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.