*(English)*Zbl 0538.53057

The authors consider isometric immersions ${\Phi}$ : ${M}^{2}\to {\mathbb{R}}^{2+k}$. Any point p in the complement W of all tangent planes canonically determines a unit normal field e on ${M}^{2}$. Result: ”Suppose that ${M}^{2}$ is complete and that ${\Phi}$ is minimal. If there exists a point $p\in W$ whose associated normal field e is parallel in the normal bundle, then ${\Phi}$ is an embedding into a 2-dimensional affine subspace.” In particular the set of tangent planes of a non-flat complete minimal surface in ${\mathbb{R}}^{3}$ covers all of ${\mathbb{R}}^{3}\xb7$

Finally the authors pass to the standard sphere ${S}^{2+k}$ and replace the tangent planes to ${M}^{2}$ by tangential great 2-spheres in ${S}^{2+k}$. In general one can only conclude that ${\Phi}$ immerses ${M}^{2}$ into a great hypersphere in ${S}^{2+k}$. However, a complete minimal surface ${M}^{2}\to {S}^{3}$ must be a great ${S}^{2}$, provided the spherical image of ${\Phi}$ lies in a closed hemisphere. All the proofs are based on some analysis of the support function associated with p and e.