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

2013/08/29 by Nabil Kahouadji, Niky Kamran, Kahouadji, Nabil +3 · 2 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1308.6545

openalex publication_date 2013/08/29 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

We consider the class of differential equations that describe\npseudo-spherical surfaces of the form u\_t=F(u,u\_x,u\_xx) and\nu\_xt=F(u, u\_x) given in Chern-Tenenblat citeChernTenenblat and\nRabelo-Tenenblat citeRabeloTenenblat90. We answer the following question:\nGiven a pseudo-spherical surface determined by a solution u of such an\nequation, do the coefficients of the second fundamental form of the local\nisometric immersion in \ℝ³ depend on a jet of finite order of u?\nWe show that, except for the sine-Gordon equation, where the coefficients\ndepend on a jet of order zero, for all other differential equations, whenever\nsuch an immersion exists, the coefficients are universal functions of x and\nt, independent of u.\n

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• A look on equations describing pseudospherical surfaces
• Local isometric immersions of pseudo-spherical surfaces and k-th order evolution equations

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