Curvature commutator on a covariant tensor

For a rank-two covariant tensor, the derivative commutator contributes one curvature action on each index:

[\nabla_{α}, \nabla_{β}] W_{μν} = - R^{ρ}_{μα β} W_{ρ ν} - R^{ρ}_{να β} W_{μ ρ} .

 Ancestors (5)