Consider the upper-half space model of hyperbolic $3$-space $\Bbb H^{+}_{3}$, the unique, simply-connected, $3$-dimensional complete Riemannian manifold with a constant negative sectional curvature equal to $-1$. From Nash embedding theorem, $\Bbb H^{+}_{3}$ can be isometrically embedded into $\Bbb R^n$ for some $n$.
Does there exist an isometric embedding $f : \Bbb H^{+}_{3}\rightarrow \Bbb R^8$? If so, how can it be constructed?
Hyperbolic 3-space embeds isometrically into a product of two hyperbolic planes (up to rescaling), see p.37 of this thesis. So there is an isometric embedding into $\mathbb{R}^{12}$. And if the hyperbolic plane could embed isometrically in $\mathbb{R}^{4}$, then you could do $\mathbb{H}^3$ in $\mathbb{R}^8$. But this seems to be open still (I found these references from @WillieWong’s comment).
This is a very long and somewhat relevant comment:$\newcommand\R{\mathbb{R}}$
In general, an embedding $\Phi: M^n \rightarrow \R^N$ is isometric if in local coordinates on $M$, it satisfies the system of PDEs $$ \partial_i\Phi\cdot\partial_j\Phi = g_{ij}, $$ where the $g_{ij}$ are the components of the metric tensor on $M$. This is in general a really really nasty system of PDEs.
Now assume $M$ is hyperbolic space $H^n$. There is one tantalizing situation where the PDEs can be written in a nicer form.
It has been known for a long time that when $n=2$ and $N=3$, an isometric embedding of $H^2$ can be used to construct a solution to the sine-Gordon equation, $$ u_{tt} - u_{xx} = \sin(u), $$ which is a pretty nice nonlinear hyperbolic PDE and has been used to study the geometric properties of the isometric embedding. In particular, it can be used to prove that there is no global isometric immersion of $H^2$ into $\R^3$, even though there are global solution to the sine-Gordon equation.
There is a less well known generalization of this to higher dimensions first studied by Élie Cartan. He showed that isometric embeddings of $H^n$ in $\R^{2n-1}$ can be used to construct solutions to a quasilinear system of PDEs for an $O(n)$-valued function, which can be called the generalized sine-Gordon equations. This is a much nicer system of equations than the general equation for isometric embeddings. Since there are more equations than unknown functions, it is overdetermined but it is involutive in the sense of Cartan-Käher. It is a well-posed overdetermined hyperbolic system of PDEs.
This system of PDEs was studied extensively by Tenenblat and Terng. I have always believed but was never able to prove that there are global solutions to this system and that they might be useful in establishing whether global isometric embeddings of $H^n$ in $\R^{2n-1}$ exist or not.
