What is the name of the hyperbolic model?

Question

I am reading about the isometries of the hyperbolic plane, but I am being somewhat confused with the names of the models: Poincaré disk, Poncaré semiplane, Lorenz model, Klein model, etc. I have the following metrics:

\begin{align*} g_{e}&=dx^2+e^{2x}dy^2\\ g_{P}&=\frac{dx^2+dy^2}{y^2}\\ g_{\cosh}&=dx^2+\cosh^2xdy^2\\ g_{D}&=4\frac{dx^2+dy^2}{(1-(x^2+y^2))^2} \end{align*}

Can someone help me to identify the names please or their exact definitions? I have already managed to verify that they are all isometric immersions of each other, but I do not know their names and I would like to know them. Thank you.

Answer 1

These metrics are known to me by the following names:

• $dx^2 + e^{2x}dy^2$ is the horocyclic coordinate system. The constant-$x$ lines are concentric horocycles and the constant-$y$ lines are unit-speed geodesics perpendicular to the horocycles.
• $(dx^2+dy^2)/y^2$ is the Poincaré half-plane model, defined only for $y > 0$. The constant-$y$ lines are concentric horocycles and the constant-$x$ lines are geodesics perpendicular to the horocycles, this time not unit speed however.
• $dx^2 + \cosh(x)^2 dy^2$ are geodesic coordinates or Fermi coordinates. The line $x=0$ is a unit-speed geodesic, and the constant-$y$ lines are unit-speed geodesics perpendicular to the first. The spherical analogue is the equirectangular projection with metric $dx^2 + \cos(x)^2 dy^2$.
• $4(dx^2+dy^2)/(1-(x^2+y^2))^2$ is the Poincaré disk model, defined only on the unit disk, where geodesics are circular arcs perpendicular to the unit circle. The spherical analogue is the stereographic projection with metric $4(dx^2+dy^2)/(1+(x^2+y^2))^2$.

More metrics you mentioned:

• $(dx^2+dy^2)/(1-(x^2+y^2))+(xdx+ydy)^2/(1-(x^2+y^2))^2$ is the Beltrami-Klein model aka Klein model. Its spherical analogue is the gnomonic projection $(dx^2+dy^2)/(1+(x^2+y^2))-(xdx+ydy)^2/(1+(x^2+y^2))^2$.
• The hyperboloid model is not a model defined by a metric on $\mathbb R^2$, but by the metric $dx^2+dy^2-dz^2$ on one sheet of the two-sheeted hyperboloid $x^2+y^2-z^2=-1$ in $\mathbb R^3$. The spherical analogue of this model is simply the embedding of the 2-sphere into euclidean 3-space.