共形变换下曲率关系的活动标架计算方法

假设$(M,g)$是黎曼流形, 令$\tilde g=e^{2\phi} g$, 这里$\phi$是$M$上一个光滑函数. 这时称$(M,g)$与$(M,\tilde g)$共形.

我们感兴趣的是, 共形变换下曲率之间的关系.

活动标架

为此, 我们用活动标架法(用自然标架计算可以参考我写的Notes). 假设$\set{e_i}$是$(M,g)$的一个幺正标架场, $\set{\omega^i}$是其对偶标架场. $\nabla,\widetilde\nabla$分别表示对应于$g,\tilde g$的黎曼联络, 相应的联络1形式记为$\omega^i_j,\widetilde\omega^i_j$. (回忆, 给定一个联络$\nabla$, 以及一个局部标架场$\set{e_i}$, 联络1形式$\set{\omega^i_j}$由下式定义:$\nabla_X(e_j)=\omega^i_j(X)e_i.$)

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伯克利资格考试与口试:PDE

Sample Questions from Past Qualifying Exams

This list may give the impression that the exams consist of a series of questions fired at the student one after another. In fact most exams have more the character of a conversation with considerable give and take. Hence this list cannot be expected to indicate accurately the difficulties involved.

The list indicates the professor associated to each question where available. Some have been in the MGSA files for a while, and this information has been lost (if it was ever there).

The listing by section is approximate, since some questions may fit under more than one heading.
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伯克利资格考试与口试:几何

Sample Questions from Past Qualifying Exams This list may give the impression that the exams consist of a series of questions fired at the student one after another. In fact most exams have more the character of a conversation with considerable give and take. Hence this list cannot be expected to indicate accurately the difficulties involved.

The list indicates the professor associated to each question where available. Some have been in the MGSA files for a while, and this information has been lost (if it was ever there).

The listing by section is approximate, since some questions may fit under more than one heading.
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别再怀疑自己的意志力

一个小故事:
多年以来, 塞利格曼一直自认为自己的脾气比较坏—-有很强的时间紧迫感, 不能很好的跟人聊天. 他的妻子和孩子们, 却非常富有活力和幸福快乐, 跟别人相处得非常融洽, 因而塞利格曼”一回到家, 就像漂浮的云朵笼罩在阳光中.”

一天下午, 塞利格曼在花园除草, 而且是像他做所有事情那样, 郑重其事的除草. 小尼奇过来”帮忙”, 把除好的草抛向空中, 不断地长啊跳啊. 这在我们看来应该是非常欢乐的一幕, 但小尼奇的举动却惹怒了塞利格曼, 他责骂了尼奇. 小尼奇悻悻地走开, 不过几分钟后她又回来了.
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共形平坦的黎曼曲面的共形函数所满足的方程

$\newcommand{\rd}{\mathrm{d}}\newcommand{\set}[1]{\left\{#1\right\}}\newcommand{\Lp}{\Delta\,}$事实上, 假设$\rd s^2=g_{ij}\rd x^i\rd x^j$是$M^2=(\Omega,g)$上的Riemann度量. 要使$M^2$ 是共形平坦的, 那么
\[
\rd s^2=g^{ij}\rd x_i\rd x_j=e^{2\lambda u}\left((\rd x_1)^2+(\rd x_2)^2\right).
\]
下面, 我们用活动标架法来计算$M^2$的高斯曲率$K$. Continue Reading