On \(n\)-dimensional surfaces in Euclidean space \(E^{n+p}\) that belong to an \((n+1)\)-dimensional plane (Q1337890)

The article is concerned with the question under which conditions an \(n\)- dimensional submanifold \(F\) of \(\mathbb{R}^{n + p}\) \((n,p \geq 2)\) is contained in an \((n + 1)\)-dimensional affine subspace of \(\mathbb{R}^{n + 1}\). According to a theorem of \textit{J. Erbacher}, a necessary and sufficient condition is that the values of the second fundamental form \(b\) of \(F\) lie in a ``parallel'', rank 1 subbundle \(\xi\) of the normal bundle of \(F\), see J. Differ. Geom. 5, 333-340 (1971; Zbl 0221.53031). Theorem 3 of the present article states that the same result where the parallelity of \(\xi\) is replaced by the condition that the covariant derivative of \(b\) is \(\xi\)-valued, too. But one can construct counterexamples for this version. The mistake in the argumentation is the choice \(\xi = n_ 1\) in the proof of the lemma, where the parallelity of \(\xi\) is used tacitly. Unfortunately Theorems 1 and 2 are corollaries of Theorem 3.