I know that if we endow $S^{n}$ with the round Riemannian metric, we will be able to join the North pole and the South pole by an unlimited number of geodesics, in particular the meridians, and indeed this is intuitive if we think about $S^2$. Yet I was now wondering how it would work for a Lorentzian metric in $S^3$, would we still have a similiar situation?
2026-03-27 21:17:53.1774646273
Geodesics on Lorentzian (2n-1)-Spheres
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I don't think it's possible to have a manifold with the topology of $S^3$ and a Lorentzian metric. The corresponding maximally-symmetric Lorentzian spaces of constant (non-zero) curvature would be deSitter space (dS) and Anti-deSitter space (AdS). These would be the closest analogs to spheres.