# 黎曼球面

${\displaystyle 1/0=\infty .}$

• 複射影直線，記為 ${\displaystyle \mathbb {CP} ^{1}}$，和
• 擴充複數平面，記為 ${\displaystyle \mathbb {\hat {C}} }$或者${\displaystyle \mathbb {C} \cup \{\infty \}}$.

## 作為複流形

${\displaystyle \zeta =1/\xi ,}$
${\displaystyle \xi =1/\zeta .}$

## 作為複射影線

${\displaystyle (\alpha ,\beta )=(\lambda \alpha ,\lambda \beta )}$

${\displaystyle (\alpha ,\beta )=(\zeta ,1).}$

${\displaystyle (\alpha ,\beta )=(1,\xi ).}$

${\displaystyle (1,\xi )=(1/\xi ,1)=(\zeta ,1)}$

## 作為球面

${\displaystyle \zeta ={\frac {x+iy}{1-z}}=\cot(\varphi /2)\;e^{i\theta }.}$

${\displaystyle \xi ={\frac {x-iy}{1+z}}=\tan(\varphi /2)\;e^{-i\theta }.}$

(兩份複數平面和平面${\displaystyle z=0}$的對應方式不同。必須使用定向翻轉來保證球面上定向的一致性，實際上複共軛使得變換映射成為全純函數。）${\displaystyle \zeta }$-座標和${\displaystyle \xi }$-座標之間的變換函數可以通過將其中一個映射和另一個的逆的複合得到。它們就是如上所述的${\displaystyle \zeta =1/\xi }$${\displaystyle \xi =1/\zeta }$。因此單位球面和黎曼球面微分同胚

## 度量

${\displaystyle ds^{2}=\left({\frac {2}{1+|\zeta |^{2}}}\right)^{2}\,|d\zeta |^{2}={\frac {4}{\left(1+\zeta {\bar {\zeta }}\right)^{2}}}\,d\zeta d{\bar {\zeta }}.}$

${\displaystyle ds^{2}={\frac {4}{\left(1+u^{2}+v^{2}\right)^{2}}}\left(du^{2}+dv^{2}\right).}$

## 自同構

${\displaystyle f(\zeta )={\frac {a\zeta +b}{c\zeta +d}},}$

${\displaystyle f(\alpha ,\beta )=(a\alpha +b\beta ,c\alpha +d\beta )={\begin{pmatrix}\alpha &\beta \end{pmatrix}}{\begin{pmatrix}a&c\\b&d\end{pmatrix}}.}$

## 參考

• Brown, James and Churchill, Ruel. Complex Variables and Applications. New York: McGraw-Hill. 1989. ISBN 0070109052.
• Griffiths, Phillip and Harris, Joseph. Principles of Algebraic Geometry. John Wiley & Sons. 1978. ISBN 0-471-32792-1.
• Penrose, Roger. The Road to Reality. New York: Knopf. 2005. ISBN 0-679-45443-8.
• Rudin, Walter. Real and Complex Analysis. New York: McGraw-Hill. 1987. ISBN 0071002766.