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 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:
- 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.
- 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.
- 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 $\mathbb R^3$?
Best Answer
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:
However, the above holds true in some very special cases. For instance:
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, or Di Scala).
I also recommend the following: