Translation Planes: Foundations and Construction PrinciplesThe 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. |
Contents
Introduction | 1 |
Spreads of 3dimensional Projective Spaces 222278 | 25 |
Kinematic Spaces | 40 |
Copyright | |
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3-dimensional projective space 4-dimensional translation planes a₁ A₁-indicator A1-indicator set AB,C affine plane affine space assume automorphism b₁ b₂ Betten bijective collineation group commutative field Dedicata defined denote desarguesian division algebras dual spread E₁ elements equation equivalent exists F-vector finite fields flock of reguli Geom Hähl Hence homeomorphic hyperbolic flock indicator set intersect inversive plane kernel kinematic mapping kinematic space Klein quadric left vector space Lemma Lenz type Let F linear mapping locally compact locally compact 4-dimensional Math matrix nt sin mt pappian parabolic flock planes of Lenz Plücker coordinates projective plane Proof Proposition quasifield real linear real projective space real vector space reguli with carrier right vector space satisfies K1 Satz skewfield F space over F spread set SPu H subspaces surjective topological spread translation plane associated Translationsebenen transversal mapping V₁