I found this local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$, given by Schur (1886) in [Über die Deformation der Räume constanten Riemannschen Krümmungsmaasses](https://eudml.org/doc/157211) as follows, $(1\leq k\leq n-1)$:
\begin{align\*}
x\_{2k-1}&=\frac{a^2}{z\_n}\cos \frac{z\_k}{a}\\
 x\_{2k}&=\frac{a^2}{z\_n}\sin \frac{z\_k}{a}\\
 x\_{2n-1}&=a\int^{z\_n}\frac{\sqrt{z\_n^2-(n-1)a^2}}{z\_n^2}dz\_n 
\end{align\*}


I'm trying to prove the following statements:


1. It is a local isometric immersion.



> 
> Here, taking $\phi:\mathbb H^n\to \mathbb R^{2n-1}$ given by $\phi(z\_1,\dots,z\_n)=(x\_1,\dots,x\_{2n-1})$ I imagine that $\phi^\*g\_{\mathbb R^{2n-1}}=g\_{\mathbb H^n}$ which would prove that is a isometric immersion, but the conditions for $x\_{2n-1}$ to be well defined make it only a local immersion.
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2. It has a constant curvature $K\equiv -1/a^2$.



> 
> This is where I have some problems: is this a consequence of the above result? I'm trying with Christoffel's symbols.
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3. Any ideas to prove that image $\phi(z\_1,\dots,z\_n)$ is not a complete surface?


I started to see this example as a coincidence but I was thinking a bit about what happens in $\mathbb R^3$: there are $3$ types of smooth surfaces of revolution with negative constant curvature given by $x(u,v)=(f(v)\cos u,f(v)\sin u,g(v))$, this is clear when solving
$$K=-\frac{f''(v)}{f(v)}.$$
Is there something similar in $\mathbb R^{2n-1}$, how many surfaces with these characteristics exist? is there a differential equation as in $\mathbb R^3$?



A local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$
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](https://www.cambridge.org/core/journals/proceedings-of-the-london-mathematical-society/article/abs/an-extension-of-the-classical-ribaucour-transformation/DCF1A5B953590BBACC770B8BDB738177) and [Tenenblat-Terng](https://annals.math.princeton.edu/1980/111-3/p04)).
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](https://eudml.org/doc/153758)). 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 $\mathbb R^{2n-1}$
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However, the above holds true in some very special cases. For instance:


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


*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](https://www.jstor.org/stable/1999793), or [Di Scala](https://academic.oup.com/blms/article-abstract/35/6/825/305609?redirectedFrom=fulltext)).*


* (weaker) If the immersion has also bounded mean curvature (see [here](https://link.springer.com/article/10.1134%2FS0001434607070024))
* (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](https://link.springer.com/article/10.1007%2Fs00013-020-01565-x))


I also recommend the following:


* The survey of Borisenko [here](https://www.ime.usp.br/%7Elymber/files/textos/RMS_56_3_R01.pdf)
* Chapter 5 of the book of Dajczer and Tojeiro *[Submanifold Theory
"Beyond an Introduction"](https://www.springer.com/gp/book/9781493996421)*


