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latentcorpse
#1
Oct12-09, 08:55 AM
P: 1,443
(i) show that [itex]R_{abcd}+R_{cdab}[/itex]
(ii) In n dimensions the Riemann tensor has [itex]n^4[/itex] components. However, on account of the symmetries
[itex]R_{abc}^d=-R_{bac}^d[/itex]
[itex]R_{[abc]}^d=0[/itex]
[itex]R_{abcd}+-R_{abdc}[/itex]
not all of these components are independent. Show that the number of independent components is [itex]\frac{n^2(n^2-1)}{12}[/itex]

not really sure how to go about this.
i think (i) follows from the 3 properties above but i cant prove it. also i dont understand what [itex]R_{[abc]}^d[/itex] means, in particular the [abc] part. is this a Lie bracket? (i havent covered these yet) so could someone explain what this is about.

thanks.
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