Constructing topological parallelisms of PG(3, ℝ) via rotation of generalized line pencils
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Dieter Betten
Abstract
Let Q be an elliptic quadric of the real projective 3-space PG(3, ℝ) and denote by Q¬i the set of non-interior points with respect to Q. A simple covering 
of Q¬i by 2-secants of Q is called generalized line star with respect to Q. In [D. Betten, R. Riesinger, Topological parallelisms of the real projective 3-space. Results Math. 47 (2005), 226–241. MR2153495 (2006b:51009) Zbl 1088.51005] the authors give a construction P such that 
is a parallelism of PG(3, ℝ); cf. Theorem 1 below. In the present article, we are mainly interested in the plane analogues of gl-stars: the gl-pencils with respect to a conic; cf. Definition 3. If a gl-star is generated by rotating a gl-pencil 
about an axis 
, then we call a latitudinal gl-star and a latitudinal parallelism. We present a general construction process for gl-pencils by giving generating functions. Along this way we prove the existence of non-Clifford latitudinal parallelisms in PG(3, ℝ); moreover, we show that each latitudinal parallelism is topological.
© de Gruyter 2008
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Articles in the same Issue
- Ruled Weingarten hypersurfaces in
- Constructing topological parallelisms of PG(3, ℝ) via rotation of generalized line pencils
- A Gauss–Bonnet formula for closed semi-algebraic sets
- Inner ideals and intrinsic subspaces of linear pair geometries
- Staircase kernels
- Twisted McFarland and Spence designs and their automorphisms
- On the relative lengths of the sides of convex polygons
- The structure of full polarized embeddings of symplectic and Hermitian dual polar spaces
- Polar spaces, BLT-sets and generalized quadrangles
- Erratum to “On the Hilbert scheme of Palatini threefolds”
