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## ３次元の場合[曲率テンソル]（４）

<<   作成日時 ： 2013/11/01 00:01   >>

 ２問目の問題を考えてみましょう。 [問題2]=========================================== ３次元擬球面 　　　$ds^{2}=d\chi^{2}+\sinh^{2}\chi\left(d\theta^{2}+\sin^{2}\theta&space;d\varphi^{2}\right)$ につき、$\left(\chi,\theta,\varphi\right)$ を $\left(x^{1},x^{2},x^{3}\right)$ 座標と見て、曲率テンソルの成分とスカラー曲率とを求めよ。 ================================================== 　　$g_{11}=\frac{1}{g^{11}}=1,\;&space;g_{22}=\frac{1}{g^{22}}=\sinh^{2}\chi,\;&space;g_{33}=\frac{1}{g^{33}}=\sinh^{2}\chi\sin^{2}\theta$ 以外の計量はゼロなので、 　　$\Gamma{^{1}}_{11}=\frac{1}{2}\frac{\partial&space;g_{11}}{\partial&space;x^{1}}=&space;0,\;\Gamma{^{1}}_{12}\left(=\Gamma{^{1}}_{21}\right)=\frac{1}{2}\frac{\partial&space;g_{11}}{\partial&space;x^{2}}=&space;0,\;$ 　　$\Gamma{^{1}}_{13}\left(=\Gamma{^{1}}_{31}\right)=\frac{1}{2}\frac{\partial&space;g_{11}}{\partial&space;x^{3}}=&space;0$ 　　$\Gamma{^{1}}_{22}=-\frac{1}{2}\frac{\partial&space;g_{22}}{\partial&space;x^{1}}=-\frac{1}{2}\frac{\partial&space;\sinh^{2}\chi}{\partial&space;\chi&space;}=-\sinh\chi\cosh\chi$ 　　$\Gamma{^{1}}_{23}\left(=\Gamma{^{1}}_{32}\right)=\frac{1}{2}\left&space;(\frac{\partial&space;g_{12}}{\partial&space;x^{3}}+\frac{\partial&space;g_{13}}{\partial&space;x^{2}}-\frac{\partial&space;g_{23}}{\partial&space;x^{1}}&space;\right)=0$ 　　$\Gamma{^{1}}_{33}=-\frac{1}{2}\frac{\partial&space;g_{33}}{\partial&space;x^{1}}=-\frac{1}{2}\frac{\partial&space;\sinh^{2}\chi\sin^{2}\theta}{\partial&space;\chi}=-\sinh\chi\cosh\chi\sin^{2}\theta$ 　　$\Gamma{^{2}}_{11}=-\frac{1}{2\sinh^{2}\chi}\frac{\partial&space;g_{11}}{\partial&space;x^{2}}=0$ 　　$\Gamma{^{2}}_{12}\left(=\Gamma{^{2}}_{21}\right)=\frac{1}{2\sinh^{2}\chi}\frac{\partial&space;g_{22}}{\partial&space;x^{1}}=\frac{1}{2\sinh^{2}\chi}\frac{\partial&space;\sinh^{2}\chi}{\partial&space;\chi&space;}=\frac{\cosh&space;\chi&space;}{\sinh\chi}$ 　　$\Gamma{^{2}}_{13}\left(=\Gamma{^{2}}_{31}\right)=\frac{1}{2\sinh^{2}\chi}\left&space;(\frac{\partial&space;g_{21}}{\partial&space;x^{3}}+\frac{\partial&space;g_{23}}{\partial&space;x^{1}}-\frac{\partial&space;g_{13}}{\partial&space;x^{2}}&space;\right)=0$　　 　　$\Gamma{^{2}}_{22}=\frac{1}{2\sinh^{2}\chi}\frac{\partial&space;g_{22}}{\partial&space;x^{2}}=0$ 　　$\Gamma{^{2}}_{23}\left(=\Gamma{^{2}}_{32}\right)=\frac{1}{2\sinh^{2}\chi}\frac{\partial&space;g_{22}}{\partial&space;x^{3}}=0$ 　　$\Gamma{^{2}}_{33}=-\frac{1}{2\sinh^{2}\chi}\frac{\partial&space;g_{33}}{\partial&space;x^{2}}=-\frac{1}{2\sinh^{2}\chi}\frac{\partial&space;\sinh^{2}\chi\sin^{2}\theta}{\partial&space;\theta}=-\sin\theta\cos\theta$ 　　$\Gamma{^{3}}_{11}=-\frac{1}{2\sinh^{2}\chi\sin^{2}\theta}\frac{\partial&space;g_{11}}{\partial&space;x^{3}}=0$ 　　$\Gamma{^{3}}_{12}\left(=\Gamma{^{3}}_{21}\right)=\frac{1}{2\sinh^{2}\chi\sin^{2}\theta}\left(\frac{\partial&space;g_{31}}{\partial&space;x^{2}}+\frac{\partial&space;g_{32}}{\partial&space;x^{1}}-\frac{\partial&space;g_{12}}{\partial&space;x^{3}}&space;\right)=0$ 　　$\Gamma{^{3}}_{13}\left(=\Gamma{^{3}}_{31}\right)=\frac{1}{2\sinh^{2}\chi\sin^{2}\theta}\frac{\partial&space;g_{33}}{\partial&space;x^{1}}=\frac{1}{2\sinh^{2}\chi\sin^{2}\theta}\frac{\partial&space;\sinh^{2}\chi\sin^{2}\theta}{\partial&space;\chi}$ 　　$=\frac{1}{2\sinh^{2}\chi}\frac{\partial&space;\sinh^{2}\chi}{\partial&space;\chi}=\frac{\cosh\chi}{\sinh\chi}$ 　　$\Gamma{^{3}}_{22}=-\frac{1}{2\sinh^{2}\chi\sin^{2}\theta}\frac{\partial&space;g_{22}}{\partial&space;x^{3}}=0$ 　　$\Gamma{^{3}}_{23}\left(=\Gamma{^{3}}_{32}\right)=\frac{1}{2\sinh^{2}\chi\sin^{2}\theta}\frac{\partial&space;g_{33}}{\partial&space;x^{2}}=\frac{1}{2\sinh^{2}\chi\sin^{2}\theta}\frac{\partial&space;\sinh^{2}\chi\sin^{2}\theta}{\partial&space;\theta}$ 　　$=\frac{1}{2\sin^{2}\theta}\frac{\partial&space;\sin^{2}\theta}{\partial&space;\theta}=\frac{\cos&space;\theta&space;}{\sin&space;\theta}$ 　　$\Gamma{^{3}}_{33}=\frac{1}{2\sinh^{2}\chi\sin^{2}\theta}\frac{\partial&space;g_{33}}{\partial&space;x^{3}}\right)=0$ ここは恒等にゼロでない独立な曲率テンソル成分を計算する。 　　$R_{1212}=-\frac{1}{2}\frac{\partial^{2}&space;g_{22}}{\partial&space;x^{1}\partial&space;x^{1}}&space;+g_{22}\Gamma{^{2}}_{21}\Gamma{^{2}}_{12}=-\frac{1}{2}\frac{\partial^2&space;\sinh^{2}\chi}{\partial&space;\chi^2}+\sinh^{2}\chi\left(\frac{\cosh&space;\chi}{\sinh&space;\chi}\right&space;)^{2}$ 　　$=-\frac{\partial&space;\sinh\chi\cosh&space;\chi}{\partial\chi}+\cosh^{2}\chi=-\cosh^{2}\chi-\sinh^{2}\chi+\cosh^{2}\chi=-\sinh^{2}\chi$ 　　$R_{1313}=-\frac{1}{2}\frac{\partial^{2}&space;g_{33}}{\partial&space;x^{1}\partial&space;x^{1}}&space;+g_{33}\Gamma{^{3}}_{31}\Gamma{^{3}}_{13}&space;=-\frac{1}{2}\frac{\partial^2&space;\sinh^{2}\chi\sin^{2}\theta}{\partial&space;\chi^2}+\sinh^{2}\chi\sin^{2}\theta\left(\frac{\cosh&space;\chi}{\sinh&space;\chi}\right)^{2}$ 　　$=-\sin^{2}\theta\frac{\partial&space;\sinh\chi\cosh\chi}{\partial&space;\chi}+\cosh^{2}&space;\chi\sin^{2}\theta$ 　　$=-\cosh^{2}\chi\sin^{2}\theta-\sinh^{2}\chi\sin^{2}\theta&space;+\cosh^{2}&space;\chi\sin^{2}\theta&space;=-\sinh^{2}\chi\sin^{2}\theta$ 　　$R_{2323}=-\frac{1}{2}\frac{\partial^{2}&space;g_{33}}{\partial&space;x^{2}\partial&space;x^{2}}-g_{11}\Gamma{^{1}}_{33}\Gamma{^{1}}_{22}&space;+g_{33}\Gamma{^{3}}_{32}\Gamma{^{3}}_{23}$ 　　$=-\frac{\sinh^{2}\chi}{2}\frac{\partial^{2}&space;\sin^{2}\theta}{\partial&space;\theta&space;^{2}}-\sinh^{2}\chi\cosh^{2}\chi\sin^{2}\theta&space;+\sinh^{2}\chi\sin^{2}\theta\left(\frac{\cos&space;\theta}{\sin&space;\theta}\right)^{2}$ 　　$=-\sinh^{2}\chi\frac{\partial&space;\sin\theta\cos&space;\theta&space;}{\partial&space;\theta&space;}-\sinh^{2}\chi\left(1+\sinh^{2}\chi\right&space;)\sin^{2}\theta&space;+\sinh^{2}\chi\cos^{2}\theta$ 　　$=-\sinh^{2}\chi\left&space;(\cos^{2}\theta&space;-\sin^{2}\theta\right)+\sinh^{2}\chi\left&space;(\cos^{2}\theta&space;-\sin^{2}\theta\right)-\sinh^{4}\chi\sin^{2}\theta$ 　　$=-\sinh^{4}\chi\sin^{2}\theta$ 　　$R_{1223}=g_{11}\left(\Gamma{^{1}}_{22}\Gamma{^{1}}_{13}-\Gamma{^{1}}_{22}\Gamma{^{1}}_{12}&space;\right)&space;+g_{22}\left(\Gamma{^{2}}_{22}\Gamma{^{2}}_{13}-\Gamma{^{2}}_{22}\Gamma{^{2}}_{12}&space;\right)&space;+g_{33}\left(\Gamma{^{3}}_{22}\Gamma{^{3}}_{13}-\Gamma{^{3}}_{22}\Gamma{^{3}}_{12}&space;\right)$ 　　$=g_{11}\Gamma{^{1}}_{22}\left(\Gamma{^{1}}_{13}-\Gamma{^{1}}_{12}\right)&space;+g_{22}\Gamma{^{2}}_{22}\left(\Gamma{^{2}}_{13}-\Gamma{^{2}}_{12}\right)&space;+g_{33}\Gamma{^{3}}_{22}\left(\Gamma{^{3}}_{13}-\Gamma{^{3}}_{12}\right)=0$ 　　$R_{2331}=\frac{1}{2}\frac{\partial^{2}g_{33}&space;}{\partial&space;x^{2}\partial&space;x^{1}}&space;+g_{11}\left(\Gamma{^{1}}_{33}\Gamma{^{1}}_{21}-\Gamma{^{1}}_{31}\Gamma{^{1}}_{23}\right)&space;+g_{22}\left(\Gamma{^{2}}_{33}\Gamma{^{2}}_{21}-\Gamma{^{2}}_{31}\Gamma{^{2}}_{23}\right&space;)$ 　　$+g_{33}\left(\Gamma{^{3}}_{33}\Gamma{^{3}}_{21}-\Gamma{^{3}}_{31}\Gamma{^{3}}_{23}\right&space;)=\frac{1}{2}\frac{\partial^{2}g_{33}&space;}{\partial&space;x^{2}\partial&space;x^{1}}&space;+g_{22}\Gamma{^{2}}_{33}\Gamma{^{2}}_{21}-g_{33}\Gamma{^{3}}_{31}\Gamma{^{3}}_{23}$ 　　$=\frac{1}{2}\frac{\partial^{2}\sinh^{2}\chi\sin^{2}\theta}{\partial&space;\chi\partial\theta}+\sinh^{2}\chi\left(-\sin\theta\cos&space;\theta\right)\left(\frac{\cosh\chi}{\sinh\chi}\right)-\sinh^{2}\chi\sin^{2}\theta\left(\frac{\cosh&space;\chi}{\sinh&space;\chi}\right)\left(\frac{\cos&space;\theta}{\sin&space;\theta}\right)$ 　　$=2\sinh\chi\cosh\chi\sin\theta\cos\theta-\sinh\chi\cosh\chi\sin\theta\cos\theta-\sinh\chi\cosh\chi\sin\theta\cos\theta$ 　　$=0$ 　　$R_{3112}&space;=g_{11}\left(\Gamma{^{1}}_{11}\Gamma{^{1}}_{32}-\Gamma{^{1}}_{12}\Gamma{^{1}}_{31}\right)&space;+g_{22}\left(\Gamma{^{2}}_{11}\Gamma{^{2}}_{32}-\Gamma{^{2}}_{12}\Gamma{^{2}}_{31}\right)&space;+g_{33}\left(\Gamma{^{3}}_{11}\Gamma{^{3}}_{32}-\Gamma{^{3}}_{12}\Gamma{^{3}}_{31}\right)$ 　　$=0$ 次にリッチテンソルを求めるが、ゼロでない計量のみを考えると 　　$R_{ik}=g^{lm}R_{limk}=g^{11}R_{1i1k}+g^{22}R_{2i2k}+g^{33}R_{3i3k}$ なので、 　　$R_{11}=g^{11}R_{1111}+g^{22}R_{2121}+g^{33}R_{3131}=g^{22}R_{1212}+g^{33}R_{1313}$ 　　$=\frac{-\sinh^{2}\chi}{\sinh^{2}\chi}+\frac{-\sinh^{2}\chi\sin^{2}\theta}{\sinh^{2}\chi\sin^{2}\theta}=-1-1=-2$ 　　$R_{12}\left(=R_{21}\right&space;)=g^{11}R_{1112}+g^{22}R_{2122}+g^{33}R_{3132}=-g^{33}R_{2331}=0$ 　　$R_{13}\left(=R_{31}\right&space;)=g^{11}R_{1113}+g^{22}R_{2123}+g^{33}R_{3133}=-g^{22}R_{1223}=0$ 　　$R_{22}=g^{11}R_{1212}+g^{22}R_{2222}+g^{33}R_{3232}=g^{11}R_{1212}+g^{33}R_{2323}$ 　　$=-\sinh^{2}\chi+\frac{-\sinh^{4}\chi\sin^{2}\theta}{\sinh^{2}\chi\sin^{2}\theta}=-2\sinh^{2}\chi$ 　　$R_{23}\left(=R_{32}\right&space;)=g^{11}R_{1213}+g^{22}R_{2223}+g^{33}R_{3233}=-g^{11}R_{3112}=0$ 　　$R_{33}=g^{11}R_{1313}+g^{22}R_{2323}+g^{33}R_{3333}=g^{11}R_{1313}+g^{22}R_{2323}$ 　　$=-\sinh^{2}\chi\sin^{2}\theta+\frac{-\sinh^{4}\chi\sin^{2}\theta}{\sinh^{2}\chi}=-2\sinh^{2}\chi\sin^{2}\theta$ となり、ゼロでないリッチテンソルをまとめると、 　　$R_{11}=-2,\;&space;R_{22}=-2\sinh^{2}\chi,\;&space;R_{33}=-2\sinh^{2}\chi\sin^{2}\theta$ となる。よってスカラー曲率は、 　　$R=g^{ik}R_{ik}=g^{11}R_{11}+g^{22}R_{22}+g^{33}R_{33}=-2+\frac{-2\sinh^{2}\chi}{\sinh^{2}\chi}+\frac{-2\sinh^{2}\chi\sin^{2}\theta}{\sinh^{2}\chi\sin^{2}\theta}$ 　　$=-2-2-2=-6$

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