In what sense is the standard hyperbolic metric on the poincare upper half plane canonical?

Question

On the Poincare upper half plane, the standard hyperbolic is given by $ds^2 = \frac{dx^2 + dy^2}{y^2}$.

In what sense is this metric canonical? E.g., why don't people use $2ds^2$? or $3ds^2$?

Answer 1

That's the metric that gives constant Gaussian curvature of $-1$.

Answer 2

This metric is the only one (up to scaling by a real)precerved by the action of $Sl_2(\mathbb{R})$. Then you take the scaling to have a curvature constant to $-1$.

Why are we looking at the action of $Sl_2(\mathbb{R})$ ? To understand this, one have to show that the only bijective holomorphic map from $\hat{\mathbb{C}}$ to $\hat{\mathbb{C}}$ are $f_(a,b,c,d): z \mapsto \frac{az+b}{cz+d}$ with $ad-bc=1$ and $(a,b,c,d) \in \mathbb{C}^4$. This is a classical exercise.

Now we want holomorphic bijection from $\mathbb{H}$ to $\mathbb{H}$. This impose $(a,b,c,d) \in \mathbb{R}^4$.

Finnaly a another exercice show that $ds$ is the only metric invariant by this maps.