Band 8, Heft 1

In this paper we classify the connected ruled Weingarten hypersurfaces in .

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.

We prove a Gauss–Bonnet formula for closed semi-algebraic sets.

We introduce the notion of intrinsic subspaces of linear and affine pair geometries , which generalizes the one of projective subspaces of projective spaces. We prove that, when the affine pair geometry is the projective geometry of a Lie algebra introduced in [W. Bertram, K.-H. Neeb, Projective completions of Jordan pairs. I. The generalized projective geometry of a Lie algebra. J. Algebra 277 (2004), 474–519. MR2067615 (2005f:17031) Zbl 02105235], such intrinsic subspaces correspond to inner ideals in the associated Jordan pair, and we investigate the case of intrinsic subspaces defined by the Peirce-decomposition which is related to 5-gradings of the projective Lie algebra. These examples, as well as the examples of general and Lagrangian flag geometries, lead to the conjecture that geometries of intrinsic subspaces tend to be themselves linear pair geometries.

Let S ⊂ ℝ 2 be a compact staircase connected set with stdiam( S ) = n . In [E. Magazanik, M. A. Perles, Staircase connected sets. Discrete Comput. Geom. 37 (2007), 587–599. MR2321743 Zbl] we showed that Ker r ( S ) is nonempty if , and for , Ker r ( S ) is staircase connected. In this paper we determine the possible values of the staircase diameter of Ker r ( S ) for , and present interesting facts about Ker r ( S ) when and .

In this note we construct symmetric designs which have the parameters of McFarland or Spence designs and whose full automorphism groups are isomorphic to an arbitrarily preassigned finite group.

Doliwka and Lassak proved that if n ≤ 5 then every convex n -gon has a relatively short side and a relatively long side. In this paper we prove that this is true for n ≤ 5 only. More precisely, for any n ≥ 6, there exist n -gons without any relatively short sides.

Let Δ be a thick dual polar space of rank n ≥ 2 and let e be a full polarized embedding of Δ into a projective space ∑. For every point x of Δ and every i ∈ {0, …, n }, let T i ( x ) denote the subspace of ∑ generated by all points e ( y ) with d( x , y ) ≤ i . We show that T i ( x ) does not contain points e ( z ) with d( x , z ) ≥ i + 1. We also show that there exists a well-defined map e i x from the set of ( i – 1)-dimensional subspaces of the residue Res Δ ( x ) of Δ at the point x (which is a projective space of dimension n – 1) to the set of points of the quotient space T i ( x )/T i –1 ( x ). In this paper we study the structure of the maps e i x and the subspaces T i ( x ) for some particular full polarized embeddings of the symplectic and the Hermitian dual polar spaces. Our investigations allow us to answer some questions asked in the literature.

We generalize an earlier construction of generalized quadrangles from polar spaces of rank 3 to polar spaces of arbitrary rank.