Second-order equations and local isometric immersions of pseudo-spherical surfaces

Abstract

We consider the class of differential equations that describe pseudo-spherical surfaces of the form $u_t=F(u,u_x,u_{xx})$ and $u_{xt}=F(u, u_x)$ given in Chern-Tenenblat \cite{ChernTenenblat} and Rabelo-Tenenblat \cite{RabeloTenenblat90}. We answer the following question: Given a pseudo-spherical surface determined by a solution $u$ of such an equation, do the coefficients of the second fundamental form of the local isometric immersion in $\mathbb{R}^3$ depend on a jet of finite order of $u$? We show that, except for the sine-Gordon equation, where the coefficients depend on a jet of order zero, for all other differential equations, whenever such an immersion exists, the coefficients are universal functions of $x$ and $t$, independent of $u$.