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

TL;DR

This paper investigates the local isometric immersions of pseudospherical surfaces defined by a specific class of third order differential equations, revealing that the second fundamental form depends only on the solution's value, not its derivatives.

Contribution

It demonstrates that the second fundamental form for these surfaces is universal and depends solely on the solution value, not on higher derivatives, clarifying the geometric structure of such surfaces.

Findings

Second fundamental form depends only on the solution value, not derivatives.

The form is universal across solutions, independent of specific solutions.

Provides insight into the geometric structure of pseudospherical surfaces.

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 $u$.