Möbius cone structures on 3-manifolds (Q1891964)

Möbius polygons are defined and their properties are studied. The torsion is a useful invariant of them and it is shown that it is additive with respect to the glueing of two Möbius polygons. A result concerning the existence of regular convex Möbius polygons with prescribed torsion is also proved. The main result of the paper is that any closed orientable 3-manifold has a Möbius cone structure with cone angle \(\alpha\), for any \(\alpha \in (0,2\pi)\). Another result states that there exists a Möbius structure with discrete monodromy group on the circle bundle having the Euler number equal to 1, giving thus a solution to a problem of \textit{N. H. Kuiper} [Publ. Math., Inst. Hautes Étud. Sci. 68, 47-76 (1988; Zbl 0692.57013)].