### Abstract

We consider the 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 pseudospherical surface determined by a solution u of such an equation, do the coefficients of the second fundamental form of the local isometric immersion in 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.

Original language | English (US) |
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Pages (from-to) | 605-643 |

Number of pages | 39 |

Journal | Communications in Analysis and Geometry |

Volume | 24 |

Issue number | 3 |

DOIs | |

State | Published - 2016 |

### Keywords

- Evolution equations
- Isometric immersions
- Nonlinear hyperbolic equations
- Pseudo-spherical surfaces

### ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Geometry and Topology
- Statistics, Probability and Uncertainty

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## Cite this

Kahouadji, N., Kamran, N., & Tenenblat, K. (2016). Second-order equations and local isometric immersions of pseudo-spherical surfaces.

*Communications in Analysis and Geometry*,*24*(3), 605-643. https://doi.org/10.4310/CAG.2016.v24.n3.a7