Local isometric immersions of pseudospherical surfaces described by a class of third order differential equations

2024/12/28 by Guo, Mingyue, Zhenhua Shi, Shi, Zhenhua
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph)

paper · pdf · doi:10.48550/arxiv.2501.01444

openalex publication_date 2024/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We discuss a specific type of pseudospherical surfaces defined by a class of third order differential equations, of the form u_t - u_xxt = λu² u_xxx + G(u, u_x, u_xx), and poses a question about the dependence of the triples \a,b,c\ of the second fundamental form in the context of local isometric immersion in 𝔼³. It is demonstrated that the triples \a,b,c\ of the second fundamental form are not influenced by a jet of finite order of u. Instead, they are shown to rely on a jet of order zero, making them universal and not reliant on the specific solution chosen for u.

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