How does the metric on the Poincaré half plane model work?

Question

Let $\mathbb{H} = \{ z = x+iy \in \mathbb{C} | \Im(z)=y>0\}$ be the upper half plane. I often see that $\mathbb{H}$ is endowed with a metric which is written as $$ ds^2 = \frac{dx^2+dy^2}{y^2}. $$

I do not understand what $s, d(\cdot)$ mean and how one can measure the distance of the points on $\mathbb{H}$ with that formula. And why do we have to take squares?

It seems like there is some background to that which I do not have. And I also do not know where to start. Could someone please explain me what all these things mean? Thank you!

Answer 1

You should first study path length integrals in Euclidean space, which you can learn about in ordinary multivariable calculus. For example, given a smooth path $\gamma(t) = (x(t),y(t))$ in the plane, $a \le t \le b$, the length of $\gamma$ is given by the expression $$\int_a^b \sqrt{\biggl(\frac{dx}{dt}\biggr)^2 + \biggl(\frac{dy}{dt}\biggr)^2} dt = \int_a^b \sqrt{(dx)^2 + (dy)^2} = \int_a^b \sqrt{ds^2} = \int_a^b ds$$ where $ds^2 = dx^2 + dy^2$.

Once you've mastered that, it's not much of a step to compute path length integrals in $\mathbb{H}$ by pretty much the same process: $$\int_a^b ds = \int_a^b \sqrt{ds^2} = \int_a^b \sqrt{\frac{dx^2+dy^2}{y^2}} = \int_a^b \frac{1}{y} \sqrt{\biggl(\frac{dx}{dt}\biggr)^2 + \biggl(\frac{dy}{dt}\biggr)^2} dt $$

To understand the underlying intuition of these expressions, in Euclidean space the formula $ds^2 = dx^2 + dy^2$ is really just the Pythagorean formula, expressed infinitesmally, whereas in the expression for the hyperbolic plane the difference is that infinitesmal lengths at each point $(x,y)$ are scaled by a factor of $\frac{1}{y}$.

Answer 2

Just to add to Lee Mosher's answer, the formula $ds^2 = \frac{dx^2+dy^2}{y^2}$ tells you how to compute lengths of paths "in the hyperbolic metric of the Poincaré half plane" (whatever that means, for now).

Now if you want to compute the distance between two points $z_1$ and $z_2$, what you do is that you are going to look at the length of the smallest path joining $z_1$ and $z_2$ in that sense. This will give you a notion of distance (actually, in this case an explicit formula can be written).

This generalizes euclidean distance, which as you know is the length of the shortest path between two points (but the "eucliden length").

If you want to place this in a more general framework, the key word is "riemannian metric".