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Definition. Riemann curvature tensor [B2D1]

The curvature RR of a Riemannian manifold (M,g)(M, g) is a tensor R∈Ω2(End(TM))R \in \Omega^2(End(TM))

R(X,Y)Z:=  ∇X∇YZ−∇Y∇XZ−∇[X,Y]Z=  [∇X,∇Y]Z−∇[X,Y]Z\begin{aligned} R(X, Y)Z :=&\; \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z \\ =&\; [\nabla_X, \nabla_Y] Z - \nabla_{[X,Y]} Z \end{aligned}

where X,Y,ZX, Y, Z are vector fields, ∇\nabla is the Levi-Civita connection of the metric gg (see Lectures on the Geometry of Manifolds Proposition 4.1.9.).

In local coordinates (x1,…,xn)(x^1, \dots, x^n) we have

Rijkm∂m=R(∂j,∂k)∂i R^m_{ijk} \partial_m = R(\partial_j, \partial_k) \partial_i

In terms of the Christoffel symbols we have

Rijkm=∂jΓikm−∂kΓijm+ΓnjmΓjkn−ΓnkmΓijn R^m_{ijk} = \partial_j \Gamma^m_{ik} - \partial_k \Gamma^m_{ij} + \Gamma^m_{nj}\Gamma^n_{jk} - \Gamma^m_{nk}\Gamma^n_{ij}

Lowering the indices we have a new tensor

Rijkl:=gimRjklm=g(R(∂k,∂l)∂j,∂i)=g(∂i,R(∂k,∂l)∂j) R_{ijkl} := g_{im}R^m_{jkl} = g(R(\partial_k, \partial_l) \partial_j, \partial_i) = g(\partial_i, R(\partial_k, \partial_l) \partial_j)