Hipersuperfícies de tipo Ribaucour

Abstract

In this work we define the Ribaucour-type surfaces (in short, RT-surfaces),These surfaces satisfy a relationship similar to the Ribaucour surfaces that are related to the Élie Cartan problem. This class furnishes what seems to be the first examples of pairs of noncongruent surfaces in Euclidean space such that, under a diffeomorphism, lines of curvatures are preserved and principal curvatures are switched. We show that every compact and connected RT-surface is the sphere with center at the origin. We obtain present a Weierstrass type representation for RT-surfaces with prescribed Gauss map which depends on two holomorphic functions. We give explicit examples of RT-surfaces. Also, we use this representation to classify the RT-surfaces of rotation. We define the GRT-surfaces which are a generalization of the RT-surfaces, show a local parameterization of this class of surfaces and classify them in the in which case they are of rotation and generalize as RT-surfaces for the case of hypersurfaces in Rn+1, display a parameterizationção for the rotational cases and analyze the general of generatrix curves when hypersurfaces and rotation behavior are 3-dimensional.