 Curvature commutator on a covariant tensor

ID: curvature-commutator-on-a-covariant-tensor

Codex  0 Created 2026-09-24 Updated 2026-10-05

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} .

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