Relations between metric on H and the disc model

In the book "Modular Forms" by Miyake one finds the definition of some obscure 'thing'. He calls it a metric on $\mathbb{H} = \{z \in \mathbb{C} : \operatorname{Im}(z) > 0 \}$. The following is defined:

$$ds^2(z) = \frac{dx^2 + dy^2}{y^2}$$

Now the first thing that bothers me is that he writes a dependency of '$z$' while $dx$ and $dy$ are processes that require whole neighbourhoods. Anyhow: for a function $\phi : [0,1] \to \mathbb{H}$ that is continuous and smooth up to a finite number of points in $[0,1]$ he defines the length of this function as $$l_{\mathbb{H}}(\phi) = \int_0^1 \sqrt{x'(t)^2 + y'(t)^2}/y(t) \, dt$$

In the unit disc $B := \{ z \in \mathbb{C} : |z| \leq 1\}$ he defines another thing which gives rise to the definition of 'length' as $$d_B s^2 = \frac{4(dx^2 + dy^2)}{(1-|w|^2)^2}$$

$\mathbb{H}$ is 'holomorphically' related to the disc by $z \mapsto \rho.z$ where $\rho = \begin{pmatrix} 1 & i \\ 1 & -i \end{pmatrix}$ and $\rho$ acts as linear fractioinal transformation, i.e. $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}.z = \frac{az + b}{cz + d}$$

Now Miyake states two things and my question is: why do they hold?

1) He says that this $ds$ is invariant under the action of $GL_2^+(\mathbb{R})$ on the upper half plane. I guess that what he means is that if we have a curve $\phi$ as above and $\alpha \in GL_2^+(\mathbb{R})$, the curve $$\psi(t) := \alpha \circ \phi(t)$$ has the same length as $\phi$ in this $ds$ length-measurement.

2) Miyake states that the new $d_Bs^2$ is the push forward of the old one, i.e. $$l_B(\phi) = l_{\mathbb{H}}(\rho^{-1} \circ \phi)$$ where $l_B(\phi) = \int_0^1 4 \frac{\sqrt{x'(t)^2 + y'(t)^2}}{1 - |\phi(t)|^2} dt$ is the measurement with respect to the new thing.

I have seen some 'proofs' of these facts and they use strange relations between the $dx, dy, dz$ that i cannot understand, for example: Miyake says that for $dz = dx + i \, dy$ and $d\overline{z} = dx - i \, dy$ we have $dx^2 + dy^2 = dz \, d\overline{z}$ and then he rewrites the length as $l(\phi) = \int_0^1 ds(\phi(t))$ which does not make any sense in my eyes because: i can accept that $ds$ is some kind of differential operator that takes functions in two real variables and sends them to a function on $\mathbb{R}$ but $\phi$ is no such function: it takes only one variable, so i beg you: if you answer the question and use such relations then please describe them in a formal precise manner...

Thank you very much,

Fabian Werner, Germany

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2022-07-25 20:44:03
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