On weak biharmonic submanifolds and 2-parallelity

Abstract

It is shown that every locally Euclidean 2-parallel submanifold of a space form has harmonic curvature vector (i.e., is weak biharmonic). In four-dimensional Euclidean space, in the class of surfaces with flat connection ∇, whose one family of curvature lines consists of geodesics, a surface is weak biharmonic if and only if it is 2-parallel.