A question about embedding hyperbolic space onto pseudosphere

Question

I have a difficulty with hyperbolic geometry.

Let $\mathbb{H}^{2}$ be a 2-dimensional hyperbolic plane.

(i.e., upper half plane in $\mathbb{R}^{2}$ with a metric $\frac{ds}{y}$)

(or, upper half plane in $\mathbb{C}$ with a metric $\frac{|dz|}{\textrm{Im}(z)}$ )

You may have heard about pseudosphere in $\mathbb{R}^{3}$. Let's denote the half-pseudosphere by $P$

This can be obtained by glueing both side($x=0$ and $x=2\pi$ parts) of

$\left[(x,y)\in\mathbb{H}^{2} : 0\leq x\leq 2\pi, y>1 \right]$

Denoting this quotient space by $A$(Note that $A$ is homeomorphic to a cylinder), we can now get a "globally isometrically embedding" map $\rho:A \rightarrow \mathbb{R}^{3}$ with $\rho(A)=P$

To be specific, $\rho(x,y) = (t-\tanh(t), \textrm{sech}(t)\ \cos(x), \textrm{sech}(t)\sin(x))$ where $t=\textrm{arccosh}(y)$

Now the question I have is following

: Is there another possible (global and isometric)embedding $\rho$ from $A$ into $\mathbb{R}^{3}$ ?

Actually I'm interested in $\rho(A)$ and by calculating, one can easily find that if $\rho(A)$ is a "surface of revolution", then it should be $P$(up to isometry of $\mathbb{R}^3$"

Thus I'm looking for $\rho(A)$ which is different from a surface of revolution.

Any idea?

Answer 1

You should be looking at the theory of Bäcklund transformations for surfaces of Gaussian curvature $K=-1$. There is a large literature on this, and there are many examples of pseudospherical immersions that are not surfaces of revolution. You should especially look at the work of Chuu-Lian Terng in this area. She, together with her husband, Richard Palais, have developed this theory quite a bit in the past 30 years, and they have some excellent graphics for these surfaces. Whether they can answer your specific question about isometrically embedding your particular domain $A$ into $\mathbb{R}^3$, I don't know, but that would be a good place to start.