Generalized null 2-type immersions in Euclidean space

Abstract

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 Δ H = f H + gC for some smooth functions f, g and a constant vector C in 𝔼 m , where Δ and H denote the Laplace operator and the mean curvature vector of a submanifold, respectively. We study developable surfaces in 𝔼3 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.