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In the Bianchi Identities of the second kind, we have ∇a R_{bcde}+ ∇b R_{cade}+ ∇c R_{abde}≡ 0

but since ∇c R_{abde}= - ∇c R_{bade}

we get :

∇a R_{bcde}+ ∇b R_{cade}- ∇c R_{bade}= 0

in the last term, we exchange the c and the b indices and we would arrive at:

∇a R_{bcde}+ ∇b R_{cade}- ∇b R_{cade}= 0

which leads to :

∇a R_{bcde}= 0 .

but this incorrect ? did I do something wrong ?

I would like actually to arrive at the formula :

2 ∇b R_{acde}− ∇a R_{bcde}≡ 0.

see formula 9 here from original Bel's article from 1938:

http://gallica.bnf.fr/ark:/12148/bp...ction+d'un+tenseur+du+quatrième+ordre;.langEN

and

(see formula 5 here : https://docs.google.com/viewer?a=v&...9JJLhB&sig=AHIEtbRUONCiZUSV_8erdxK9YSMiouUrjA)

Thanks,

cheers,

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# Bianchi Identity-where did I go wrong?

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