Comparing metric tensors of the Poincare and the Klein disk models of hyperbolic geometry I was trying to compare the metric tensor at the wikipedia pages of the 
Beltrami Klein model https://en.wikipedia.org/wiki/Klein_disk_model
and the metric tensor of the Poincare disk model at https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model
A the moment (22 may 2015) 
The formulas for the metric tensor of the Beltrami Klein model is given as:
$$ g (x, dx) = \frac{4 (x \cdot dx)^2}{(1 - \left\Vert x \right\Vert^2)^2} + \frac{4 \left\Vert dx \right\Vert^2}{(1 - \left\Vert x \right\Vert^2)}. $$
While for Poincare disk model the metric tensor is given as:
$$ ds^2 = 4 \frac{\sum_i dx_i^2}{(1-\sum_i x_i^2)^2}$$
(where the $x_i$ are the Cartesian coordinates of the ambient Euclidean space)
While I understand the two metric tensers should be different, here there also seems to be a difference in notation,  and that makes it for me impossible even to compare the two tensors.
How can I convert one tensor into the notation of the other?
OR (maybe more practical) : 
What are the metric tensors of  the Beltrami Klein model in the notation $ds^2  = ... $
What are the metric tensors of  the Poincare disk model in the  notation $ g (x, dx) = ... $
OR:
Are are these imposssible questions and are the tensors incommensurable.? (the tensors are just not the same kind of beast)   
PS I now hardly anything about tensors, (let alone metric tensors) so any basic info about them should be welcome. (what is the easiest formula?)

The two notations are only slightly different.  The metric tensor in the Klein model is
$$
ds^2 \;=\; \frac{{dx_1}^2 + \cdots + {dx_n}^2 }{1 - {x_1}^2-\cdots - {x_n}^2}+\frac{(x_1\,dx_1 + \cdots + x_n\, dx_n)^2}{\bigl(1-{x_1}^2-\cdots-{x_n}^2\bigr)^2}
$$
and the metric tensor in the Poincaré model is
$$
ds^2 \;=\; 4\frac{{dx_1}^2+ \cdots + {dx_n}^2}{\bigl(1-{x_1}^2-\cdots -{x_n}^2\bigr)^2}
$$
If we let $\textbf{x} = (x_1,\ldots,x_n)$ and $\textbf{dx} = (dx_1,\ldots,dx_n)$, the metric tensor for the Klein model can be written
$$
ds^2 \;=\; \frac{\|\mathbf{dx}\|^2}{1-\|\mathbf{x}\|^2} + \frac{(\textbf{x}\cdot\textbf{dx})^2}{\bigl(1-\|\mathbf{x}\|^2\bigr)^2}
$$
and the metric tensor for the Poincaré model can be written
$$
ds^2 \;=\; \frac{4\|\mathbf{dx}\|^2}{\bigl(1-\|\mathbf{x}\|^2\bigr)^2}
$$
If you prefer to think in terms of matrices/components, the $ij$'th component of the metric tensor for the Klein model is
$$
g_{ij} \;=\; \frac{\delta_{ij}}{1-{x_1}^2-\cdots-{x_n}^2} + \frac{x_ix_j}{\bigl(1-{x_1}^2-\cdots-{x_n}^2\bigr)^2}
$$
where $\delta_{ij}$ is the Kronecker delta, and the  $ij$'th component of the metric tensor for the Poincaré model is
$$
g_{ij} \;=\; \frac{4\,\delta_{ij}}{\bigl(1-{x_1}^2-\cdots-{x_n}^2\bigr)^2}
$$
