Generalized null 2-type immersions in Euclidean space

Abstract

We define generalized null 2-type submanifolds in the m-dimensional Euclidean space E-m. Generalized null 2-type submanifolds are a generalization of null 2-type submanifolds defined by B.-Y. Chen satisfying the condition Delta H = fH + gC for some smooth functions f, g and a constant vector C in E-m, where Delta and H denote the Laplace operator and the mean curvature vector of a submanifold, respectively. We study developable surfaces in E-3 and investigate developable surfaces of generalized null 2-type surfaces. As a result, all cylindrical surfaces are proved to be of generalized null 2-type. Also, we show that planes are the only tangent developable surfaces which are of generalized null 2-type. Finally, we characterize generalized null 2-type conical surfaces.