Immersions with a parallel normal field (Q1580067)

Let \(f:M^m\to\mathbb R^{m+k}\) be a \(C^{\infty}\) immersion of a connected \(C^{\infty}\) \(m\)-dimensional manifold \(M\) without boundary into Euclidean \(\left( m+k\right)\)-space. For \(p\in M\) denote by \(\nu_p\) the affine normal \(k\)-plane to \(f\left( M\right)\) at \(f\left( p\right)\), and by \(F_p\) the set of focal points of \(f\) with base \(p\). The main theorem of the present paper: Let \(f\) be an immersion with \(k\geq 2\) admitting a parallel normal field \(\psi :M^m\to\mathbb R^{m+k}\). For \(p\in M\) let \(Q\subset\nu_p\) be a 2-subplane containing \(f\left( p\right)\) and \(f_{\psi}\left( p\right)\). Then either \(Q\cap F_p=\emptyset\) or \(Q\cap F_p\) is the union of at most \(m\) straight lines.