Local Isometric Immersions of Pseudo-Spherical Surfaces and Evolution Equations

  • Nabil Kahouadji
  • Niky KamranEmail author
  • Keti Tenenblat
Part of the Fields Institute Communications book series (FIC, volume 75)


The class of differential equations describing pseudo-spherical surfaces, first introduced by Chern and Tenenblat (Stud. Appl. Math. 74, 55–83, 1986), is characterized by the property that to each solution of a differential equation within the class, there corresponds a two-dimensional Riemannian metric of curvature equal to − 1. The class of differential equations describing pseudo-spherical surfaces carries close ties to the property of complete integrability, as manifested by the existence of infinite hierarchies of conservation laws and associated linear problems. As such, it contains many important known examples of integrable equations, like the sine-Gordon, Liouville and KdV equations. It also gives rise to many new families of integrable equations. The question we address in this paper concerns the local isometric immersion of pseudo-spherical surfaces in E3 from the perspective of the differential equations that give rise to the metrics. Indeed, a classical theorem in the differential geometry of surfaces states that any pseudo-spherical surface can be locally isometrically immersed in E3. In the case of the sine-Gordon equation, one can derive an expression for the second fundamental form of the immersion that depends only on a jet of finite order of the solution of the pde. A natural question is to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudo-spherical surfaces. In an earlier paper (Kahouadji et al., Second-order equations and local isometric immersions of pseudo-spherical surfaces, 25 pp. [arXiv:1308.6545], to appear in Comm. Analysis and Geometry (2015), we have shown that this property fails to hold for all other second order equations, except for those belonging to a very special class of evolution equations. In the present paper, we consider a class of evolution equations for u(x, t) of order k ≥ 3 describing pseudo-spherical surfaces. We show that whenever an isometric immersion in E3 exists, depending on a jet of finite order of u, then the coefficients of the second fundamental form are universal, that is they are functions of the independent variables x and t only.


Local Isometric Immersion Pseudo-spherical Surfaces Tenenblat sine-Gordon Equation Fundamental Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Research partially supported by NSERC Grant RGPIN 105490-2011 and by the Ministério de Ciência e Tecnologia, Brazil, CNPq Proc. No. 303774/2009-6.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  3. 3.Departamento de MatemàticaUniversidade de BrasiliaBrasíliaBrazil

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