# Topology of (Anti) de Sitter space

1. Oct 3, 2013

### Slereah

Almost every book that I can think of derives the de Sitter spacetime as a hyperboloid in a (4+1) D spacetime, with the topology $ℝ \times S^3$, but I'm having trouble finding something justifying it. "Exact solutions of Einstein's Field Equations" gives what seems like an $ℝ^{4}$ version :

$ds^2 = \frac{dt^2 - dx^2 - dy^2 - dz^2}{1+\frac{K}{4}(t^2 - x^2 - y^2 - z^2}$

Which makes sense, since this is basically the Lorentzian version of the stereographic projection. My guess is that this suffers from geodesic incompleteness for the same reason that the stereographic projection does.

The most natural choice would be $S^{4}$ of course, but beyond the obvious CTC problems, it doesn't even admit a Lorentzian metric.

So why $ℝ \times S^3$ in particular? Are there any other nicely behaved topologies for that metric, other than whatever quotient manifolds of it, I guess?