Elliptic, hyperbolic and parabolic geometry

The three models of surfaces can be seen within a common framework.
Let's consider the space $M^{4}$ , that is, the manifold $R^{4}$ with coordinates $(x, y, z, t)$ but with the pseudo-Riemannian metric given in each tangent space by

d s^{2} = d x^{2} + d y^{2} + d z^{2} - d t^{2} .

This space with this pseudo-metric is called Minkowski space, and can be also denoted by $R^{3, 1}$ .
This metric is left invariant by the group $O (3, 1)$ when acting on $M^{4}$ . It also leaves invariant the cone

t^{2} - x^{2} - y^{2} - z^{2} = 0

so is logical that we pay attention to it, because it is very "physical": the group $O (3, 1)$ represents all the change of point of view of a physical observer, that is, rotations and Lorentz boost.
Now, following [TRTR] from Penrose page 423, we can consider the family of hyperplanes inside $M^{4}$ given by

z + t + λ (t - z) = 2

intersecting our cone in different 2-dimensional submanifolds $P_{λ}$ of $M^{4}$ .
Here we have a picture from that book, where we consider only $(x, z, t)$ for purposes of drawability.

If we analyse the case $λ = 0$ , we observe that we obtain the submanifold $P_{0}$ , $E$ in the picture with a shape of a parabola, given by the embedding

ϕ : R^{2} ⟶ M^{4}

(u_{1}, u_{2}) \mapsto (u_{1}, u_{2}, 1 - \frac{u_{1}^{2} + u_{2}^{2}}{4}, 1 + \frac{u_{1}^{2} + u_{2}^{2}}{4})

We can look for the (possibly pseudo) Riemannian metric inherited by $P_{0}$ . Since

d ϕ = (\begin{matrix} 1 & 0 \\ 0 & 1 \\ - \frac{u_{1}}{2} & - \frac{u_{2}}{2} \\ \frac{u_{1}}{2} & \frac{u_{2}}{2} \end{matrix})

we conclude that $P_{0}$ consists of $R^{2}$ with the euclidean metric:

g = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})

That is, although we see it like a parabola, intrinsically is only the Euclidean plane, and that is the reason the latter is sometimes called parabolic geometry.
But, what happens for others $λ$ ? Let's see. We have a far more complicated parametrization of our embedded manifold:

ϕ (u_{1}, u_{2}) = (u_{1}, u_{2}, \frac{- 1 + λ + \sqrt{- (1 + λ)^{2} (- 1 + λ (u_{1}^{2} + u_{2}^{2}))}}{2 λ},

\frac{2 + λ + \sqrt{- (1 + λ)^{2} (- 1 + λ (u_{1}^{2} + u_{2}^{2})) + \frac{1 - \sqrt{- (1 + λ)^{2} (- 1 + λ (u_{1}^{2} + u_{2}^{2}))}}{λ}}}{2 (1 + λ)})

(computations made with Mathematica).
If we compute the (possibly pseudo) Riemannian metric in this chart we obtain the inherited metric in $R^{2}$

g_{λ} = (\begin{matrix} \frac{1 - λ u_{2}^{2}}{1 - λ (u_{1}^{2} + u_{2}^{2})} & \frac{λ u_{1} u_{2}}{1 - λ (u_{1}^{2} + u_{2}^{2})} \\ \frac{λ u_{1} u_{2}}{1 - λ (u_{1}^{2} + u_{2}^{2})} & \frac{1 - λ u_{1}^{2}}{1 - λ (u_{1}^{2} + u_{2}^{2})} \end{matrix})

Next step is to show that for $λ > 0$ we have an isometry into the usual sphere (a model for spherical geoemtry), and that for $λ < 0$ we have an isometry into a model for hyperbolic geometry. This would justify the terms elliptic and hyperbolic for those geometries, as you can see at the picture above ( $S$ and $H$ respectively). In order to keep it simple we can work with $λ = 1$ and $λ = - 1$ .
For $λ = 1$ , we have

g = (\begin{matrix} \frac{1 - u_{2}^{2}}{1 - (u_{1}^{2} + u_{2}^{2})} & \frac{u_{1} u_{2}}{1 - (u_{1}^{2} + u_{2}^{2})} \\ \frac{u_{1} u_{2}}{1 - (u_{1}^{2} + u_{2}^{2})} & \frac{1 - u_{1}^{2}}{1 - (u_{1}^{2} + u_{2}^{2})} \end{matrix})

But if we look at a typical chart of the usual sphere in $E^{3}$

ψ : (x, y) ⟼ (x, y, \sqrt{1 - x^{2} - y^{2}})

and compute the metric we obtain just

g = (\begin{matrix} \frac{1 - y^{2}}{1 - (x^{2} + y^{2})} & \frac{x y}{1 - (x^{2} + y^{2})} \\ \frac{x y}{1 - (x^{2} + y^{2})} & \frac{1 - x^{2}}{1 - (x^{2} + y^{2})} \end{matrix})

so they are the same.
And for $λ = - 1$ we have

g = (\begin{matrix} \frac{1 + u_{2}^{2}}{1 + u_{1}^{2} + u_{2}^{2}} & \frac{- u_{1} u_{2}}{1 + u_{1}^{2} + u_{2}^{2}} \\ \frac{- u_{1} u_{2}}{1 + u_{1}^{2} + u_{2}^{2}} & \frac{1 + u_{1}^{2}}{1 + u_{1}^{2} + u_{2}^{2}} \end{matrix})

But if we look at the typical model for hyperbolic plane that consists of the pseudosphere embedded in $M^{3}$ :

x^{2} + y^{2} - z^{2} = - 1

with a chart given by

ψ : (x, y) ⟼ (x, y, \sqrt{1 + x^{2} + y^{2}})

we obtain the inherited metric in $R^{2}$ (keep an eye: inherited from the Minkowski metric, not the Euclidean one):

g = (\begin{matrix} \frac{1 + y^{2}}{1 + x^{2} + y^{2}} & \frac{- x y}{1 + x^{2} + y^{2}} \\ \frac{- x y}{1 + x^{2} + y^{2}} & \frac{1 + x^{2}}{1 + x^{2} + y^{2}} \end{matrix})

so we are done.