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olgerm

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## g^{\nu m_2}g^{\mu m_1}g^{j_1 m_3} \frac{\partial \Gamma_{m_3 m_2 m_1}}{\partial x^{j_1}}=\frac{\partial{\Gamma}^{j1\nu\mu}}{\partial x^{j_1}}##

##g^{i_2 m_2}g^{i_1 m_1}g^{j_1 m_3}{\Gamma_{m_3 j_1 j_2}}{\Gamma_{m_4 m_2 m_1}}={\Gamma^{j_1}}_{j_1 j_2}\Gamma^{j_2 i_2 i_1 }##

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##\Gamma## is christoffel symbol.

##g## is metric tensor.

contrvariance and covariance of chistoffel symbol indices have same meaning like in ricci tensor##

{\displaystyle R_{\alpha \beta }=2{\Gamma ^{\rho }}_{\alpha [\beta ,\rho ]}+2{\Gamma ^{\rho }}_{\lambda [\rho }{\Gamma ^{\lambda }}_{\beta ]\alpha }.}={R^{\rho }}_{\alpha \rho \beta }=\partial _{\rho }{\Gamma ^{\rho }}_{\beta \alpha }-\partial _{\beta }{\Gamma ^{\rho }}_{\rho \alpha }+{\Gamma ^{\rho }}_{\rho \lambda }{\Gamma ^{\lambda }}_{\beta \alpha }-{\Gamma ^{\rho }}_{\beta \lambda }{\Gamma ^{\lambda }}_{\rho \alpha }##.