Über die Anzahl der ordnungsgeometrischen Scheitel von Kurven. II: Verallgemeinerungen des Vier-Scheitel-Satzes. (On the number of order geometric vertices of curves. II: Generalizations of the Four-vertex theorem) (Q1822764)

[For part I see Aequationes Math. 34, 82-88 (1987; Zbl 0634.51008).] Juel's correspondences and Haupt's order characteristic are combined as follows. For \(k>2\) and a curve K, which must not lie in a bigger space, consider a continuous map \(\phi\) : \(K^ k\to X\), X a metric space, which has certain properties. The following theorem is proved: If K has order \(r\geq k+1\) \((r=k+1)\), then K has at least (exactly) r vertices. Three special cases are treated, where X is the space of 1. order characteristics in the euclidean plane, 2. hyperplanes in k-dimensional real projective space, 3. hyperspheres in (k-1)-dim real euclidean space. Unfortunately the author's style is sensitive to misprints and the paper contains a lot of them. So it is a painful task to follow the details.