Dieter Betten

Topological parallelisms of the real projective 3-space

A parallelism of a projective 3-space Π is a family P of spreads such that each line of Π is contained in exactly one spread of P. A parallelism is said to be totally regular, if all its members are regular spreads. By a generalized line star with respect to an elliptic quadric Q of a classical projective 3-space we understand a set % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

Projektive Darstellung der Moulton-Ebenen

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Constructing topological parallelisms of PG(3, ℝ) via rotation of generalized line pencils

of 2-secants of Q such that each non-interior point of Q is incident with exactly one line of % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

Die Projektivitätengruppe der Moulton-Ebenen

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Four-dimensional compact projective planes with two fixed points

. From each generalized line star we can construct a totally regular parallelism which we do in essential by the Thas-Walker construction. A parallelisms of the real projective 3-space PG(3, ℝ) is called topological, if the operation of drawing a line parallel to a given line through a given point is continuous. Clifford parallelisms are topological. Using generalized line stars we exhibit examples of non-Clifford topological parallelisms and of non-topological parallelisms.

Einbettung von topologischen Raumgeometrien auf R3 in den reellen affinen Raum

The Moulton planes can be characterized as 2-dimensional topological projective planes having a 4-dimensional collineation group, which fixes exactly one nonincident point-line-pair a∋w. We give a representation of these geometries on the real protective plane such that a and W coincide with the origin and the line of infinity. This representation shows that the collineation groups of nonisomorphic Moulton planes act differently, although they are isomorphic as topological groups.

Die Projektivitätengruppe einer Klasse 4-dimensionaler Translationsebenen

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.

Generalized Line Stars and Topological Parallelisms of the Real Projective 3-Space

The group of projectivities is calculated for the Moulton-Planes Mk, k>1: it does not depend on k and consists of all “piecewise projective” mappings of the circle onto itself.

Orbits in 4-dimensional compact projective planes

We determine all 4-dimensional compact projective planes with a solvable 6-dimensional collineation group fixing two distinct points, and acting transitively on the affine pencils through the fixed points. These planes form a 2-parameter family, and one exceptional member of this family is the dual of the exceptional translation plane with 8-dimensional collineation group.

Collineation groups of topological parallelisms

Betten [1] had defined topological spatial geometries on R3: In R3 a system L of closed subsets homeomorphic to R (the lines) and a system ℰ of closed subsets homeomorphic to R2 (the planes) are given such that through any two different points passes exactly one line and through any three non-collinear points passes exactly one plane. Furthermore, ℒ and ℰ carry topologies such that the operations of joining and intersection are continuous. It is proved that any topological spatial geometry on R3 can be imbedded into R3 as an open convex subset K such that the lines in ℒ (planes in ℰ) are mapped onto intersections of lines (planes) of R3 with K. The collineation group of the geometry is isomorphic to the subgroup of the colineation group of real projective space consisting of the automorphisms that map K into itself. In particular, it is a Lie group of dimension ⩽12.