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} $.