The Poincare upper half-plane

The upper half-plane, $\mathbb{H}$, with coordinates $(x,y)$ ($x\in \mathbb{R}$ and $y>0$) with the Poincar\'e metric

(1)
\begin{align} ds^2 =\frac{dx^2+dy^2}{y^2}\ , \end{align}

is called the Poincar\'e half-plane.It is also known as the Lobachevsky plane in the Russian literature. It is known that all Riemann surfaces with genus $>1$ can be obtained as quotients of this space by a discrete subgroup of $PSL(2,\mathbb{R})$ — this is a consequence of the uniformization theorem.

The Christoffel connection for the