(I believe that) de Sitter space is the only maximally symmetric Lorentzian spacetime, and that for $n$ spacetime dimensions, it has the hypercylindrical topology $\mathbb{R} \times S^{n-1}$.
This is already a little counterintuitive to me, because I would have thought that for a maximally symmetric spacetime, space and time would be on a quite equal footing, so it seems a bit strange that the spatial dimensions are compact but the time dimension isn't. I understand how it's logically possible though; the space and time directions are mathematically distinct because of their different signs in the metric signature, and moreover differential geometry is full of non-obvious constraints that purely local requirements can impose on the global topology of manifolds (e.g. the Gauss-Bonnet theorem).
My question is how this generalizes to pseudo-Riemannian manifolds with arbitrary metric signature. Specifically, my question is twofold:
If we consider the $n$-dimensional pseudo-Riemannian manifolds with some fixed (not necessarily Lorentzian) metric signature, is there a unique maximally symmetric manifold with positive scalar curvature? (By maximally symmetric manifold, I mean one with the maximum possible number $\frac{1}{2} n (n+1)$ of linear independent Killing fields on the whole manifold.)
If so, what is its global topology? A guess is that if the metric signature is $(p,m)$, then the global topology is $R^{\min(p,m)} \times S^{\max(p,m)}$, but I have no idea if that's correct.