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So I'm currently studying the hyperbolic plane with curvature $\kappa$. I'm mainly working with the hyperboloid model, which is defined as

$\mathbb{H}(\kappa) = \{(x,y,z) \in \mathbb{R}^3 ~|~ z > 0\}$ with metric $ds^2 = dx^2+dy^2-dz^2$.

Now I'm also interested in some different models, mainly the Poincare disk and Poincaré half-plane models. I have found plenty of references that give their definitions and explain the relation between the models but they are only defined in the case where $\kappa = -1$.

I think that the 'generalization' of the upper half-plane model is just the upper half-plane but now equipped the metric $ds^2 = \dfrac{dx^2+dy^2}{-\kappa y^2}$. However, I'm mainly struggling with how I can define the Poincaré disk model in the general case, as well as how the stereographic projection would then be defined.

Mee98
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  • Why can't you just do something very similar with the usual formula for the metric of the Poincare disk model, namely throw in a factor of $\frac{1}{\kappa}$? – Lee Mosher Nov 08 '20 at 19:44
  • I see that this works but I'm mainly wondering how you would define the stereographic projection from the hyperboloid model to the Poincaré disk model. Do we still project on the unit disk or do we take another radius? – Mee98 Nov 08 '20 at 19:51

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