The authors consider isometric immersions : . Any point p in the complement W of all tangent planes canonically determines a unit normal field e on . Result: ”Suppose that is complete and that is minimal. If there exists a point whose associated normal field e is parallel in the normal bundle, then is an embedding into a 2-dimensional affine subspace.” In particular the set of tangent planes of a non-flat complete minimal surface in covers all of
Finally the authors pass to the standard sphere and replace the tangent planes to by tangential great 2-spheres in . In general one can only conclude that immerses into a great hypersphere in . However, a complete minimal surface must be a great , provided the spherical image of lies in a closed hemisphere. All the proofs are based on some analysis of the support function associated with p and e.