Riemann Normal Coordinates

A manifold is locally like the n-dimensional real space and we expect spacetime to locally be like the flat space of special relativity. So, there should be some coordinate system where the metric at least at a point p isMinkowski . In such a coordinate system, $R_{\rho \mu \nu }^{\lambda }={\partial }_{\mu }{\Gamma }_{\nu \rho }^{\lambda }-{\partial }_{\nu }{\Gamma }_{\mu \rho }^{\lambda }\text{since }{\partial }_{\lambda }{g}_{\mu \nu }=0⟹{\Gamma }_{\nu \rho }^{\mu }=0$ Then, $R_{\rho \mu \nu }^{\lambda }=\frac{1}{2}{g}^{\lambda \sigma }({\partial }_{\mu }{\partial }_{\rho }{g}_{\nu \sigma }+{\partial }_{\nu }{\partial }_{\sigma }{g}_{\mu \rho }-{\partial }_{\mu }{\partial }_{\sigma }{g}_{\nu \rho }-{\partial }_{\nu }{\partial }_{\rho }{g}_{\mu \sigma })$ It is convenient to lower the index with the metric, to write $R_{\beta \rho \mu \nu }=g_{\beta \lambda }{R}_{\rho \mu \nu }^{\lambda }=\frac{1}{2}({\partial }_{\mu }{\partial }_{\rho }{g}_{\nu \beta }+{\partial }_{\nu }{\partial }_{\beta }{g}_{\mu \rho }-{\partial }_{\mu }{\partial }_{\beta }{g}_{\nu \rho }-{\partial }_{\nu }{\partial }_{\rho }{g}_{\mu \beta })$