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As my title says above, I am trying to find answers to and also good online reference where I can find complete description of projective models of hyperbolic space, de-Sitter space and anti-de-Sitter space?

I understand that for hyperboloid model of hyperbolic $3$-space sitting inside $\mathbb{R}^{3,1}$ for example, we take each point $p$ in $\mathbb{H}^3$, draw the unique line through $p$ and $0$, denote by $f(p)$ the point where it intersects the plane $\{x_4=1\}$. So $f(x)=\frac{x}{x_4}$. Thus $f$ defines a projection from $\mathbb{H}^3$ to $\{x_4=1\}$. But there is also another projective model of $\mathbb{H}^3$: where you just consider the map : $g:\mathbb{H}^3 \to \mathbb{RP}^3$ defined by: $g(p)=[p]\in \mathbb{RP}^3$. The thing which is not clear to me is why these two models are equivalent? I understand that in the case of $f$, $f(\mathbb{H}^3)$ lies in a single chart of $\mathbb{RP}^3$, so is it true $g(\mathbb{H}^3)$ lies in single chart in $\mathbb{RP}^3$? Are these two projective models equivalent?

How exactly do we define projective models for the anti-de-Sitter spaces? We can define projections from $AdS_3=\{x\in \mathbb{R}^{4,2}:x_1^2+x_2^2-x_3^2-x_4^2=-1\}$ to $\mathbb{RP}^3$ by $g_1(p)=[p]\in \mathbb{RP}^3$. $g_1$ is similar to $g$, but is there a $f_1$ similar to $f$, where it projects to a certain plane?

I have seen in papers that $g_1(AdS_3)\subset \mathbb{RP}^3$ is the interior of a quadric $Q\subset \mathbb{RP}^3$ of signature $(1,1)$. I am not sure what exactly is meant by quadric of signature $(1,1)$ in $\mathbb{RP}^3$? I would appreciate a definition. And also, why is $g_1(AdS_3)$ a the interior of quadric of signature (1,1)?

Also, what is meant by "$g_1(AdS_3)$ is a quadric foliated by two families of projective lines, which isotropy curves of the Lorenz conformal structure on $\partial_\infty{AdS_3}$"? And why and how is it true?

Thak you for your time and if you could answer and mention also some resources, that will be great.

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  • $\begingroup$ I don't see much difference between $f$ and $g$. Using $g$ you take the point on the hyperboloid and treat it as a representant of a point in $\mathbb{RP}^3$. Using $f$ you do the same but then choose a pecific representant of that point. So $f$ is useful to visualize things in an affine way, but $g$ is enough to get things into $\mathbb{RP}^3$. You could do the same for $AdS_3$ I guess: simply project onto any affine space. $\endgroup$
    – MvG
    Jun 3, 2013 at 11:17
  • $\begingroup$ Thanks for your comment, I think $f,g$ are equivalent too. But for $H^3$, there is a canonical choice of the hyperplane (namely $\{x_4=1\}$) where you can project $H^3$ (the hyperboloid model), and also the image of the projection becomes unit ball in that hyperplane. But for $AdS_3$, is there such a canonical choice? Anyway, my main question is: why is $g_1(AdS_3)$ a the interior of quadric of signature (1,1)? And what is meant by "$g_1(AdS_3)$ is a quadric foliated by two families of projective lines, which isotropy curves of the Lorenz conformal structure on $\partial_\infty{AdS_3}$"?How? $\endgroup$ Jun 3, 2013 at 12:58

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