Transformations of flat Lagrangian immersions and Egoroff nets

Abstract

We associate a natural $\lambda$-family ($\lambda \in \R \setminus \{0\} $) of flat Lagrangian immersions in $\C^n$ 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 $\frac{\U(n) \ltimes \C^n}{\OO(n) \ltimes \R^n}$. An interesting observation is that the family degenerates to an Egoroff net on $\R^n$ when $\lambda \to 0$. 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 $\C P^{n-1}$ and $\p$-invariant Egoroff nets follow nicely via a spherical restriction and Hopf fibration.