Submanifolds with nonparallel first normal bundle revisited (Q2438272)

For an isometric immersion \(f: M^n \to \mathbb{R}^N\) of a locally substantial connected Riemannian manifold, whose first normal spaces have constant dimension \(p < N - n\), the authors define an index \(0 \leq s \leq p\) that vanishes if the first normal bundle is parallel. For \(0 < s\), it can be viewed as a measure of the non-parallelity of the first normal bundle \(N_1^f\). The main result states that submanifolds with constant \(s\) and \(0 < s < n\), \(s \leq 6\) is (part of) a ruled submanifold for which a certain subbundle \(\mathcal S\) of rank \(s\) of \(N_1^f\) is constant along the rulings and the rulings' dimension is bounded from below. By an example the authors show that this result cannot be improved. The low-dimensional cases \(p \in \{1,2,3\}\) have already been discussed in an earlier paper [\textit{M. Dajczer} and \textit{L. Rodríguez}, Bull. Lond. Math. Soc. 19, 467--473 (1987; Zbl 0631.53042)]