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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, because a pair of transverse line fields on a surface can always be integrated locally, e.g., see p. 167 of Spivak Vol I. Furthermore, in the case where $S$ has constant curvature, an asymptotic parametrization may be constructed globally. This is established in the course of the usual proof 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?

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, because a pair of transverse line fields on a surface can always be integrated locally, e.g., see p. 167 of Spivak Vol I. It is also well known that when $S$ has constant curvature, an asymptotic parametrization may be constructed globally. This is the basis for the proof 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?

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?

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, because a pair of transverse line fields on a surface can always be integrated locally, e.g., see p. 167 of Spivak Vol I. Furthermore, in the case where $S$ has constant curvature, an asymptotic parametrization may be constructed globally. This is established in the course of the usual proof 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?