Natural metric on unit disk

Question

I am new to Riemann surface and I am struggling to understand the induced metric by the conformal structure. I am following the book Riemann Surface by Farkas and Kra and in the chapter IV.8 it talks about Riemannian metrics. It indroduces the iperbolic metric on the unit disk as $\frac{2}{1-|z|^2} |dz|$. My question is: can we put the spherical or the euclidean metric on the unit disk?

It seems that the only natural metric on the unit disk is the hyperbolic one. I think the answer is negative because the metric is induced by the Riemann surface structure and the disk and the plane are not biholomorphic, but I don't see any obstacle to put other metrics.

Answer 1

You can put a lot of different metrics on the unit disk. The restriction of the euclidean metric on $\mathbb{R}^2$ to the disk is one example.

Answer 2

As in @quarague's answer, we can put many different Riemannian structures on the open unit disk.

The choice of hyperbolic structure is that this is the structure that is invariant/equivariant under the (bi-) holomorphic maps of the disk to itself, namely, the linear fractional maps given by the indefinite-signature unitary group $U(1,1)$.

(In contrast, the Euclidean structure on the open unit disk only has rotations as symmetries, for example, ...)