On the unit disk $|x|<1$ in $\mathbb{R}^n$, let's give a hyperbolic metric $$ g_{ij}=\frac{4\delta_{ij}}{(1-|x|^2)^2}. $$ I want to compute sectional curvature $K(\partial_i\wedge\partial_j)$ by computing it in the coordinate expressions, but somehow I cannot get the correct answer. Please let me know where I got wrong.
First let me compute the first derivatives of $g_{ij}$ : $$ g_{ij,l}=\frac{16\delta_{ij}x^l}{(1-|x|^2)^3}. $$
Then the Christoffel symbols are computed as : $$ \Gamma_{ij}^k=\frac{1}{2}g^{kl}(g_{il,j}+g_{jl,i}-g_{ij,l}) =\frac{2}{1-|x|^2}(\delta_{ik}x^j+\delta_{jk}x^i-\delta_{ij}x^k). $$
From this, we can compute the $R_{ijij}$:(now $i,j$ are distinct and fixed) $$ R_{ijij}=\frac{1}{2}(\partial_{ij}g_{ij}+\partial_{ij}g_{ij}-\partial_{ii}g_{jj}-\partial_{jj}g_{ii})+g_{jp}(\Gamma_{ii}^{m}\Gamma_{jm}^{p}-\Gamma_{ij}^{m}\Gamma_{im}^{p}) =\frac{-1}{2}(\partial_{ii}g_{jj}+\partial_{jj}g_{ii})+\frac{4\delta_{jp}}{(1-|x|^2)^2}(\Gamma_{ii}^{m}\Gamma_{jm}^{p}-\Gamma_{ij}^{m}\Gamma_{im}^{p}) $$
For $\partial_{jj}g_{ii}$, we get $$ \partial_{jj}g_{ii}=\partial_j 16\frac{x^j}{(1-|x|^2)^3}=16\frac{1}{(1-|x|^2)^3}+16\frac{6(x^j)^2}{(1-|x|^2)^4}=16\frac{1-|x|^2+6(x^j)^2}{(1-|x|^2)^4}. $$
And for the Christoffel symbols part, we compute: $$ \Gamma_{ii}^m=\frac{2}{1-|x|^2}(2\delta_{im}x^i-x^m), \Gamma_{jm}^j=\frac{2}{1-|x|^2}(x^m), \Gamma_{ij}^m=\frac{2}{1-|x|^2}(\delta_{im}x^j+\delta_{jm}x^i), \Gamma_{im}^j=\frac{2}{1-|x|^2}(\delta_{jm}x^i-\delta_{im}x^j). $$ Hence $$ \Gamma_{ii}^{m}\Gamma_{jm}^{j}-\Gamma_{ij}^{m}\Gamma_{im}^{j}=\frac{4}{(1-|x|^2)^2}(2(x^i)^2-|x|^2-(x^i)^2+(x^j)^2)=\frac{4}{(1-|x|^2)^2}(-|x|^2+(x^i)^2+(x^j)^2). $$ Note that we changed $p$ into $j$ by $\delta_{jp}$ multiplied in front of them.
Combining these results, we conclude that $$ R_{ijij}=\frac{16}{(1-|x|^2)^4}(-1+|x|^2-3(x^i)^2-3(x^j)^2-|x|^2+(x^i)^2+(x^j)^2)=\frac{16}{(1-|x|^2)^4}(-1-2(x^i)^2-2(x^j)^2). $$
Then $$ K(\partial_i\wedge\partial_j)=\frac{R_{ijij}}{g_{ii}g_{jj}}=-1-2(x^i)^2-2(x^j)^2\not\equiv -1, $$ contradicting the constant sectional curvature $=-1$ property of the Hyperbolic space.
Where did I do wrong?
$$ $$ Added: I found out that I can get the correct answer by using the following formula for the (3,1)-curvature tensor : $$ R_{ijk}^l=\partial_j \Gamma_{ik}^l-\partial_i \Gamma_{jk}^l+\Gamma_{ik}^{m}\Gamma_{jm}^{l}-\Gamma_{jk}^{m}\Gamma_{im}^{l} $$ while I was using different formula for (4,0)-curvature tensor above(which is given by the book "The Ricci Flow in Riemannian Geometry" by Andrews and Hopper, section 2.7.4). $$ R_{ijkl}=\frac{1}{2}(\partial_{jk}g_{il}+\partial_{il}g_{jk}-\partial_{ik}g_{jl}-\partial_{jl}g_{ik})+g_{lp}(\Gamma_{ik}^{m}\Gamma_{jm}^{p}-\Gamma_{jk}^{m}\Gamma_{im}^{p}) $$
I tried to verify the second formula from the first one, but I get the extra terms involving first derivatives of $g$, namely $g^{pq}_{,i}, g_{pq,j}$ etc. So I guess that the second formula is incorrect. Am I right about this point?