I am looking at a manifold of dimension $n$ (And I am considering a local co-ordinates system $x_1,x_2,\ldots x_n$) and the metric defined by the components $g_{ij} = \frac{\delta_{ij}}{x_1^2}$. I'm wanting to find the components of the corresponding Riemann tensor. This should be a pretty straight forward task but I cannot seem to match my workings to the actual answer.
Of course $g^{ij} = x_1^2\delta^{ij}$ and I have that $R^l_{ijk} = \frac{\partial}{\partial x^j} \Gamma^l_{ik} - \frac{\partial}{\partial x^k} \Gamma^l_{ij} + \Gamma^s_{ik}\Gamma^l_{sj} - \Gamma^s_{ij}\Gamma^l_{sk}$
And I also have derived that $\Gamma^l_{ij} = \frac{1}{2}g^{lm}(g_{im,j}+g_{jk,i}-g_{ij,m})$
So I've done the hard stuff, and it should just be an easy calculation to get the the components of the Riemann tensor. But I can't get my answer to match.
Could someone please help me with the calculations please?
Many thanks!