Defining Euclidean global $\rm AdS$ 
*

*How does one see that the Euclidean $\rm AdS$ is the same as the hyperbolic space at the same dimension ie $EAdS_n = \mathbb{H}_n = SO_0(n,1)/SO(n)$?
Or is this to be seen as the definition of Euclidean AdS?


*Now one can see two kinds of global coordinates metric on this $EAdS_n$,

*

*as given in equation 4.4 (page 13) of this paper.


*as given in equation 3.11 (page 18) of this paper.
Is there any natural relationship between the two?
And are they describing the same space - Euclidean AdS?


*In what sense is the description in the second link a foliation of $AdS_{n+1}$ by $\mathbb{R} \times \mathbb{H}_{n-1}$? And is the existence of such a foliation by hyperbolic cylinders anyhow related to the fact that Euclidean AdS is itself the same as the hyperbolic plane?
 A: 1) The general idea is to see maximally symmetric manifolds of dimension $n$ embedded in a manifold of dimension $n+1$, with a constraint : 
$$\epsilon_{-1} x_{-1}^2+\epsilon_0 x_{0}^2 +\sum_{i=1}^{n-1}x_i^2= \epsilon_{-1} R^2 \tag{1}$$
where $\epsilon_{-1} =\pm1, \epsilon_0 =\pm1$
with the metrics :
$$\epsilon_{-1} dx_{-1}^2+\epsilon_0 dx_{0}^2 +\sum_{i=1}^{n-1}dx_i^2\tag{2}$$
for the embedding manifold.
[EDIT]
The maximally symmetric manifold could be written $M_n = G/L$, where $G$  is the "global" symmetry (inherited from the embedding manifold), and $L$ is a "local" symmetry. The "global" symmetry is $SO(p,q)$, where $p$ is the number of space coordinates and $q$ the number of time coordinates. The local symmetry could be obtained from the global symmetry by fixing the coordinate $x_{-1}$, which is a time-coordinate if $\epsilon_{-1}=-1$, and a space-coordinate if $\epsilon_{-1}=1$. So, the "local symmetry" is $SO(p,q-1)$ if $\epsilon_{-1}=-1$, and $SO(p-1,q)$ if $\epsilon_{-1}=1$
[/EDIT]
So, such a maximally symmetric manifold $M_n$could be written, with a general formula : 
$$M_n = SO(n + \frac{1}{2}(\epsilon_{-1} + \epsilon_0),1-\frac{1}{2}(\epsilon_{-1} + \epsilon_0))\\/SO(n+\frac{1}{2}(\epsilon_0-1),\frac{1}{2}(1-\epsilon_0))\tag{3}$$
The $AdS_n$ manifold corresponds to the case $\epsilon_{-1}=\epsilon_0=-1$, so 
$$AdS_n = SO(n-1,2)/SO(n-1,1)\tag{4}$$
The euclidean version of the  $AdS_n$ manifold, corresponds to  a change of sign of $\epsilon_0$, that is $\epsilon_{-1}=-1,\epsilon_0=+1$, this is the manifold :
$$H_n = SO(n,1)/SO(n)\tag{5}$$
[EDIT]
The euclideanization can be simply viewed as transferring one time degree of freedom to one space degree of freedom for both the "global" symmetry and the "local symmetry"
[/EDIT]
2) The two papers are different. The second paper gives the metrics for an AdS space, while the first paper is doing a construction for appearing $SO(N,1)$ as an analytic continuation of $SO(N+1)$, and more precisely, starting from the metrics of a sphere, and making some analytic continuation on the coordinates to get the metrics of the hyperbolic space.
3) In the second paper, the expression of the $AdS_{n+1}$ metrics could be considered at $\tilde \tau$ constant. At fixed $\tilde \tau$, the spatial slice is a product of a real dimension $R$ ($\rho$) by the  hyperbolic $H_{n-1}$ metrics:
$ds^2(\tilde \tau$ constant) $= \frac{d\rho^2}{(\rho^2/L^2-1)}+\rho^2 \underbrace{(du^2+sinh^2(u)d\Omega^2_{(d-2)})}_{H_{n-1} metrics}$$\tag{6}$

4) "And is the existence of such a foliation by hyperbolic cylinders
  anyhow related to the fact that Euclidean AdS is itself the same as
  the hyperbolic plane?"

I have no real good answer about this last point, but note that the dimensions are not the same, the euclidean version  is $H_{d+1}$, while the foliation is $R*H_{d-1}$
