# Embedding of the hyperbolic 3-space into the 4-dimensional Euclidean space

Is there a global isometric immersion of the 3-dimensional hyperbolic space into the 4-dimensional Euclidean space?

(I am aware of Hilbert's theorem, but that is on the embedding of $H^2$ in $E^3$. I have also seen the thesis of Brander 2003, but there I could not find an answer to my question. Furthermore, other posts in stackexchange do not refer to $H^3$ in $E^4$.)

• Did you have a look in this survey article? I don't think such an embedding is possible. A Riemannian $n$-manifold of constant negative sectional curvature cannot be isometrically immersed in $E^{2n−2}$. – Dietrich Burde Feb 24 '18 at 16:54
• Thank you! I did not know this survey. – kaffeeauf Feb 24 '18 at 16:59