Which of the (non-)Euclidean planes can we embed into non-Euclidean 3-space?
I read the answers to this very interesting question and saw that we can in fact embed the Euclidean plane into hyperbolic 3-space using what is called a horosphere. However, as Hilbert showed us, the reverse is not true; we cannot embed the hyperbolic plane into Euclidean 3-space. This made me interested in considering the other non-Euclidean geometry: elliptic geometry. We can embed the plane from elliptic geometry into Euclidean 3-space - the result is spherical geometry - but:
a) is the reverse true: can we embed the Euclidean plane into elliptic 3-space?
b) Furthermore, is it possible to embed the elliptic plane into hyperbolic 3-space?
c) What about the reverse: can we embed the hyperbolic plane into elliptic 3-space?
Solution 1:
Here is the detailed answer to eliminate the confusion. Let $H^n, R^n, S^n$ denote the $n$-dimensional spaces of sectional curvature $-1, 0$ and $1$ respectively. Then the following hold (Items 1 and 2 are immediate, but items 3 and 4 are not):
For every $n$, $S^n$ embeds isometrically in $R^{n+1}$ and $H^{n+1}$ as a metric sphere of certain radius.
$R^n$ isometrically embeds in $H^{n+1}$ as a horosphere.
$H^2$ does not isometrically embed in $R^3$ (Hilbert's theorem). However, $H^2$ does embed (isometrically) in $R^6$.
$R^n$ and $H^n$ do not isometrically embed in $S^k$ for any $n$ and $k$.
David Brander wrote a UPenn thesis in 2003 summarizing the results on isometric embeddings between various constant curvature spaces. See http://davidbrander.org/penn.pdf for details (in particular, he explains what happens if one considers other dimensions and other constant curvature values, including embeddings between spaces with the same curvature sign).