In this paper we consider M a fixed hypersurface in Euclidean space and we introduce two types of spaces relative to M, of type I and type II. We observe that when M is a hyperplane, the two geometries coincides with the isotropic geometry. By applying the theory to a Dupin hypersurface M, we define a relative Dupin hypersurface M of type I and type II , we provide necessary and sufficient conditions for a relative hypersurface M 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 M. We provide explicit examples of the Dupin hypersurface relative to a hyperplane, torus, S1  x  Rn-1 and  S2 x  Rn-2, in both spaces.

In this paper, we consider a method of constructing flat surfaces based on Ribaucour transformations in S3. By applying the theory to the flat torus, we obtain a family of complete flat surfaces in S3 which is determined by several parameters. We provide explicit examples.

In this paper, we consider a method of constructing isothermic surfaces in S3 based on Ribaucour transformations. By applying the theory to the flat torus, we obtain two family of complete isothermic surfaces in S3. One four-parameter family of complete isothermic surfaces that contains n-bubble surfaces inside and outside of the torus. We also get another four-parameter of complete isothermic surfaces which are Dupin surfaces. As aplication we obtain explicit solutions of the Calapso equation.

In this paper, we provide a class of surfaces called ϵ-isothermic surface in the pseudo-Euclidean 3-space and we introduce the pseudo-Calapso equation. We prove that for each ϵ-isothermic surface, we can associate two solutions to the pseudo-Calapso equation. In particular, we associate solutions to the Calapso, Zoomeron and Davey-Stewartson III equations. In sequence, we classify the Dupin surfaces in pseudo-Euclidean 3-space having distinct principal curvatures and provide explicit coordinates for such surfaces. As application of the theory, we obtain explicit solutions to the pseudo-Calapso equation and from these solutions, we provide new explicit solutions of the Zoomeron and Davey-Stewartson III equations. Moreover, we also provide explicit solutions to these equations that depend on ϵ2−holomorphic functions.

In this paper, we study oriented surfaces S in R3, called surfaces with quadratic support function (in short QSF-surfaces). We obtain a Weierstrass type representation for the QSF-surfaces which depends on two holomorphic functions. Moreover, classify the QSF-surfaces of rotation. Also, we give some explicit examples of this class of surfaces.

In this paper, we define surfaces with mean of the hyperbolic curvature radii of double harmonic type (in short DHRMC-surfaces) in the hyperbolic space, these surfaces include the generalized Weingarten surfaces of the harmonic type (HGW-surfaces). We give a characterization of DHRMCsurfaces. Given a real function, we will present a family of DHRMC-surfaces that depend on two holomorphic functions. Moreover, we classify the DHRMC-surfaces of rotation.

In this work, using the Laplace invariants theory we give other proof for the following result: A proper Dupin hypersurfaces Mn for n ≥ 4 in Rn+1 with n distinct principal curvatures andconstant mobius curvature, cannot be parametrized by lines of curvature. Also, we study special classes of hypersurfaces Mn; n ≥ 3; in Rn+1, parametrized by lines of curvature with n distinct principal curvatures and we obtain a geometric relation when the Laplace invariants are vanish, we show that the foliations of Mn are umbilical hypersurfaces if and only if mijk = 0. Moreover, the foliations of Mn are Dupin hypersurfaces if and only if mij = 0.

In this work, using the Laplace invariants theory we give other proof for the following result: A proper Dupin hypersurfaces Mn for n ≥ 4 in Rn+1 with n distinct principal curvatures andconstant mobius curvature, cannot be parametrized by lines of curvature. Also, we study special classes of hypersurfaces Mn; n ≥ 3; in Rn+1, parametrized by lines of curvature with n distinct principal curvatures and we obtain a geometric relation when the Laplace invariants are vanish, we show that the foliations of Mn are umbilical hypersurfaces if and only if mijk = 0. Moreover, the foliations of Mn are Dupin hypersurfaces if and only if mij = 0.

In this work, we present explicit parameterizations of hypersurfaces parameterized by lines of curvature with prescribed Gauss map and we characterize the hypersurfaces with planar curvature lines. As an applicationwe obtain a classification of isothermic surfaces with respect to the third fundamental form with two planar curvature lines. Also, we present a class of surfaces with one family of planar curvature lines and generalize these results to present classes of hypersurfaces with families of planar curvature lines.

In this paper we introduce the generalized Helmholtz equation and present explicit solutions to this generalized Helmholtz equation, these solutions depend on three holomorphic functions. As an application we present explicit solutions to the Helmholtz equation. We note that these solutions are not necessarily limited to certain domains of the complex plane C.

In this work, we present explicit parameterizations of hypersurfaces parameterized by lines of curvature with prescribed Gauss map and we characterize the hypersurfaces with planar curvature lines. As an applicationwe obtain a classification of isothermic surfaces with respect to the third fundamental form with two planar curvature lines. Also, we present a class of surfaces with one family of planar curvature lines and generalize these results to present classes of hypersurfaces with families of planar curvature lines.

In this paper we introduce the generalized Helmholtz equation and present explicit solutions to this generalized Helmholtz equation, these solutions depend on three holomorphic functions. As an application we present explicit solutions to the Helmholtz equation. We note that these solutions are not necessarily limited to certain domains of the complex plane C.

In this paper, we present a method for obtaining solutions to the Calapso and Zoomeron equations using Ribaucour transformations. We provide explicit formulas for the solutions of these equations. By applying this method to the generalized cylinder, we provide new solutions of these equations, which are determined by a free function and by two other functions, each one defined in a given variable. As a consequence, we provide new solutions for the Davey-Stewartson III equation.

In this research we study harmonic surfaces immersed in R3. We dened Harmonic surfaces of graphic type and showed that a harmonious surface graphic type is minimal if and only if it is part ofa plane or a helix. Also, we give a characterization of harmonic surfaces graphic type parameterized by asymptotic lines and some examples.

In this research we study harmonic surfaces immersed in R3. We dened Harmonic surfaces of graphic type and showed that a harmonious surface graphic type is minimal if and only if it is part ofa plane or a helix. Also, we give a characterization of harmonic surfaces graphic type parameterized by asymptotic lines and some examples.

In this work, we dene the hypersurfaces of the spherical type degenerated (in short DST-hypersurfaces), these hypersurfaces has the geometric property that the middle spheres pass through the origin of the Euclidean space. We present a representation for these hypersurfaces in the case where the stereographic projection of the Gauss map N is given by the identity application. We characterizethe DST-hypersurfaces through a diferential equation and we give an explicit example of a two-parameter family of DST-hypersurfaces with planar lines of curvature foliated by (n-1)-dimensional spheres. Moreover, we classify the DST-hypersurfaces of rotation.

In this work, we dene the hypersurfaces of the spherical type degenerated (in short DST-hypersurfaces), these hypersurfaces has the geometric property that the middle spheres pass through the origin of the Euclidean space. We present a representation for these hypersurfaces in the case where the stereographic projection of the Gauss map N is given by the identity application. We characterizethe DST-hypersurfaces through a diferential equation and we give an explicit example of a two-parameter family of DST-hypersurfaces with planar lines of curvature foliated by (n-1)-dimensional spheres. Moreover, we classify the DST-hypersurfaces of rotation.

In this work, we study biharmonic surfaces that are parameterized by biharmonic coordinate functions. We study a class of biharmonic surfaces called graph-type biharmonic surfaces. Also, we define a class of surfaces associated to two harmonic functions (FH2A-surfaces), these surfaces satisfy a relation between the Gaussian curvature, the projection of the Gauss map on a fixed plane and two harmonic functions. We show that a particular class of graph-type biharmonic surfaces are FH2A-surfaces. Finally, we classify the FH2A-surfaces of rotation.

In this work, we study biharmonic surfaces that are parameterized by biharmonic coordinate functions. We study a class of biharmonic surfaces called graph-type biharmonic surfaces. Also, we define a class of surfaces associated to two harmonic functions (FH2A-surfaces), these surfaces satisfy a relation between the Gaussian curvature, the projection of the Gauss map on a fixed plane and two harmonic functions. We show that a particular class of graph-type biharmonic surfaces are FH2A-surfaces. Finally, we classify the FH2A-surfaces of rotation.