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, 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 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, 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 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.

En este articulo estudiamos superficies parametrizadas por líneas de curvatura. Obtenemos una caracterización de una clase de superficies con líneas de curvatura planas, esta caracterización depende de ciertas funciones holomorfas. Presentamos una clase de superficies con dos familias de líneas de curvatura planas, esta clase incluye las superficies de Dupin. También damos algunos ejemplos explícitos.