2017/06/07 by D. P. Guimarães, Ruy Tojeiro, Guimarães, Daniel +1
Mathematics · Medicine · Physics and Astronomy · #53B25 #Advanced Differential Geometry Research #Automotive and Human Injury Biomechanics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.1706.02405
openalex publication_date 2017/06/07 · openalex created_date 2022/09/23 · openalex updated_date 2026/07/28
We obtain a reduction of the vectorial Ribaucour transformation that\npreserves the class of submanifolds of constant sectional curvature of space\nforms, which we call the L-transformation. It allows to construct a family of\nsuch submanifolds starting with a given one and a vector-valued solution of a\nsystem of linear partial differential equations. We prove a decomposition\ntheorem for the L-transformation, which is a far-reaching generalization of\nthe classical permutability formula for the Ribaucour transformation of\nsurfaces of constant curvature in Euclidean three space. As a consequence, we\nderive a Bianchi-cube theorem, which allows to produce, from k initial scalar\nL-transforms of a given submanifold of constant curvature, a whole\nk-dimensional cube all of whose remaining 2k-(k+1) vertices are\nsubmanifolds with the same constant sectional curvature given by explicit\nalgebraic formulae. We also obtain further reductions, as well as corresponding\ndecomposition and Bianchi-cube theorems, for the classes of n-dimensional\nflat Lagrangian submanifolds of \ℂn and n-dimensional Lagrangian\nsubmanifolds with constant curvature c of the complex projective space\n mathbb C mathbb Pn(4c) or the complex hyperbolic space mathbb C mathbb\nHn(4c) of complex dimension n and constant holomorphic curvature~4c.\n