Curvature transformation

A mapping $R ( X, Y)$ of the space ${\mathcal T} ( M)$ of vector fields on a manifold $M$, depending linearly on $X, Y \in {\mathcal T} ( M)$ and given by the formula

$$R ( X, Y) Z = \ \nabla _ {X} \nabla _ {Y} Z - \nabla _ {Y} \nabla _ {X} Z - \nabla _ {[ X, Y] } Z;$$

here $\nabla _ {X}$ is the covariant derivative in the direction of $X$ and $[ X, Y]$ is the Lie bracket of $X$ and $Y$. The mapping

$$R \equiv \ R ( X, Y) Z: {\mathcal T} ^ {3} ( M) \rightarrow {\mathcal T} ( M)$$

is the curvature tensor of the linear connection defined by $\nabla _ {X}$.