The Ribaucour transformation for flat Lagrangian submanifolds.
The authors study Ribaucour transformations for flat Lagrangian submanifolds of (flat) complex \(n\)-space and complex projective space. They show that such submanifolds of complex \(n\)-space are in correspondence with the solutions of a completely integrable system of PDEs which describe a system of globally defined principal coordinates on the submanifold: this system is apparently one of the type which Darboux named an ``E-system''. The Ribaucour transformations generate new submanifolds from old. The corresponding theory for submanifolds of \(\mathbb{C} \text{P}^n\) makes use of the fact that flat Lagrangian submanifolds of \(\mathbb{C} \text{P}^n\) are precisely the images, under Hopf fibration, of flat Lagrangian submanifolds of complex \(n+1\)-space which also lie in \(S^{2n+1}\).
scientific article; zbMATH DE number 1542078