= Riemannian curvature two-form
{c}
{title2=$R\in\Omega^2(M;\operatorname{End}(TM))$}
For the <Levi-Civita connection>, the <curvature form of a connection> is the endomorphism-valued two-form
$$
R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z.
$$
It has <tensoriality> in all three arguments. Its <first Bianchi identity> follows from torsion-freeness: the cyclic sum is $\sum_{\mathrm{cyc}}(\nabla_X[Y,Z]-\nabla_{[Y,Z]}X)=\sum_{\mathrm{cyc}}[X,[Y,Z]]=0$ by the <Jacobi identity>.
Back to article page
