Is Hilbert's theorem generalizable to $H^3$ immersion in $\mathbb{R}^4$?

Question

The Hilbert theorem in differential geometry concerns the immersion of the hyperbolic plane in $\mathbb{R}^3$. Is it valid for $H^3$ in $\mathbb{R}^4$?, and for all $H^n$ in $\mathbb{R}^{n+1}$?

Answer 1

It is known that

• There is no isometric immersion of $\mathbb{H}^n$ into $\mathbb{R}^{2n-2}$, even a local one. There is a local immersion into $\mathbb{R}^{2n-1}$. This is due to E. Cartan (1919-20) and T. Otsuki (1955).

• There is a complete isometric immersion of $\mathbb{H}^n$ into $\mathbb{R}^{4n-3}$. This was proved recently (1980s) by W. Henke.

• There is an isometric embedding of $\mathbb{H}^n$ into $\mathbb{R}^{6n-6}$. For $n=2$ this was found by D. Blanusa in 1955.

All of this, with references and proofs, can be found in Master's thesis Isometric Embeddings between Space Forms by David Brander.

Specifically for the three-dimensional hyperbolic space $\mathbb{H}^3$:

• there is no local immersion into $\mathbb{R}^4$
• there is a local immersion into $\mathbb{R}^5$
• there is a complete immersion into $\mathbb{R}^9$
• there is an embedding into $\mathbb{R}^{12}$

For all we know, $\mathbb{H}^3$ could have an isometric embedding into $\mathbb{R}^5$ rather than $\mathbb{R}^{12}$...