Integrable systems in projective differential geometry

1999/03/25 by E. V. Ferapontov, Ferapontov, E. V.
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Nonlinear Waves and Solitons #math.DG

paper · pdf · doi:10.48550/arxiv.math/9903150

arxiv created 1999/03/25 · openalex publication_date 1999/03/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Some of the most important classes of surfaces in projective 3-space are reviewed: these are isothermally asymptotic surfaces, projectively applicable surfaces, surfaces of Jonas, projectively minimal surfaces, etc. It is demonstrated that the corresponding projective "Gauss-Codazzi" equations reduce to integrable systems which are quite familiar from the modern soliton theory and coincide with the stationary flows in the Davey-Stewartson and Kadomtsev-Petviashvili hierarchies, equations of the Toda lattice, etc. The corresponding Lax pairs can be obtained by inserting a spectral parameter in the equations of the Wilczynski moving frame.

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