I'm given a metric $$ds^2=g_{\mu\nu}dx^\mu dx^\nu=\left(1-\frac{r_g}{r}\right)c^2dt^2-\left(1+\frac{r_g}{r}\right)dr^2-r^2d\theta^2-r^2\sin^2\theta d\phi^2$$ where $r_g$ is a constant.
I'm trying to find, If $$T^k_{ij}\equiv \Gamma^k_{ij}-\Gamma^k_{ji}$$ But the formula that I'm using that is $$\Gamma^m_{ik}=\frac{1}{2}g^{ml}\left(\frac{\partial g_{li}}{\partial x^k}+\frac{\partial g_{lk}}{\partial x^i}-\frac{\partial g_{ik}}{\partial x^l}\right)$$
already assumes that the torsion tensor is zero. Can anyone help me out, How to see if $T^k_{ij}$ is zero or not?