(i) show that [latex]R_{abcd}+R_{cdab}[/latex]
(ii) In n dimensions the Riemann tensor has [latex]n^4[/latex] components. However, on account of the symmetries
[latex]R_{abc}^d=R_{bac}^d[/latex]
[latex]R_{[abc]}^d=0[/latex]
[latex]R_{abcd}+R_{abdc}[/latex]
not all of these components are independent. Show that the number of independent components is [latex]\frac{n^2(n^21)}{12}[/latex]
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 [latex]R_{[abc]}^d[/latex] 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.
