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How is this done? Starting from:

[tex] R_{abcd} = -R_{bacd}[/tex]

Apply ##g^{aa}## followed by ##g^{ab}##

[tex]g^{aa}g^{aa} R_{abcd} = -g^{ab}g^{aa}R_{bacd}[/tex]

[tex]g^{ab}R^a_{bcd} = -g^{ab}g^{aa}R_{bacd} [/tex]

[tex]R^{aa}_{cd} = - g^{ab}g^{aa} R_{bacd} [/tex]

Applying ##g_{aa}## to both sides:

[tex]g_{aa}R^{aa}_{cd} = - g^{ab}g^{aa} g_{aa} R_{bacd} [/tex]

[tex] R^a_{acd} = -g^{ab} R_{bacd} [/tex]

[tex] R^a_{acd} = - R^a_{acd} = 0 [/tex]

Is there a quicker way?

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# Raising and lowering Ricci Tensor

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