2006/11/02 by Chuu-Lian Terng, Terng, Chuu-Lian, Erxiao Wang +1 · 1 citation
Mathematics · Physics and Astronomy · #37K35 #53D12 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #math-ph #math.AP #math.DG #math.MP #msc:37K35 #msc:53D12
paper · pdf · doi:10.48550/arxiv.math/0611063
23 pages, submitted
arxiv created 2006/11/02 · openalex publication_date 2006/11/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We associate a natural λ-family (λ∈ \R ∖ \0\ ) of flat Lagrangian immersions in \Cn with non-degenerate normal bundle to any given one. We prove that the structure equations for such immersions admit the same Lax pair as the first order integrable system associated to the symmetric space (\U(n) \ltimes \Cn)/(\OO(n) \ltimes \Rn). An interesting observation is that the family degenerates to an Egoroff net on \Rn when λ→ 0. We construct an action of a rational loop group on such immersions by identifying its generators and computing their dressing actions. The action of the generator with one simple pole gives the geometric Ribaucour transformation and we provide the permutability formula for such transformations. The action of the generator with two poles and the action of a rational loop in the translation subgroup produce new transformations. The corresponding results for flat Lagrangian submanifolds in \C Pn-1 and \p-invariant Egoroff nets follow nicely via a spherical restriction and Hopf fibration.