Takatani Note

# $S^2$ のリッチ曲率

この記事では球面 $S^2$ の
・リッチ曲率 $R_{ij}$
・スカラー曲率 $S$
・ガウス曲率 $K_{\sigma}$
を求める.

$\R^3$ におけるユークリッド計量:
$g=dx^2+dy^2+dz^2$ を極座標 $(r,\theta,\v)$ で変換すると, $g=dr^2+r^2d\theta^2+r^2\sin^2\theta\ d\v^2$ と表せる. (計算の詳細はリーマン幾何学の公式集を参照.) この計量を $\R^3$ から球面 $S^2$ に制限したときの誘導計量は,
$g=d\theta^2+\sin^2\theta \ d\v^2$ である. ( $\because\$ $r=1$ より, $r$ は定数なので, $dr=0.$)

$S^2=\{(x,y,z)\in \R^3 \mid x^2+y^2+z^2=1\}$ にリーマン計量
$\ \ \ g=d\theta^2+\sin^2\theta \ d\v^2$
を入れる. このとき, 次を示せ.
$\ \ \ R_{11}=1,$
$\ \ \ R_{22}=\sin^2\theta,$
$\ \ \ S=2,$
$\ \ \ K_{\sigma}=1.$

ただし, $\d_1=\dd{}{\theta},\ \d_2=\dd{}{\v},$ $\ R_{ij}=\mathrm{Ric}(\d_i, \d_j)$ とする.

公式
$\ \G_{ij}^k=\dfrac{1}{2}g^{kl}(\d_ig_{jl}+\d_jg_{il}-\d_lg_{ij}) \ \ \cdots (a)$
$\ R_{ijk}^{\ \ \ \ \ l}=\d_i\G_{jk}^l-\d_j\G_{ik}^l+\G_{jk}^m \G_{im}^l-\G_{ik}^m\G_{jm}^l \ \ \cdots(b)$
$\ R_{ij}=R_{hij}^{\ \ \ \ \ \ h} \ \ \cdots (c)$

$g$ を行列で表すと次のようになる.
$$$(g_{ij})= \begin{pmatrix} 1 & 0 \\ 0 & \sin^2\theta \\ \end{pmatrix},$$$
$$$(g^{ij})=(g_{ij})^{-1}= \begin{pmatrix} 1 & 0 \\ 0 & 1/\sin^2\theta \\ \end{pmatrix}.$$$

$\G_{12}^2=\G_{21}^2=\dfrac{\cos\theta}{\sin\theta},$
$\G_{22}^1=-\sin\theta\cos\theta,$
$\G^1_{11}=\G^1_{12}=\G^1_{21}=\G^2_{11}=\G^2_{22}=0.$
(上の詳しい計算については以下を参照せよ.)

$\ \G_{ij}^k$ の計算
$\ \G_{ij}^k=\dfrac{1}{2}g^{kl}(\d_ig_{jl}+\d_jg_{il}-\d_lg_{ij})$ より,

$\G^1_{11}=\dfrac{1}{2}g^{11}(\d_1g_{11}+\d_1g_{11}-\d_1g_{11}) +\dfrac{1}{2}g^{12}(\d_1g_{12}+\d_1g_{12}-\d_2g_{11})$
$\ \ \ \ \ \ =\dfrac{1}{2}g^{11}(\d_1g_{11})+0$
$\ \ \ \ \ \ =0.$
$\G^1_{12}=\dfrac{1}{2}g^{11}(\d_1g_{21}+\d_2g_{11}-\d_1g_{12}) +\dfrac{1}{2}g^{12}(\d_1g_{22}+\d_2g_{12}-\d_2g_{12})$
$\ \ \ \ \ \ =\dfrac{1}{2}g^{11}\cdot 0+0$
$\ \ \ \ \ \ =0.$
$\G^1_{22}=\dfrac{1}{2}g^{11}(\d_2g_{21}+\d_2g_{21}-\d_1g_{22}) +\dfrac{1}{2}g^{12}(\d_2g_{22}+\d_2g_{22}-\d_2g_{22})$
$\ \ \ \ \ \ =\dfrac{1}{2}g^{11}\cdot (0+0-2\sin\t\cos\t)+0$
$\ \ \ \ \ \ =-\sin\t\cos\t.$
$\G^2_{11}=\dfrac{1}{2}g^{21}(\d_1g_{11}+\d_1g_{11}-\d_1g_{11}) +\dfrac{1}{2}g^{22}(\d_1g_{12}+\d_1g_{12}-\d_2g_{11})$
$\ \ \ \ \ \ =0+\dfrac{1}{2}g^{22}(0+0-0)$
$\ \ \ \ \ \ =0.$
$\G^2_{12}=\dfrac{1}{2}g^{21}(\d_1g_{21}+\d_2g_{11}-\d_1g_{12}) +\dfrac{1}{2}g^{22}(\d_1g_{22}+\d_2g_{12}-\d_2g_{12})$
$\ \ \ \ \ \ =0+\dfrac{1}{2}g^{22}(2\sin\t\cos\t+0-0)$
$\ \ \ \ \ \ =\dfrac{\cos\t}{\sin\t}.$
$\G^2_{22}=\dfrac{1}{2}g^{21}(\d_2g_{21}+\d_2g_{21}-\d_1g_{22}) +\dfrac{1}{2}g^{22}(\d_2g_{22}+\d_2g_{22}-\d_2g_{22})$
$\ \ \ \ \ \ =0+\dfrac{1}{2}g^{22}(0+0-0)$
$\ \ \ \ \ \ =0.$

レビチビタ接続のとき, $\G^k_{ij}=\G^k_{ji}$ であるので,
$\G^1_{21}=\G^1_{12}=0.$
$\G^2_{21}=\G^2_{12}=\dfrac{\cos\t}{\sin\t}.$

$\G^1_{22}=-\sin\t\cos\t,$
$\G^2_{12}=\G^2_{21}=\dfrac{\cos\t}{\sin\t},$
$\G^1_{11}=\G^1_{12}=\G^1_{21}=\G^2_{11}=\G^2_{22}=0.$

$R_{122}^{\ \ \ \ \ \ 1}=\sin^2\theta,$
$R_{211}^{\ \ \ \ \ \ 2}=1.$
(上の詳しい計算については以下を参照せよ.)

$\ R_{122}^{\ \ \ \ \ 1}$ と $R_{211}^{\ \ \ \ \ \ 2}$ の計算
$\ R_{ijk}^{\ \ \ \ \ l}=\d_i\G_{jk}^l-\d_j\G_{ik}^l+\G_{jk}^m \G_{im}^l-\G_{ik}^m\G_{jm}^l$ より,

$\ R_{122}^{\ \ \ \ \ 1}=\d_1\G_{22}^1-\d_2\G_{12}^1+\G_{22}^m \G_{1m}^1-\G_{12}^m\G_{2m}^1$
$\ \ \ \ \ \ \ \ \ = \d_1\G_{22}^1-\d_2\G_{12}^1 +(\G_{22}^1\G_{11}^1+\G_{22}^2\G_{12}^1) -(\G_{12}^1\G_{21}^1+\G_{12}^2\G_{22}^1)$
$\ \ \ \ \ \ \ \ \ = (-\cos^2\t+\sin^2\t)-0+(0+0)-(0+(-\cos^2\t))$
$\ \ \ \ \ \ \ \ \ = \sin^2\t.$

$\ R_{211}^{\ \ \ \ \ 2}=\d_2\G_{11}^2-\d_1\G_{21}^2+\G_{11}^m \G_{2m}^2-\G_{21}^m\G_{1m}^2$
$\ \ \ \ \ \ \ \ \ = \d_2\G_{11}^2-\d_1\G_{21}^2 +(\G_{11}^1\G_{21}^2+\G_{11}^2\G_{22}^2) -(\G_{21}^1\G_{11}^2+\G_{21}^2\G_{12}^2)$
$\ \ \ \ \ \ \ \ \ = 0-\dfrac{-1}{\sin^2\t}+(0+0)-(0+\dfrac{\cos^2\t}{\sin^2\t})$
$\ \ \ \ \ \ \ \ \ = 1.$

ゆえに, $R_{iik}^{\ \ \ \ \ l}=0.$
よって, $R_{111}^{\ \ \ \ \ \ 1}=R_{222}^{\ \ \ \ \ \ 2}=0.$

$R_{11}=R_{h11}^{\ \ \ \ \ \ h} =R_{111}^{\ \ \ \ \ \ 1}+R_{211}^{\ \ \ \ \ \ 2}=1,$
$R_{22}=R_{h22}^{\ \ \ \ \ \ h} =R_{122}^{\ \ \ \ \ \ 1}+R_{222}^{\ \ \ \ \ \ 2}=\sin^2\theta.$
よって,
$S=g^{ij}R_{ij}=g^{11}R_{11}+g^{22}R_{22}$ $=1\cdot 1+(1/\sin^2\theta)\cdot\sin^2\theta=2.$
$(\because\ g^{12}=g^{21}=0.)$

$R_{1221}=R_{122}^{\ \ \ \ \ \ 1}g_{11} +R_{122}^{\ \ \ \ \ \ 2}g_{21}$
$\ \ \ \ \ \ \ \ \ =R_{122}^{\ \ \ \ \ \ 1}g_{11} =\sin^2\theta.$
よって, $K_\sigma=1.$ $\ \ \ \square$