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Riemann Tensor

A mathematical object that quantifies the curvature of spacetime, appearing prominently in the Einstein Field Equations.
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The statement of the theorem

Let gμνg_{\mu\nu} be the metric tensor defining the spacetime manifold (M,g)(M, g). The Riemann curvature tensor RρσμνR^{\rho}{}_{\sigma\mu\nu} is defined by the commutator of covariant derivatives acting on a vector field VV: \n\nRρσμνVσ=[∇μ,∇ν]Vρ−RρσμνVσR^{\rho}{}_{\sigma\mu\nu} V^{\sigma} = \left[ \nabla_{\mu}, \nabla_{\nu} \right] V^{\rho} - R^{\rho}{}_{\sigma\mu\nu} V^{\sigma} \n\nAlternatively, using the Christoffel symbols Γμρ\Gamma^{\rho}_{\mu} derived from gμνg_{\mu\nu}, the components are given by:\n\nRρσμν=∂μΓνρσ−∂νΓμρσ+ΓνλσΓμρ−ΓμλσΓνρR^{\rho}{}_{\sigma\mu\nu} = \partial_{\mu}\Gamma^{\rho}_{\nu}\sigma - \partial_{\nu}\Gamma^{\rho}_{\mu}\sigma + \Gamma^{\lambda}_{\nu}\sigma\Gamma^{\rho}_{\mu} - \Gamma^{\lambda}_{\mu}\sigma\Gamma^{\rho}_{\nu}
Source: Wikipedia