How to convert from Euclidean to Hyperbolic Space using the Poincare Ball?

Question

The Poincaré ball is $\mathcal{B}^d = \{x \in \mathbb{R}^d : \|x\|<1 \}$. Then the Poincaré model is given by the ball and the Riemannian metric tensor $$ g_x = \left(\frac{2}{1-\|x\|}\right)^2g^E$$ where $x \in \mathcal{B}^d$ and $g^E$ denotes the Euclidean metric tensor.

The hyperbolic distance metric is $$ d(u,v) = \operatorname{arcosh}\left( 1+ 2 \frac{\|u-v\|^2}{(1-\|u\|)^2(1-\|v\|)^2}\right)$$

My first question is, what is $g_x$ telling us if $d(u,v)$ is telling us the distance?

Then, I'm unsure of what the "Euclidean metric tensor" is, one source referred to a $n\times n$ identity matrix, is that it?

As a couple of examples:

(a) Convert Euclidean to Hyperbolic in $\mathbb{R}$:

$$ g_x = \left(\frac{2}{1-|x|}\right)^2$$ And the distance between $x_1$ and $x_2$ is

$$ d = \operatorname{arcosh}\left( 1+ 2 \frac{|x_1 - x_2|^2}{\left(1-|x_1|\right)^2\left(1-|x_2|\right)^2}\right) $$ (b) Convert Euclidean to Hyperbolic in $\mathbb{R}^2$:

$$ g_x = \left(\frac{2}{1-\sqrt{\sum_{i=1}^n x_i}}\right)^2\times \pmatrix{1&0\\0&1} $$

And the distance between $x$ and $y$ here is $$ d = \operatorname{arcosh}\left( 1+ 2 \frac{\sum_{i=1}^n (x_i - y_i)^2}{\left(1-\sqrt{\sum_{i=1}^n x_i}\right)^2\left(1-\sqrt{\sum_{i=1}^n y_i}\right)^2}\right) $$

Have I got the notation correct?

Answer 1

The Wikipedia article states

In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.

The key part is the last sentence, where the tensor can be integrated to give length of curves. Essentially it defines the differential element of arc length at any point in the space. Thus, the equation $$ g_x = \left(\frac{2}{1-\|x\|)}\right)^2g^E $$ is stating that the Euclidean and Hyperbolic tensors at a point $\,x\,$ differ by a constant factor which depends only on $\,x\,$ and not the two tangent vectors.