# Is there a simple classification of maximally symmetric spaces?

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.

• The third is anti-de Sitter space. The sphere (with the usual metric) is not a pseudo-Riemannian manifold. In fact, de Sitter space is the spacetime analogue of the sphere, whilst anti-de Sitter space is the spacetime analogue of hyperbolic space (and Minkowski space is, of course, the analogue of Euclidean space). Commented Mar 6, 2018 at 19:51
• @gj255 Actually I also mean to include Riemannian manifolds. But thanks to your comment I see that my list of three (including the sphere) did not make much sense, indeed, because clearly there are way more. Commented Mar 6, 2018 at 21:11
• @Sjorszini : If I understand you require the space to possess a nondegenerate metric? Because there have recently been works generalizing this concept to also include galilean and carrollian spacetimes. Commented Aug 8, 2019 at 9:30
• @ungerade Yes I do require the metric to be nondegenerate. Commented Aug 29, 2019 at 13:11

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).

• This subgroups $\Gamma$ would not be respecting the Killing fields globally. Commented Mar 6, 2018 at 20:32
• What do you mean? Commented Mar 6, 2018 at 21:02
• Thanks for the answer. When you say that `there are three max. symmetric geometries', do you mean that every max. symm. manifold is locally isometric to one of those three? Also, your answer focusses on Lorentz signature (I see now that my question suggested this, but that was not my intention). Does something analogous hold for all signatures? So that for instance in 3 dimensions there is a total of $2\cdot 3= 6$ different max. symm. geometries, namely 3 for 1+2 (=2+1) signature and 3 for 0+3 (=3+0) signature? Commented Mar 6, 2018 at 21:05
• @Slereah: Example: Your torus spacetime no longer has rotations, so it does not have requisite 10 Killing VFs. Commented Mar 6, 2018 at 21:07
• Yes, that is what I mean. And yes, spacetimes of different signatures cannot be isometric, by Sylvester's theorem. Commented Mar 6, 2018 at 21:07