I was wondering if someone could explain the interpretation of the following results. In hyperbolic geometry, we say that lengths are invariant under the action of Mob($\mathbb{H}$) if given any piecewise-differentiable curve $f:[a,b]\rightarrow \mathbb{H}$ and an element $\gamma \in $ Mob($\mathbb{H})$
$length_\rho (f) = length_\rho (\gamma \circ f)$,
namely the arclength of a curve $f$ in $\mathbb{H}$ with respect to some metric $\rho(z)$ is the same even after $\gamma$ has been applied to it. So I am looking for some conditions on this metric $\rho$, and since I know that $length_\rho (f) = \int_f \rho(z) |dz|$, after doing some manipulations I arrive at the condition
$\rho(z) - \rho\Big(\gamma(z)\Big) |\gamma'(z)| = 0$.
Now I know that the group of all Möbius transformations that preserve $\mathbb{H}$ is generated by $g(z) = az+b$, $a,b \in \mathbb{R}$, $a>0$, $h(z) = -\frac{1}{z}$ and $B(z) = -\bar{z}$.
Now we look at the case when $\gamma$ is a translation by $b$ units, and we arrive at the fact that
$\rho(z) = \rho(z+b)$, namely the metric $\rho(z)$ in $\mathbb{H}$ is invariant under any translation by a real number, and so dependso only on $Im(z)$. Doing next with $\gamma = az$, we get the condition combined with the previous one that
$\rho(z) = \frac{c}{Im(z)}$, where $c$ is some constant. We can show that this form of $\rho(z)$ is consistent with $h(z) = -\frac{1}{z}$ and $B(z) = -\bar{z}$, but we'll assume that for now.
So here's the question. Does this result mean that if lengths in the hyperbolic plane are invariant under Möbius transformations, then they are precisely the ones that have the element of arc length as $\frac{c}{Im(z)} |dz|$?
But then I can arrive at this metric for the hyperbolic plane without invoking Möbius transformations, namely looking at the pseudosphere and see how if I try to map this pseudo sphere into something flat. A natural coordinate system to choose for the pseudo sphere would be coordinates $(\theta, \rho)$, namely latitude and longitude. So say for a given latitude the radius of my pseudo sphere is $R$ then the length it subtends on the surface by an angle $d\rho$ is $R d\rho$. But if we map it to something flat then the length is just $d\rho$, so the lengths must have been "shrunk" by a factor of $R$. Further investigation reveals that I can arrive at the same hyperbolic metric without invoking Möbius transforms.
What's the Connection?
Ben