Surfaces of finite type with respect to the third fundamental form

In this article, we consider surfaces in the 3dimensional Euclidean space E³without parabolic points which are of finite III-type, that is, they are of finite type, in the sense of B.-Y. Chen, with respect to the third fundamental form. We present an important family of surfaces, namely, surfaces of revolution in E³. We study a special case of this family, namely, surfaces of revolution where the sum of the radii of the principal curvature R is constant.