# Coordinate transformation for the hyperbolic plane to the pseudo sphere

There are three common ways to represent the hyperbolic plane. One usually starts with the hyperboloid $$x_1^2+x_2^2-x_3^2=-1$$ embedded in a Minkowski space with metric $$ds^2 = dx_1^2+dx_2^2-dx_3^2$$. If the hyperboloid is parameterized with the embedding

$$x_1(t,\phi)=\sinh t \cos\phi\\ x_2(t,\phi)=\sinh t\sin\phi\\ x_3(t,\phi)=\cosh t$$

one finds the induced metric for the hyperbolic plane $$ds^2=dt^2+\sinh^2t\,\,d\phi^2$$. Two further familiar coordinate systems include the projection onto the Poincare disk, given by the transformations

$$y_i = \frac{x_i}{1+x_3} \text{ for } i \in \{1,2\}$$

The metric on the Poincare disk is given by $$ds^2 = \frac{2}{1-\vec y^2}(dy_1^2+dy_2^2).$$ Although it is called projection, it is actually invertible and therefore a regular diffeomorphism.

Finally, inversion on a circle with radius $$r=\sqrt{2}$$ centered at $$(0,-1)$$ projects the Poincare disk onto the upper half plane. The transformation is again a diffeomorphism

$$z_1 = 2\frac{y_1}{y_1^2+(y_2+1)^2}\\ z_2 = 2\frac{y_2+1}{y_1^2+(y_2+1)^2}-1$$

For the sake of completeness, the metric on the upper half plane is given by $$ds^2 = \frac{1}{z_2^2}(dz_1^2+dz_2^2).$$

I am interested in a fourth representation of the hyperbolic plane. It is only an immersion, only a patch of the hyperbolic plane is covered. But on the other hand it is immersed into a regular Euclidean space with positive signature $$ds^2 = dx_1^2+dx_2^2+dx_3^2$$.

The immersion is described by $$x_1(T,\Phi) = \sqrt{1-{\rm e}^{2T}}-\text{arctanh}\sqrt{1-{\rm e}^{2T}}\\ x_2(T,\Phi) = {\rm e}^T \cos\Phi\\ x_3(T,\Phi) = {\rm e}^T \sin\Phi$$

and the corresponding metric is given by $$ds^2 = dT^2+{\rm e}^{2T} d\Phi^2$$.

Does anyone know the coordinate transformation $$T(t,\phi)$$ and $$\Phi(t,\phi)$$, that directly transforms from the metric $$ds^2 = dt^2+\sinh^2 t\,\,d\phi^2$$ to the metric $$ds^2 = dT^2+{\rm e}^{2T} d\Phi^2$$? It doesn't look like an overwhelmingly difficult problem, but the resulting set of partial differential equations doesn't offer any clue for a possible solution:

$$t_{,T}^2+\sinh^2 t\,\, \phi_{,T}^2 = 1\\ t_{,T}t_{,\Phi} +\sinh^2 t\,\, \phi_{,T}\phi_{,\Phi}=0\\ t_{,\Phi}^2+\sinh^2 t\,\, \phi_{,\Phi}^2 = {\rm e}^{2T}$$

Is it possible to perform a Wick rotation and get from the first embedding to the immersion? It also has some resemblance to the different slices of de-Sitter space-time for open and flat three-dimensional sections, but it is not a much help either to me, unfortunately.

• Second sentence: this is not Euclidean space, but Minkowski space. Apr 21, 2021 at 7:26

In the pseudosphere representation you wrote down, $$T\in (-\infty,0)$$.
If you let $$R = e^{-T}$$ (which now takes values in $$(1,\infty)$$), your metric becomes $$R^{-2} (dR^2 + d\Phi^2)$$ and conects to the upper half plane model. From there you can invert your transformation to get to the Poincare disk model, which has a known formula relating to the hyperboloid model.