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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 "$g_{ij}$".

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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 $g_{ij}$.

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