Weak helix submanifolds of euclidean spaces

Let M⊂ℝ ⁿ be a submanifold of a euclidean space. A vector d∈ℝ ⁿ is called a helix direction of M if the angle between d and any tangent space T _p M is constant. Let ℋ(M) be the set of helix directions of M. If the set ℋ(M) contains r linearly independent vectors we say that M is a weak r-helix. We say that M is a strong r-helix if ℋ(M) is a r-dimensional linear subspace of ℝ ⁿ . For curves and hypersurfaces both definitions agree. The object of this article is to show that these definitions are not equivalent. Namely, we construct (non strong) weak 2-helix surfaces of ℝ⁴. The author is supported by the Project M.I.U.R. “Riemann Metrics and Differentiable Manifolds” and by G.N.S.A.G.A. of I.N.d.A.M., Italy.