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.