Second-order equations and local isometric immersions of pseudo-spherical surfaces
Résumé
We consider a class of differential equations that describe pseudo-spherical surfaces of the form $u_t=F(u,u_x,u_{xx})$ and $u_{xt}=F(u, u_x)$. 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 $u$.
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