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The Vectorial Ribaucour Transformation for Submanifolds of Constant Sectional Curvature
| Content Provider | Semantic Scholar |
|---|---|
| Author | Guimarães, Damaris Tojeiro, Rita |
| Copyright Year | 2017 |
| 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 $$2^k-(k+1)$$2k-(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 $${\mathbb {C}}^n$$Cn and n-dimensional Lagrangian submanifolds with constant curvature c of the complex projective space $${\mathbb {C}}{\mathbb {P}}^n(4c)$$CPn(4c) or the complex hyperbolic space $${\mathbb {C}}{\mathbb {H}}^n(4c)$$CHn(4c) of complex dimension n and constant holomorphic curvature 4c. |
| Starting Page | 1903 |
| Ending Page | 1956 |
| Page Count | 54 |
| File Format | PDF HTM / HTML |
| DOI | 10.1007/s12220-017-9892-2 |
| Volume Number | 28 |
| Alternate Webpage(s) | https://arxiv.org/pdf/1706.02405v1.pdf |
| Alternate Webpage(s) | https://export.arxiv.org/pdf/1706.02405 |
| Alternate Webpage(s) | https://doi.org/10.1007/s12220-017-9892-2 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
