Null 2-type submanifolds of the Euclidean space E⁵ with non-parallel mean curvature vector

Abstract.

Let M be a 3-dimensional submanifold of the Euclidean space E⁵ such that M is not of 1-type. We show that if M is flat and of null 2-type with constant mean curvature and non-parallel mean curvature vector then the normal bundle is flat. We also prove that M is an open portion of a 3-dimensional helical cylinder if and only if M is flat and of null 2-type with constant mean curvature and non-parallel mean curvature vector.