Differential Geometry – 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?

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 $R*H_{d-1}$