# 广义正交群

## 矩阵定义

${\displaystyle \eta =\mathrm {diag} (\underbrace {1,\cdots ,1} _{p},\underbrace {-1,\cdots ,-1} _{q}).\,}$

${\displaystyle M^{-1}=\eta ^{-1}M^{T}\eta .\,}$

## 拓扑

O(p,q)和SO(p,q)都不是连通的，分别有4个和2个分支${\displaystyle \pi _{0}(O(p,q))\cong C_{2}\times C_{2}}$克莱因四元群，每个分支保持或改变p维正定或q维负定子空间的定向。特殊正交群有分支${\displaystyle \pi _{0}(SO(p,q))=\{(1,1),(-1,-1)\}}$，同时保持或同时改变两个定向。

O(p,q)的单位分支常记作SO+(p,q)，能和SO(p,q)中同时保持两个定向的元素的集合等价起来。

${\displaystyle \pi _{1}({\mbox{SO}}^{+}(p,q))}$ ${\displaystyle p=1}$ ${\displaystyle p=2}$ ${\displaystyle p\geq 3}$
${\displaystyle q=1}$ {1} ${\displaystyle \mathbf {Z} }$ ${\displaystyle \mathbf {Z} _{2}}$
${\displaystyle q=2}$ ${\displaystyle \mathbf {Z} }$ ${\displaystyle \mathbf {Z} \times \mathbf {Z} }$ ${\displaystyle \mathbf {Z} \times \mathbf {Z} _{2}}$
${\displaystyle q\geq 3}$ ${\displaystyle \mathbf {Z} _{2}}$ ${\displaystyle \mathbf {Z} _{2}\times \mathbf {Z} }$ ${\displaystyle \mathbf {Z} _{2}\times \mathbf {Z} _{2}}$

## 参考文献

• Joseph A. Wolf, Spaces of constant curvature, (1967) 335页。