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 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.