Is there a simple classification of maximally symmetric spaces?

Question

By maximally symmetric space I mean a (pseudo-) Riemannian manifold of dimension $n$ that has $n(n+1)/2$ linearly independent Killing vector fields. I seem to remember that there are only three kinds, one of them being Minkowski space, and another being de Sitter space. And the third probably being the sphere. But I'm not quite sure this is true in any dimension. Can someone shed some light on this issue? I would also very much appreciate references.

EDIT: Although the question above might have suggested it (because I wasn't thinking straight), I do not mean to focus on Lorentzian manifolds only. Indeed, as mentioned in the comments, the sphere I mention above is Riemannian, while the other two mentioned manifolds are Lorentzian, so that did not make very much sense on my part, because clearly there are more geometries then three (in at least 2 dimensions), thinking of Euclidean space.

Answer 1

It is true that there are only three (classes) of maximally symmetric geometries, classified by their curvature. But there are more spacetimes than that. If you pick one of the three maximally symmetric spaces $(\mathbb R^n, \eta)$ (Minkowski space), $(\mathbb R \times S^{n-1}, d)$ (de Sitter space) and $(\mathbb R^n, a)$ (anti-de Sitter space), then consider any subgroup $\Gamma$ of the symmetry group that acts on those spacetimes smoothly, freely, and properly, then the spacetime $M / \Gamma$ is also a maximally symmetric spacetime.

There are many such spacetimes. A whole laundry list of them exist for Minkowski space : the timelike cylinder and spacelike cylinder, both with topology $\mathbb R^n / \mathbb Z$, the torus spacetime $\mathbb R^n / \mathbb Z^n$, the Klein bottle spacetime, Misner space, moebius strip spacetimes, the non-time orientable cylinder $(\mathbb R^n / \mathbb Z) / (I \times T)$ and so forth.

Many variations of anti-de Sitter space also exist (actually what is classically referred to as AdS isn't what I told you, but one of those quotient). The classic AdS has topology $\mathbb R^{n-1} \times S^1$. There are variants of AdS that fail to be causal, orientable, time orientable, etc.

Same with de Sitter space, which includes among its famous topologies the polyhedral universes, using one of the discrete rotation group.

Edit : As A.V.S. points out, those spaces are not maximally symmetric, but only locally maximally symmetric (they fail to be rotationally invariant on the whole manifold). The complete classification of spacetimes of constant curvatures can be found in Wolf (chapter 11), and it is the following :

All homogeneous isotropic connected pseudo-Riemannian manifolds (of signature $(1, n-1)$) are classified thusly :

• If the manifold is flat, then $M$ is isometric to $\mathbb{R}^{1, n-1}$ (theorem 11.6.8).
• If the manifold is of constant curvature $K > 0$, then it is a cover of $S^{1, n-1} / \{ \pm I\} = \mathbb{R}^1 \times S^{n-1} / \{ \pm I\}$ (theorem 11.6.7)
• If the manifold is of constant curvature $K < 0$, then it is a cover of $\mathbb{H}^{1, n-1} / \{ \pm I\} = S^1 \times \mathbb{R}^{n-1} / \{ \pm I\}$ (theorem 11.6.7)

Those correspond to Minkowski space, quotients of de Sitter spacetime (this include such spaces as de Sitter spacetime itself, as well as the elliptic de Sitter spaces), and quotients (and covers) of anti-de Sitter space.

If you want to include Riemannian manifolds, then the classification is the following. By theorem 8.12.2, all two-points homogeneous Riemannian manifolds (equivalent to homogeneous and isotropic) are isometric to one of these :

• Euclidian space $\mathbb{R}^n = \mathrm{E}^n / \mathrm{O}(n)$
• The $n$-sphere $S^n = \mathrm{SO}(n + 1) / \mathrm{SO}(n)$
• The real projective space $\mathbb{R}\mathrm{P}^n = \mathrm{SO}(n + 1) / \mathrm{O}(n)$
• The complex projective space $\mathbb{C}\mathrm{P}^n = \mathrm{SU}(n + 1) / \mathrm{U}(n)$
• The quaternionic projective space $\mathbb{H}\mathrm{P}^n = \mathrm{Sp}(n + 1) / \mathrm{Sp}(n) \times \mathrm{Sp}(1)$
• The Cayley projective plane $\mathrm{CayP}^2 = \mathrm{F}_4 / \mathrm{Spin}(9)$
• The real hyperbolic space $\mathbb{H}^n(\mathbb{R}) = \mathrm{SO}(1, n-1) / \mathrm{SO}(n)$
• The complex hyperbolic space $\mathbb{H}^n(\mathbb{C}) = \mathrm{SU}(1, n-1) / \mathrm{U}(n)$
• The quaternionic hyperbolic space $\mathbb{H}^n(\mathbb{H}) = \mathrm{Sp}(1, n-1) / \mathrm{Sp}(n) \times \mathrm{Sp}(1)$
• The Cayley hyperbolic plane $\mathbb{H}^n(\mathrm{Cay}) = \mathrm{F}^*_4 / \mathrm{Spin}(9)$

(a bit of a notation collision : $\mathbb{H}^n$ is the $n$-dimensional hyperbolic space while $\mathbb{H}$ is the field of quaternions)

The cases of interest in physics (the $3$-manifolds used in the FRW metric) are the Euclidian space, $3$-sphere, hyperbolic space and real projective space (the real projective space has the same curvature as the $3$-sphere, but a different topology). This is due to all of the manifolds built from complex groups being even-dimensional (such as $\mathbb{C}\mathrm{P}^n$ being $(2n + 2)$-dimensional).