Translation Planes: Foundations and Construction Principles
The book discusses various construction principles for translation planes and spreads from a general and unifying point of view and relates them to the theory of kinematic spaces. The book is intended for people working in the field of incidence geometry and can be read by everyone who knows the basic facts about projective and affine planes.
The methods developed work especially well for topological spreads of real and complex vector spaces. In particular, a complete classification of all semifield spreads of finite dimensional complex vector spaces is obtained.
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Spreads of 3dimensional Projective Spaces
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3-dimensional projective space 4-dimensional translation planes Ab,c affine plane affine space Ai-indicator set assume automorphism Betten bijective collineation group commutative field compact 4-dimensional translation Dedicata defined denote desarguesian division algebras dual spread elements equation equivalent exists F-vector finite fields flock of reguli Geom Grassmann manifold group of shears Hahl Hence homeomorphic hyperbolic flock indicator set intersect inversive plane ip(m kernel kinematic kinematic mapping kinematic space Klein quadric left vector space Lemma Lenz type linear mapping locally compact 4-dimensional Math matrix pappian parabolic flock planes of Lenz projective plane Proof Proposition quadratic extension quasifield quaternion real linear real projective space real vector space reguli with carrier right vector space satisfies Kl Satz skewfield F space over F spread set SPuH subspaces surjective topological spread translation plane associated Translationsebenen transversal mapping x o m