Converse construction for admissible one-dimensional factors

Determine whether every product of extrinsic factors of the types specified in Theorem 1.2, including one-dimensional admissible factors whose curvature normals do not have constant length, can serve as the generating hypersurface of a locally irreducible Euclidean submanifold whose curvature normals with respect to the flat part of the normal bundle have constant length.

Background

The paper characterizes the local structure of submanifolds with curvature normals of constant length by showing that their generating hypersurfaces decompose into extrinsic factors. Factors of dimension at least two are spherical rank-one submanifolds, whereas one-dimensional factors need only satisfy the weaker condition of admissibility: after embedding the curve as a non-full submanifold, the projection of its curvature normal onto the orthogonal complement of a suitable parallel normal field has constant length.

Section 7 constructs examples when every one-dimensional factor has a curvature normal of constant length. The construction therefore establishes only a partial converse to the structural theorem. The unresolved issue is whether the construction, or some other construction, works for the broader class of admissible one-dimensional factors whose full curvature normals are not of constant length.