TITLE: Isometric embedding of a genus g surface
QUESTION [5 upvotes]: Can a genus $g$ surface with constant negative curvature and $g>1$ be isometrically embedded in $\mathbb{R}^4?$

REPLY [7 votes]: 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.