The classification of hypersurfaces of a Euclidean space with parallel higher order fundamental form (Q1119913)

We show that a hypersurface of a Euclidean space \({\mathbb{R}}^{n+1}\) with parallel higher order fundamental form \((\nabla^ kh=0)\) is either parallel \((\nabla h=0)\), and therefore an open part of a hypersphere, a hyperplane, or a product of a sphere and an affine subspace, or else it is a cylinder on a plane curve whose curvature is a polynomial function of the arc length.