Construction for admissible factors without constant curvature-normal length

Construct a generating-hypersurface construction for products containing a one-dimensional factor that is admissible but whose curvature normal does not have constant length, and thereby determine whether such a product generates a locally irreducible Euclidean submanifold with curvature normals of constant length.

Background

The general examples section proves that products of spherical rank-one factors and one-dimensional spherical factors with constant-length curvature normals generate the desired locally irreducible submanifolds. The theorem preceding this construction permits one-dimensional factors that are merely admissible, so the construction does not cover all cases allowed by the structural classification.

The authors explicitly state that they do not know how to extend their construction to an admissible one-dimensional factor whose curvature normal lacks constant length. This is the concrete unresolved construction problem left by the paper.