I don't understand how you start when calculating the Christoffel symbols for the Levi-Civita connection of the hyperbolic metric in the upper half-plane model for $\mathbb{H}^2$.
Specifically, let $\mathbb{H}^2=\{(x,y)\in\mathbb{R}^2\mid y>0\}$ and $$g=\frac{(dx)^2+(dy)^2}{y^2}$$ be the hyperbolic metric.
I have this equation $$\Gamma_{ij}^k=\frac{1}{2}g^{kl}\Big(\frac{\partial g_{jl}}{\partial x^i}+\frac{\partial g_{il}}{\partial x^j}-\frac{\partial g_{ij}}{\partial x^l}\Big)$$
for the Christoffel symbols of the Levi-Civita connection corresponding to $g$, but I don't understand how to get these "$g_{ij}$".