Abstract: Generalized constant slope surfaces(GCSSs)， whose inner product of the unit position vector and the unit mean curvature vector is constant， are studied in four dimension Euclidean space.Firstly， the position vector is divided into the tangent and normal parts. Then， two special GCSSs， one is that the parallel normal vector field has the mean curvature vector characteristics and the other is that the normal parts has the mean curvature vector characteristics， are studied， respectively， with the existence and classification of the two special GCSSs obtained. The expression and performance of a GCSS of which the normal plane is parallel to the mean curvature vector are discussed， proving that the GCSS can be parameterized with the principal lines and defined as a Chen surface.Finally， some examples of these surfaces and 3-D projection of the surfaces are given by MATLAB.

Key words: generalized constant slope surface; parallel normal vector field; Chen surface; E⁴; inner product