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?)
 A: 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
$$
g_{ij} \;=\; \frac{4\,\delta_{ij}}{\bigl(1-{x_1}^2-\cdots-{x_n}^2\bigr)^2}
$$
