Solution

Using the Leibniz rule for the connection, or first checking decomposable tensors and extending linearly, the commutator acts on each covariant index:

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

(427)

Choose

W_{μν} = g_{μν}

. Metric compatibility makes the left side zero. Applying the formula with commutator indices

β, γ

gives

0 = - R^{ρ}_{μ β γ} g_{ρ α} - R^{ρ}_{α β γ} g_{μ ρ} = - R_{αμ β γ} - R_{μα β γ} .

(428)

Hence

R_{μα β γ} = - R_{αμ β γ} .

(429)

Solved by gpt-5.6-sol high.

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