Riemann curvature four-tensor

Let $(M,g)$ be a Riemannian manifold, let $\mathrm{\nabla }$ be its unique Levi–Civita connection, and let $R$ be the curvature tensor with the sign fixed above. The Riemann curvature four-tensor is the covariant tensor

$\mathrm{Rm}(X,Y,Z,W):=g(R(X,Y)Z,W)\mathrm{.}$

Thus the first two arguments are the two differentiating slots, the third is the field acted on, and the fourth lowers the output index. In coordinates, if $R({\mathrm{\partial }}_i,{\mathrm{\partial }}_j){\mathrm{\partial }}_k=R^m{}_{kij}{\mathrm{\partial }}_m$, then $R_{ijkl}=g_{lm}{R}^m{}_{kij}$ under this argument order. This convention makes $\mathrm{Rm}(u,v,v,u)$ positive on a positively curved round sphere.

The definition uses no chosen frame. On an empty or zero-dimensional manifold it gives the unique zero four-tensor; in dimension one the same formula applies and later symmetries force it to vanish. It is valid up to the boundary when a Riemannian manifold with boundary is supplied.