The Vectorial Ribaucour Transformation for Submanifolds of Constant Sectional Curvature

Abstract

We obtain a reduction of the vectorial Ribaucour transformation that preserves the class of submanifolds of constant sectional curvature of space forms, which we call the L-transformation. It allows to construct a family of such submanifolds starting with a given one and a vector-valued solution of a system of linear partial differential equations. We prove a decomposition theorem for the L-transformation, which is a far-reaching generalization of the classical permutability formula for the Ribaucour transformation of surfaces of constant curvature in Euclidean three space. As a consequence, we derive a Bianchi-cube theorem, which allows to produce, from k initial scalar L-transforms of a given submanifold of constant curvature, a whole k-dimensional cube all of whose remaining $$2^k-(k+1)$$ vertices are submanifolds with the same constant sectional curvature given by explicit algebraic formulae. We also obtain further reductions, as well as corresponding decomposition and Bianchi-cube theorems, for the classes of n-dimensional flat Lagrangian submanifolds of $${\mathbb {C}}^n$$ and n-dimensional Lagrangian submanifolds with constant curvature c of the complex projective space $${\mathbb {C}}{\mathbb {P}}^n(4c)$$ or the complex hyperbolic space $${\mathbb {C}}{\mathbb {H}}^n(4c)$$ of complex dimension n and constant holomorphic curvature 4c.