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Riemann Tensor

This tensor is the fundamental source of curvature in general relativity.
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The statement of the theorem

Define the Riemann curvature tensor RR for a manifold with metric gμνg_{\mu\nu} as:\nRiem(X,Y)Z=∇X∇YZ−∇Y∇XZ−∇[X,Y]Z\text{Riem}(X, Y) Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z \nIn local coordinates, the components are given by:\nRρσμν=∂Γνρ∂xμ−∂Γμρ∂xν+ΓνλΓλρ−ΓμλΓλρR^{\rho}{}_{\sigma\mu\nu} = \frac{\partial \Gamma^{\rho}_{\nu}}{\partial x^{\mu}} - \frac{\partial \Gamma^{\rho}_{\mu}}{\partial x^{\nu}} + \Gamma^{\lambda}_{\nu} \Gamma^{\rho}_{\lambda} - \Gamma^{\lambda}_{\mu} \Gamma^{\rho}_{\lambda}
Source: Wikipedia