This is one of Do Carmo's excersices and I got it as homework. Part (a) is easy and I include it here for the sake of completness. But I am entirely lost on part (b).
A function $g:\mathbb{R} \rightarrow \mathbb{R}$ given by $g(t)=yt+x$, $t,x,y \in \mathbb{R}, y>0$ is called an affine proper function. The subset of all such functions with respect to the usual composition law forms a Lie group $G$.As a differentiable manifold $G$ is just the half upper plane with the differentiable structure induced from $\mathbb{R}^{2}$. Prove that:
(a) The left-invariant Riemannian metric of $G$ which at the neutral element $e=(0,1)$ coincides with the euclidean metric ($g_{11}=g_{22}=1$, $g_{12}=0$) is given by $g_{11}=g_{22}=\frac{1}{y^{2}}$, $g_{12}=0$.
(b) Putting $(x,y)= z=x + iy$, $i=\sqrt{-1}$, the transformation z\rightarrow z'= \frac{az+b}{cz+d}, $a,b,c,d \in \mathbb{R}$, $ad-bc=1$ is an isometry of $G$.
Hint: Observe that the first fundamental form can be written as:
$ds^{2}= \frac{dx^{2} + dy^{2}}{y^{2}} = \frac{4dzd\overline{z}}{(z- \overline{z})^{2}}$.