How to proof that
εμνρσ Rμνρσ =0 ?
Use the symmetries of the Riemann tensor!
I'm not sure Rμνρσ = Rμ(νρ)σ or not? If so, the problem solved for me.
I thought those symmetries would help, but it seems they don't. Start with the first Bianchi identity and contract all the indices with a Levi-Civita symbol. Rearranging the indices of the Levi-Civita symbol in two of the terms will give you what you want.
OK, I know use the Bianchi identity
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