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We know that there exists a canonical projection $\pi:\mathbb{S}^2\to\mathbb{R}\mathbb{P}^2$. Are there similar transformations for hyperbolic plane $\mathbb{H}^2_1$ or pseudosphere $\mathbb{S}^2_1$ in Minkowski 3-space?

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  • $\begingroup$ Similar and canonical in what sense? The elliptic plane is a quotient space of the sphere, likewise the hyperbolic plane has quotient spaces too, some of which are important (e.g. Klein Quartic) but a bit less canonical (a Klein quartic has only 168 automorphisms which is also maximal for a surface of that genus, while the elliptic plane can be translated and rotated in any way and you get the same). $\endgroup$ – Zeno Rogue Jan 28 at 3:11
  • $\begingroup$ As for the projection which identifies $x$ and $-x$, in the hyperbolic plane you do that by default (but you could also not do it and get two disconnected hyperbolic planes), I do not know much about the pseudosphere, but it seems that such a projection should work there. $\endgroup$ – Zeno Rogue Jan 28 at 3:14
  • $\begingroup$ Other than that, I think you would basically project the space to itself via identity. $\endgroup$ – Zeno Rogue Jan 28 at 3:20

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