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Riemann curvature tensor

= Ricci identity
{c}
{synonym}

For a rank-two covariant tensor, the derivative commutator contributes one curvature action on each index:
$$
[\nabla_\alpha,\nabla_\beta]W_{\mu\nu}
=-R^\rho{}_{\mu\alpha\beta}W_{\rho\nu}
-R^\rho{}_{\nu\alpha\beta}W_{\mu\rho}.
$$


Curvature commutator on a covariant tensor

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 37D / d / Solution

Curvature commutator on a covariant tensor

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Curvature commutator on a covariant tensor

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