Authors: Richard Bödi
Publish Date: 2013/04/17
Volume: 31, Issue: 3-4, Pages: 300-321
Abstract
This paper deals with smooth stable planes which generalize the notion of differentiable affine or projective planes 7 It is intended to be the first one of a series of papers on smooth incidence geometry based on the Habilitationsschrift of the author It contains the basic definitions and results which are needed to build up a foundation for a systematic study of smooth planes We define smooth stable planes and we prove that point rows and line pencils are closed submanifolds of the point set and line set respectively Theorem 16 Moreover the flag space is a closed submanifold of the product manifold Ptimes cal L Theorem 114 and the smooth structure on the set P of points and on the set cal L of lines is uniquely determined by the smooth structure of one single line pencil In the second section it is shown that for any point p te P the tangent space TpP carries the structure of a locally compact affine translation plane cal A p see Theorem 25 Dually we prove in Section 3 that for any line L in cal L the tangent space rm T Lcal L together with the set cal rm S L=lbrace rm T Lcal L pmid p in Lrbrace gives rise to some shear plane It turned out that the translation planes cal A p are one of the most important tools in the investigation of smooth incidence geometries The linearization theorems 39 311 and 44 can be viewed as the main results of this paper In the closing section we investigate some homogeneity properties of smooth projective planes
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