# Christoffel Symbols for the Hyperbolic upper half plane

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 "$$g_{ij}$$".

The values $$g_{ij}$$ of the metric tensor are given by the coefficients of the metric $$g=g_{ab}dx^adx^b$$. So in your case, $$g_{10}=g_{01}=0$$ and $$g_{00}=g_{11}=\frac{1}{y^2}$$. The contravariant components $$g^{ij}$$ are given by the inverse of the matrix $$g_{ij}$$.