So my textbook definitions of the ricci tensor and ricci scalar are:(adsbygoogle = window.adsbygoogle || []).push({});

##R_{ab}=R_{acbd}g^{cd}## - I note the contraction is over the 2nd and 4th index. and the 1st and 3rd.

##R=R_{ab}g^{ab}##

Now, I'm trying to show that ##g^{ad}g^{ce}(\bigtriangledown_{a}R_{bcde}+\bigtriangledown_{b}R_{cade}+\bigtriangledown_{c}R_{abde})=2\bigtriangledown^{a}R_{ab}-\bigtriangledown_{b}R##

It's obvious that the ##R## term must come from ##g^{ad}g^{ce}\bigtriangledown_{b}R_{cade}## . But, I'm not seeing this properly,if the ricci vector is defined for the contraction to only be over specific indicies -or can it be any 2? As here the summation is over indicies consecutive to each other, but in the definition given above its over the 1st and 3rd, and 2nd and 4th.

Also, I'm totally clueless as to where a minus sign is coming from?

Thanks very much !!

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# Quick Ricci tensor/scalar contraction manipulation question

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