Thread: Riemann Curvature Tensor View Single Post

## Riemann Curvature Tensor

(i) show that $R_{abcd}+R_{cdab}$
(ii) In n dimensions the Riemann tensor has $n^4$ components. However, on account of the symmetries
$R_{abc}^d=-R_{bac}^d$
$R_{[abc]}^d=0$
$R_{abcd}+-R_{abdc}$
not all of these components are independent. Show that the number of independent components is $\frac{n^2(n^2-1)}{12}$

i think (i) follows from the 3 properties above but i cant prove it. also i dont understand what $R_{[abc]}^d$ means, in particular the [abc] part. is this a Lie bracket? (i havent covered these yet) so could someone explain what this is about.