On 4-flattening theorems and the curves of Carathéodory, Barner and Segre (Q1407636)

Three classes of closed space curves, called Barner, Segre and Carathéodory class, resp., are studied. It is proved: (A) There is a non-empty open set of Carathéodory curves in \(\mathbb{R}^3\) which are not Segre curves. (B) There is a non-empty open set of Segre curves in \(\mathbb{R}^3\) which are not Carathéodory curves. (C) Any Barner curve in \(\mathbb{R}^3\) is a Carathéodory curve and a Segre curve at the same time. A new class of closed curves containing the Segre class is also introduced. It is conjectured that any curve of the new class has at least 4 flattenings (points at which the torsion vanishes).