TY - JOUR AU - Corro, Armando AU - Lopes Ferro, Marcelo PY - 2022/12/30 Y2 - 2024/03/29 TI - Relatives Geometries JF - Selecciones Matemáticas JA - Sel.Mat. VL - 9 IS - 02 SE - DO - 10.17268/sel.mat.2022.02.03 UR - https://revistas.unitru.edu.pe/index.php/SSMM/article/view/4501 SP - 243 - 257 AB - <p>In this paper we consider <strong>M</strong> a fixed hypersurface in Euclidean space and we introduce two types of spaces relative to <strong>M</strong>, of type I and type II. We observe that when <strong>M</strong> is a hyperplane, the two geometries coincides with the isotropic geometry. By applying the theory to a Dupin hypersurface <strong>M</strong>, we define a relative Dupin hypersurface <strong><em>M</em></strong> of type I and type II , we provide necessary and sufficient conditions for a relative hypersurface <strong><em>M</em></strong> to be relative Dupin parameterized by relative lines of curvature, in both spaces. Moreover, we provides a relationship between the Dupin hypersurfaces locally associated to M by a Ribaucour transformation and the type II Dupin hypersurfaces relative <strong><em>M</em></strong>. We provide explicit examples of the Dupin hypersurface relative to a hyperplane, torus, S<sup>1</sup>  x  R<sup>n-1</sup> and  S<sup>2</sup> x  R<sup>n-2</sup>, in both spaces.</p> ER -