Let $S$ be a complete simply connected negatively curved surface immersed in Euclidean space $\textbf{R}^3$. Does there exist a parametrization $f\colon\textbf{R}^2\to\textbf{R}^3$ for $S$ such that the curves $t\mapsto f(t,s)$ and $s\mapsto f(t,s)$ are asymptotic, i.e., their second derivatives are tangent to $S$?

Such a parametrization would be called a global asymptotic net. It is well-known that asymptotic nets exist locally. Furthermore, if $S$ has constant curvature then it has a global asymptotic net. This is established in the course of usual proofs of Hilbert's theorem on the nonexistence of complete surfaces of constant negative curvature in $\textbf{R}^3$.

If it is not possible in general to construct a global asymptotic net, then when does one exist?