The circle space of a spherical circle plane
The authors prove that the circle space of every spherical circle plane \({\mathcal C}=({\mathcal S},{\mathcal K})\) is homeomorphic to the punctured real projective \(3\)-space, and thus generalize a result due to \textit{K. Strambach} [Monatsh. Math. 78, 156--163 (1974; Zbl 0297.50023)]. Their proof presents a tricky introduction to a kind of polar coordinates on \(\mathcal S\) which is based on the flock concept. Another important tool of their proof is the following Theorem. In a spherical circle plane \({\mathcal C}=({\mathcal S},{\mathcal K})\), let \(K\) and \(L\) be disjoint circles, and consider a point \(x\) in the annulus bounded by \(K\cup\,L\). Then the set of lines \({\mathcal K}_x\) of the derived plane at \(x\) contains exactly four circles touching both \(K\) and \(L\). Exactly two of these separate \(K\) from \(L\), that is with exception of the points of touching, \(K\) and \(L\) lie in different complementary components with respect to these circles. (See also \textit{H. Groh} [Geom. Dedicata 1, 65--84 (1972; Zbl 0241.50025)]). Moreover, the authors study the extended circle space \(\widetilde{{\mathcal K}}\) of a spherical circle plane \({\mathcal C}=({\mathcal S},{\mathcal K})\) and the flag space of an embeddable spherical circle plane.
scientific article; zbMATH DE number 6305501