The Vectorial Ribaucour Transformation for Submanifolds of Constant Sectional Curvature

• Published in: The Journal of geometric analysis 2018-07, Vol.28 (3), p.1903-1956
• Main Authors: Guimarães, D., Tojeiro, R.
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Convex and Discrete Geometry; Curvature; Decomposition; Differential Geometry; Dynamical Systems and Ergodic Theory; Euclidean geometry; Formulas (mathematics); Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Manifolds (mathematics); Mathematics; Mathematics and Statistics; Partial differential equations; Theorems; Transformations (mathematics)
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-017-9892-2

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 C n and n -dimensional Lagrangian submanifolds with constant curvature c of the complex projective space C P n ( 4 c ) or the complex hyperbolic space C H n ( 4 c ) of complex dimension n and constant holomorphic curvature 4c.