# Canonical projections for $\mathbb{H}^2_1$ and $\mathbb{S}^2_1$

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?

• 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). – Zeno Rogue Jan 28 at 3:11
• 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. – Zeno Rogue Jan 28 at 3:14
• Other than that, I think you would basically project the space to itself via identity. – Zeno Rogue Jan 28 at 3:20