Isometric embedding of a genus g surface Can a genus $g$ surface with constant negative curvature and $g>1$ be isometrically embedded in $\mathbb{R}^4?$
 A: I don't know the answer to your question, and I wouldn't be surprised if it were open. You may be interested to know that any compact Riemannian two-manifold $(V, g)$ admits a $C^{\infty}$ isometric embedding $V \to \mathbb{R}^5$. See Gromov's Partial Differential Relations, pages 298 - 303.
A: The Nash-Kuiper Theorem implies the answer is yes if you only require the embedding to be of class $C^1$.  However, I believe the actual visualization problem for $g\geq 2$ (that is, producing the embedding in this case) is an open problem (unlike the visualization of the $C^1$-embedding of the flat torus, which has a $C^\infty$-isometric embedding into $\mathbb{R}^4$).
Since the smallest known $C^\infty$-embedding for the hyperbolic plane is $\mathbb{R}^6$ I would guess the answer is no for a genus 2 hyperbolic surface (but as far as I know it is open).  Note it is a theorem of Hilbert that the hyperbolic plane cannot be $C^r$-embedded into $\mathbb{R}^3$ for $r\geq 2$.  Later Efimov generalized this to closed hyperbolic surfaces.
I believe these facts and references may be found in:
Isometric Embedding of Riemannian Manifolds in Euclidean Spaces by Qing Han, and Jia-Xing Hong.
