"리만 곡률 텐서"의 두 판 사이의 차이

2012년 1월 16일 (월) 05:56 판

접속 (connection)\(\nabla\)이 정의되어 있다고 하자
세 개의 벡터장 X,Y,Z 가 주어지면, 새로운 벡터장 R(X,Y)Z 를 얻는다
\(R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z\)
covariant tensor

\({R^\rho}_{\sigma\mu\nu} = dx^\rho(R(\partial_{\mu},\partial_{\nu})\partial_{\sigma})\)
텐서 표현
\(R(\partial_{i},\partial_{j})={R^l}_{kij} dx^{k}\otimes \frac{\partial}{\partial x^{l}}\)
크리스토펠 기호 를 이용한 성분의 계산
\({R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}\)
\({R^r}_{msq} = \partial_s\Gamma^r_{qm} - \partial_q\Gamma^r_{sm} + \Gamma^r_{s\lambda}\Gamma^\lambda_{qm} - \Gamma^r_{q\lambda}\Gamma^\lambda_{sm}\)
\({R^l}_{kij} = \partial_i\Gamma^l_{jk} - \partial_j\Gamma^l_{ik} + \Gamma^l_{is}\Gamma^s_{jk} - \Gamma^l_{js}\Gamma^s_{ik}\)
\(R_{\rho\sigma\mu\nu} = g_{\rho \zeta} {R^\zeta}_{\sigma\mu\nu} .\)

\(R(X,Y)\partial_{j}=\Omega_{j}^{s}(X,Y)\partial_s\)

\(\Omega_i^j =\frac{1}{2} R_{kli}^j \phi^k \wedge \phi^l \)

\(\Omega_i^j = d\omega_i^j - \omega_i^k \wedge \omega_k^j \)

@@ 19번째 줄: / 19번째 줄: @@
 * <math>{R^\rho}_{\sigma\mu\nu} = dx^\rho(R(\partial_{\mu},\partial_{\nu})\partial_{\sigma})</math>
 *  텐서 표현 <br><math>R(\partial_{i},\partial_{j})={R^l}_{kij} dx^{k}\otimes \frac{\partial}{\partial x^{l}}</math><br>
-* [[크리스토펠 기호]] 를 이용한 성분의 계산<br><math>{R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma}     - \partial_\nu\Gamma^\rho_{\mu\sigma}     + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}     - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}</math><br><math>{R^l}_{kij} = \partial_i\Gamma^l_{jk} - \partial_j\Gamma^l_{ik} + \Gamma^l_{is}\Gamma^s_{jk} - \Gamma^l_{js}\Gamma^s_{ik}</math><br><math>R_{\rho\sigma\mu\nu} = g_{\rho \zeta} {R^\zeta}_{\sigma\mu\nu} .</math><br>
+* [[크리스토펠 기호]] 를 이용한 성분의 계산<br><math>{R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma}     - \partial_\nu\Gamma^\rho_{\mu\sigma}     + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}     - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}</math><br><math>{R^r}_{msq} = \partial_s\Gamma^r_{qm} - \partial_q\Gamma^r_{sm} + \Gamma^r_{s\lambda}\Gamma^\lambda_{qm} - \Gamma^r_{q\lambda}\Gamma^\lambda_{sm}</math><br><math>{R^l}_{kij} = \partial_i\Gamma^l_{jk} - \partial_j\Gamma^l_{ik} + \Gamma^l_{is}\Gamma^s_{jk} - \Gamma^l_{js}\Gamma^s_{ik}</math><br><math>R_{\rho\sigma\mu\nu} = g_{\rho \zeta} {R^\zeta}_{\sigma\mu\nu} .</math><br>