(anti)-symmetries of the Riemann curvature tensor

[quasar987]

The Riemannian curvature tensor has the following symmetries:
(a) R_ijkl=-R_jikl
(b) R_ijkl=-R_ijlk
(c) R_ijkl=R_klij
(d) R_ijkl+R_jkil+R_kijl=0

This is surely trivial, but I do not see how to prove that

R_ijk^l=-R_jil^k.

:(

Thanks.

 

[quasar987]

This is supposed to be obvious, according to a commentary in a book.
I can get

R_ijk^l=-R_ji^l_k

(from memory) but not the required identity...

 

[fzero]

Contravariant and covariant indices must match on both sides of an equality (else the left and right hand sides do not transform the same way), so the original identity makes no sense. The identity in post #2 is as close as you can get.

 

[quasar987]

True!

So do you know how we can deduce from identities (a)-(d) that

R_ijm^m=0

(summation over m implied)?

 

[fzero]

True!

So do you know how we can deduce from identities (a)-(d) that

R_ijm^m=0

(summation over m implied)?

Yes, use (b). Specifically use the fact that the metric is symmetric to show that

{R_{ijm}}^m = g^{mn} R_{ijmn} = - {R_{ijm}}^m .

 

[quasar987]

mh, I see! Thanks fzero.