U\cap V=\theta$, then nothing.</p>
<p>When i) and ii) are satisfied, we name U and V are $C^{r}$<br />
consistent.</p>
<p>If we want to concern the movement of point on a manifold, and to<br />
analyze such movement, we'd better let the functions<br />
$\phi(U)\circ\varphi(V)^{-1}$ and $\varphi(V)\circ\phi(U)^{-1}$<br />
have good enough analytical property, because their analytical<br />
property restrict the analyzable of the manifold.</p>
<p>Then, as soon as we get enough map charts $(U, \phi(U))$ of<br />
manifold M, we can analyze the point movement on it. the "enough"<br />
means:</p>
<p>i) $\mathbbm{A}=\{U, V, W,...\}$ is a open cover of M;</p>
<p>ii) Any two of $\mathbbm{A}$ are $C^{r}$ consistent;</p>
<p>iii) $\mathbbm{A}$ is the maximum.</p>
<p>then, we name $\mathbbm{A}$ a $C^{r}$ differential structure of M,<br />
correspondingly, M is a $C^{r}$ differential manifold.</p>
<p>Let's treat some familiar set as manifold, and to see how does<br />
adding various differential structures on them means.<br />
Euclid space $R^{n}$<br />
Sphere $S^{n}$<br />
Projective space $P^{n}$<br />
Milnor monstrous ball.</p></p>
<p></p>http://krsna.lamost.org/engine/node/729#commentshttp://krsna.lamost.org/engine/crss/node/729数学几何观点几何流形Wed, 20 Sep 2006 18:39:09 +0000yijun729 at http://krsna.lamost.org/engine三维流形的分类
http://krsna.lamost.org/engine/node/157
<p><strong>何谓分类？</strong><br />
已经找到2维流形的清晰分类：<br />
任何一个闭2维曲面，都可以由3种基本曲面通过离散群作用求商集得到。<br />
这三基本曲面是2维球面，欧氏平面，双曲盘面。<br />
所以分类落实在曲面之间的拓扑变换关系。 </p>
<p>那么3维情形呢？</p>
<p><strong>Poincaré还是从球面入手</strong><br />
有理由认为球面具有非常尊贵的几何地位，Poincaré提出如下猜想：<br />
如果一个3维闭流形与3维球面同调，那么它必定与之同胚。<br />
不久他找到一个反例SO(3)/I_60，于是再问：<br />
如果一个3维闭流形具有平凡基本群，那么它是否一定与3维球面同胚？</p>
http://krsna.lamost.org/engine/node/157#commentshttp://krsna.lamost.org/engine/crss/node/157数学拓扑的观点几何流形Wed, 29 Dec 2004 08:19:05 +0000yijun157 at http://krsna.lamost.org/engine