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 $S$?

Asymptotic parametrizations exist locally, and in case $S$ has constant curvature may be constructed globally as well. 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 parametrization, then when does one exist?