Surfaces in the Euclidean space $Bbb{E}^4$ with pointwise 1-type Gauss map

Abstract

In this article we study surfaces in Euclidean space $Bbb{E}^4$ with pointwise 1-type Gauss map. We give a characterization of surfaces in $Bbb{E}^4$ with a pointwise 1-type Gauss map of the first kind. We conclude that an oriented non-minimal surface M in $Bbb{E}^4$ has a pointwise 1-type Gauss map of the first kind if and only if M is a surface in a 3-sphere of $Bbb{E}^4$ with constant mean curvature. We also obtain a characterization for non-planar minimal surfaces in $Bbb{E}^4$ with pointwise 1-type Gauss map of the second kind. Further we give a partial classification of surfaces in $Bbb{E}^4$ in terms of the pointwise 1-type Gauss map of the second kind.