# Lie groups acting transitively (and isometrically) on anti de Sitter spaces

I hope this question is not deemed too localised.

Recall that anti de Sitter space is the lorentzian analogue of hyperbolic space; that is, a simply-connected lorentzian manifold of constant negative sectional curvature. More precisely, the $n$-dimensional anti de Sitter space $\mathrm{AdS}_n$ is the universal cover of the one-sheeted hyperboloid in $\mathbb{R}^{n+1}$ cut out by the following quadric: $$x_1^2 + x_2^2 + \dots + x_{n-1}^2 - x_n^2 - x_{n+1}^2 = - R^2$$ where $R>0$ is the radius of curvature. The ambient metric is flat with signature $(n-1,2)$ and the induced metric on the hyperboloid has constant negative sectional curvature proportional to $1/R$.

The group $\mathrm{SO}(n-1,2)$ acts transitively and isometric on the hyperboloid; indeed, the hyperboloid is diffeomorphic to $\mathrm{SO}(n-1,2)/\mathrm{SO}(n-1,1)$.

This is analogous to the better known fact that the round $n$-sphere is diffeomorphic to $\mathrm{SO}(n+1)/\mathrm{SO}(n)$. However, it happens that for some spheres, a proper subgroup $G \subset \mathrm{SO}(n+1)$ already acts transitively on the sphere and in some cases such $G$ is (locally) isomorphic to $\mathrm{SO}(p)$ for some $p\lt n+1$. A case in point is the round 7-sphere, on which $\mathrm{Spin}(6)$ acts transitively with stabiliser subgroup $\mathrm{SU}(3)$.

I am interested in knowing whether something like that can happen for anti de Sitter space. In particular, I would like to know whether $\mathrm{SO}(n-1,2)$ (or some other group with isomorphic Lie algebra) can act transitively on $\mathrm{AdS}_p$ for some $p\gt n$. We already know it acts on $\mathrm{AdS}_n$ and it cannot act (effectively) on $\mathrm{AdS}_p$ for $p\lt n$ by dimension. As in the case of the spheres, this can perhaps only happen for low values of $n$ and in fact, to be perfectly honest, I am presently only interested in $n=4$.

Hence let me ask the following

Question

Can $\mathrm{SO}(n-1,2)$ act locally transitively (and isometrically!) on $\mathrm{AdS}_p$ for some $p\gt n$? In particular, can this happen for $n=4$?

Epilogue

I should have mentioned that I have checked by explicit calculation that $\mathrm{SO}(3,2)$ cannot act locally transitively and isometrically on $\mathrm{AdS}_5$, but before trying my hand at $\mathrm{AdS}_6$ or higher, I thought of asking here first.

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I couldn't write this as a comment and I might be off base, but in Berger's paper Theorem 6 he proves that the list of real Lie groups acting transitively on the quadric $$x_1^2 + \dots + x_{n-h}^2 - x^2_{n-h+1} - \dots - x^2_{n} = 1$$ induced from a linear action on $\mathbb{R}^n$ are modulo a finite number of exceptions
1) $SO(n-h,h)$ $n \geq 2$

2) $T^1 \times SU(m/2-h/2,h/2)$, $SU(m/2-h/2,h/2)$ for some $m$ $m/2 \geq 2$

3) $SP(1) \times SP(m/4-h/2,h/2)$, $T^1 \times SP(m/4-h/2,h/2)$, $SP(m/4-h/2,h/2)$ $m/4 \geq 2$

Edit: since I still think the answer to your question is in Berger's list, I transcribe it completely as it can be found in [2]

signature (p,q): $SO(p,q)$
signature (2p, 2q) $U(p,q)$ and $SU(p,q)$
signature (4p,4q) $SP(1)SP(p,q)$ $SP(p,q)$
signature (n,n) $SO(n,\mathbb{C})$
signature (2n,2n) $SO(n, \mathbb{H})$

$G_2^\mathbb{C}$ $G_2$ and $G_2^2$ signatures (7,7), (7,0) and (4,3)

$Spin(7,\mathbb{C})$ $Spin(7)$ and Spin(4,3) signatures (8,8) (8,0) and (4,4).

you of course are interested in signature (p,2)

The point is that irreducible holonomies are the ones that act transitively on the unit sphere in the tangent space. And this seems to me as the problem you are asking for.

[2] Proc of Symp in pure mathematics 54 (1993) part 2. On the Holonomy of Lorentzian manifolds. Bergery and Ikemakhen

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Thanks. I'm precisely asking about the finite number of exceptions! –  José Figueroa-O'Farrill Oct 16 '11 at 18:20
Just added the exceptional cases then, but none of them is interesting to your case it seems. –  Reimundo Heluani Oct 16 '11 at 20:09
Thanks again. I am not sure that what I am asking is exactly the same thing as Berger's problem. I do not care whether the action on AdS is induced from a linear action in the ambient space. –  José Figueroa-O'Farrill Oct 17 '11 at 8:33
my bad, there are theorems to this respect to check when the action of an algebraic group comes from a linear action, but I don't think they are applicable here. –  Reimundo Heluani Oct 17 '11 at 11:06

In my paper

Isometry groups of Lobachevskian spaces, similarity transformation groups of Euclidean spaces and Lorentzian holonomy groups. Rend. Circ. Mat. Palermo (2) Suppl. No. 79 (2006), 87–97.

I classify transitive group of isometries of the real Hyperbolic space. These groups are $SO(n,1)$ and subgroups of

$$Sim(n-1)=(\mathbb{R}_+\times SO(n-1))\times \mathbb{R}^{n-1}$$

that preserve an isotropic line in the Minkowski space and do not preserve any non-degenerate subspace of the Minkowski space (weakly-irreducible subgroups), in addition they have non-trivial projection on $\mathbb{R}_+$.

The connected group of isometries of AdS_n is $SO(n-1,2)$, hence, if a connected $G$ acts isometrically, it is contained in $SO(n-1,2)$. From the transitivity it follows that the representation is weakly irreducible. Some of weakly irreducible subgroups of $SO(n-1,2)$ can be found in

Ikemakhen, A. Sur l'holonomie des variétés pseudo-riemanniennes de signature (2,2+n). (French) [Holonomy of pseudo-Riemannian manifolds with signature (2,2+n)] Publ. Mat. 43 (1999), no. 1, 55–84.

I think that it is possible to classify all transitively acting subgroups.

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