Local isometric immersions of pseudospherical surfaces described by a class of third order differential equations

Abstract

We discuss a specific type of pseudospherical surfaces defined by a class of third order differential equations, of the form $u_t-u_{xxt}=\lambda u^2{u}_{xxx}+G(u,u_x,u_{xx})$, and poses a question about the dependence of the triples $\{a,b,c\}$ of the second fundamental form in the context of local isometric immersion in ${\mathbb{E}}^3$. It is demonstrated that the triples $\{a,b,c\}$ of the second fundamental form are not influenced by a jet of finite order of $u$. Instead, they are shown to rely on a jet of order zero, making them universal and not reliant on the specific solution chosen for $u$.

This submission has been superseded by arXiv:2505.19728. Please refer to and cite the updated version