Local isometric immersions of pseudo-spherical surfaces and evolution equations

Nabil Kahouadji, Niky Kamran*, Keti Tenenblat

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

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]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 pseudospherical 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;

Original languageEnglish (US)
Pages (from-to)369-381
Number of pages13
JournalFields Institute Communications
Volume75
DOIs
StatePublished - 2015

ASJC Scopus subject areas

  • General Mathematics

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