SURFACES IN THE EUCLIDEAN SPACE E4 WITH POINTWISE 1-TYPE GAUSS MAP

In this article we study surfaces in Euclidean space E4 with pointwise 1-type Gauss map. We give a characterization of surfaces in E4 with a pointwise 1-type Gauss map of the first kind. We conclude that an oriented non-minimal surface M in E4 has a pointwise 1-type Gauss map of the first kind if and only if M is a surface in a 3-sphere of E4 with constant mean curvature. We also obtain a characterization for non-planar minimal surfaces in E4 with pointwise 1-type Gauss map of the second kind. Further we give a partial classification of surfaces in E4 in terms of the pointwise 1-type Gauss map of the second kind.