The isometric immersion that you describe above is the higher dimensional pseudosphere. Now, concerning your final question, I presume that you need to search about isometric immersions of the hyperbolic space $\mathbb H^n$ by means of a warped product representation (of $\mathbb H^n$) into the Euclidean space.  

Now, some additional things that you might be interested to: 

 1. There are many (explicit in some cases) local isometric immersions from $\mathbb H^n$ to $\mathbb R^{2n-1}$. These can be constructed by using either the Ribaucour or the Bäcklund transformation (for instance, see the papers by [Dajczer-Tojeiro][1] and [Tenenblat-Terng][2]). 
 2. Local isometric immersions of the hyperbolic plane $\mathbb H^2$ into $\mathbb R^3$ imply "local" solutions, that is, solutions that are not defined on the whole $\mathbb R^2$, of the sine-Gordon equation and vice versa. Therefore, it follows from Hilbert's theorem that there is no "global" solution, that is, a solution defined on the whole plane $\mathbb R^2$, of the sine-Gordon equation. Just like in the case of dimension two, the same also happens in the higher dimensional case where now you will end up with a system of PDES (see for instance [Dajczer-Tojeiro][3]). We can have local solutions to this system but we don't know if there exists any global. The existence of a global solution would imply the existence of a global isometric immersion of $\mathbb H^n$ into $\mathbb R^{2n-1}$, which would give a non affirmative answer to the major still open problem (in submanifolds) up to this day, which is the following conjectured extension of Hilbert's theorem:

> There is no global isometric immersion from $\mathbb H^n$ to $\mathbb R^{2n-1}$

However, the above holds true in some very special cases. For instance:
 

 - If the immersion is also minimal (the mean curvature vanishes) (see [Moore][4]). 

*I should also mention here that $\mathbb H^2$ admits no minimal immersion in any Euclidean space. (for a proof of this fact see either "*Lectures on minimal submanifolds*" by Lawson, or [Bryant][5], or [Di Scala][6]).*

 - (weaker) If the immersion has also bounded mean curvature (see [here][7])
 - (even weaker) If also the length of the mean curvature of the immersion does not go to infinity too fast, that is, exponentially fast (see [here][8])


  [1]: https://www.cambridge.org/core/journals/proceedings-of-the-london-mathematical-society/article/abs/an-extension-of-the-classical-ribaucour-transformation/DCF1A5B953590BBACC770B8BDB738177
  [2]: https://annals.math.princeton.edu/1980/111-3/p04
  [3]: https://eudml.org/doc/153758
  [4]: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2ahUKEwjjvoWztq7zAhUp_7sIHatZBGIQFnoECAIQAQ&url=https%3A%2F%2Fprojecteuclid.org%2Fjournals%2Fpacific-journal-of-mathematics%2Fvolume-40%2Fissue-1%2FIsometric-immersions-of-space-forms-in-space-
  [5]: https://www.jstor.org/stable/1999793
  [6]: https://academic.oup.com/blms/article-abstract/35/6/825/305609?redirectedFrom=fulltext
  [7]: https://link.springer.com/article/10.1134%2FS0001434607070024
  [8]: https://link.springer.com/article/10.1007%2Fs00013-020-01565-x