Let $S=[(x,y)\in\mathbb{H}^{2}:0< x< 2\pi]$ where $\mathbb{H}^{2}$ is a hyperplane with standard metric. I.e., a strip whose boundary components are geodesics, both approaching a common infinite point.


My simple question is about whether it can be isometrically embedded in $\mathbb{R}^{3}$.


I don't remember exactly but I read some articles about an isometric embedding from a portion of $\mathbb{H}^{2}$ into $\mathbb{R}^{3}$. For example, it might have been proved that infinite polygons of some type can be isometrically embedded but I could not catch the meaning of the type the author said about. Or, I read in another article that every(?) equidistant strip in $\mathbb{H}^{2}$ can be embedded in $\mathbb{R}^{3}$. If this is true, the answer to my question would be positive.


Is there anyone who know about this content precisely?


How about $T=[(x,y)\in\mathbb{H}^{2}:0<\sqrt{{x}^{2}+{y}^{2}}<{e}^{2\pi}]$? This strip also has geodesic boundary components but both components are approaching different infinite point.



Embedding Again
The comment given by Joseph gives indeed the answer...
In Borisenko's survey (iop.org/EJ/abstract/0036-0279/56/3/R01), one can find in section 2.4 the following result due to Poznyak:
"Any polygon belonging to one of the classes $M\_1$ and $M\_2$ can be regularly and isometrically embedded in $E^3$".


The polygons considered are intersections of a finite or denumerable set of closed halfplanes whose boundaries have no points in common.
The class $M\_1$ (which is the only one needed here) is defined as the class of all polygons for which there is a horocycle $O$ in the plane such that the greatest lower bound of the lengths of the orthogonal projections of the sides on this horocycle is positive. 


As observed by Poznyak in his original article, any polygon with finitely many sides which does not contain a halfplane belongs to the class $M\_1$.
The band $S$ you described in your question is of this type, hence it can be isometrically immersed in the euclidean space $E^3$.


Here is the reference to the english translation of Poznyak's article:


“Isometric embedding in $E^3$ of certain non-compact domains of the
Lobachevsky plane”, Mat. Sb. 102 (1977), 3–12; English transl., Math. USSR-Sb. 31
(1977), 1–8.


Concerning your surface $T$, it is a half-plane and thus cannot be isometrically immersed in $E^3$ (as explained in section 2.4 of the survey).


