Submanifolds with nonparallel first normal bundle revisited. (English) Zbl 1290.53006

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 [M. Dajczer and L. Rodríguez, Bull. Lond. Math. Soc. 19, 467–473 (1987; Zbl 0631.53042)]


53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53B25 Local submanifolds


Zbl 0631.53042
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