## Generalized line stars and topological parallelisms of the real projective 3-space.(English)Zbl 1168.51003

A simple covering $$\mathfrak S$$ of the non-interior points of an elliptic quadric $$Q$$ in real projective three-space is called a generalized line star. It is continuous, if the mapping that assigns a line of $$\mathfrak S$$ to a (non-interior) point is continuous. A family $$\mathbf P$$ of spreads such that every line is contained in exactly one spread is called a parallelism. It is called topological if the mapping that assigns to a point $$p$$ and a line $$L$$ a line incident with $$p$$ and in the same spread as $$L$$ is continuous. In this case, $$\mathfrak S$$ is also called topological.
From a given generalized line star $$\mathfrak S$$ a parallelism $${\mathbf P}({\mathfrak S})$$ can be constructed [D. Betten and R. Riesinger, Result. Math. 47, No. 3–4, 226–241 (2005; Zbl 1088.51005)]). In Section 2 the authors prove that continuity of $$\mathfrak S$$ implies that $${\mathbf P}({\mathfrak S})$$ is topological.
Section 3 is dedicated to a generalization of a construction of a topological generalized line star $$\mathfrak A$$ that was originally presented in [D. Betten and R. Riesinger, Adv. Geom. 8, No. 1, 11–32 (2008; Zbl 1149.51002)]. This line star is axial: any line of $$\mathfrak A$$ intersects a fixed line $$A$$. In Section 4 the authors construct the apparently first example of a generalized line star $$\mathfrak S$$ that is topological but not axial. The elaborate construction requires a quadratic cone $$\Gamma$$ with vertex $$o$$ in the interior of $$Q$$, a quadric $$\Psi$$ tangent to $$\Gamma$$ along a conic $$T$$ contained in the interior of $$Q$$ and a covering of parts of $$\Psi$$ by a continuous one-parametric family of conics $$C(t)$$. The lines of $$\mathfrak S$$ are obtained as tangents of $$Q$$ that meet the poles $$C(t)$$ with respect to $$\Psi$$ plus certain lines incident with $$o$$. Limiting cases and generalizations of this construction are considered as well.

### MSC:

 51H10 Topological linear incidence structures 51A40 Translation planes and spreads in linear incidence geometry 51M30 Line geometries and their generalizations

### Citations:

Zbl 1088.51005; Zbl 1149.51002
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