Riemann metric in the open disk

I am currently studying The Princeton companion to mathematics.

According the book, "A more precise definition is that the open unit disk is the set of all points $(x,y)$ such that $x^2 + y^2 < 1$ and that the Riemannian metric on this disk is given by the expression $(dx^2+dy^2)/(1−x^2−y^2)$. This is how we define the square of the distance between $(x,y)$ and $(x + dx,y + dy)$." If the Riemann metric for the Poincare disk is just directly defined as $(dx^2+dy^2)/(1−x^2−y^2)$ without any proof, why can't we randomly use any expression as the Riemann metric? Is there any reason for choosing this specific notion of distance?

Also can there be multiple Riemannian metrics for a given manifold? How can there be multiple metrics for the same manifold?

• This en.wikipedia.org/wiki/Poincar%C3%A9_disk_model might help. The thing is that Poincare disk is modelled as a hyperbolic space, and the metric you are talking about enforces a negative constant curvature -1 (hence, hyperbolic space). Commented Mar 23, 2015 at 18:24
• The Poincaré disk model is hyperbolic space, under the right identifications (isometries). Commented Mar 23, 2015 at 23:43

Yes, given an underlying smooth manifold, there are generally many choices of metric. For instance, $dx^2 + dy^2$ (the standard Euclidean metric) is another perfectly acceptable choice. In this sense, there is no single "correct" metric to put on a particular smooth manifold, and it is just a definition.
Now, that said, we might arrive at the same Riemannian manifold via two different constructions. That is, we might start with $M_1$, give it a metric $g_1$, consider a distinct manifold $M_2$, give it some other metric $g_2$, and then show that $(M_1,g_1)$ is isometric to $(M_2,g_2)$.