Local Isometric Immersions of Pseudospherical Surfaces Described by a Class of Third‐Order Partial Differential Equations

ABSTRACT In this paper, we study the problem of local isometric immersion of pseudospherical surfaces determined by the solutions of a class of third‐order nonlinear partial differential equations with the type . We prove that there are two subclasses of equations admitting a local isometric immersion into the three‐dimensional Euclidean space for which the coefficients of the second fundamental form depend on a jet of finite order of , and furthermore, these coefficients are universal, namely, they are functions of and , independent of . Finally, we show that the generalized Camassa–Holm equation describing pseudospherical surfaces has a universal second fundamental form.