Smooth projective translation planes
type: Item
instance of: scholarly article
title: Smooth projective translation planes
zbMATH Open document ID: 0838.51008
DOI: 10.1007/BF01265639
author: Joachim Otte
published in: Geometriae Dedicata
publication date: 1995-12-18T00:00:00Z
review text: A projective or affine plane is called smooth if point and line space are endowed with smooth structures such that the geometric operations of joining and intersecting are smooth. The author of the paper under review has produced examples of smooth projective planes and of smooth affine translation planes. However, the latter examples do not extend to smooth projective planes, except if one has the plane over one of the real algebras \(\mathbb{R}, \mathbb{C}, \mathbb{H}\), or \(\mathbb{O}\). In the paper under review, it is shown that in fact these four planes are the only smooth projective translation planes. The proof starts with a characterization of those elements of \(\text{GL}(n,\mathbb{R})\) that extend to diffeomorphisms of the sphere \(\mathbb{R}^n \cup \{\infty\}\), with its natural smooth structure. This characterization is used to show that the group of projectivities of a line in a smooth projective translation plane is comparatively small; a result of Grundhöfer and Strambach then yields the theorem.
Mathematics Subject Classification ID: 51H25
zbMATH DE Number: 826781
zbMATH Keywords: smooth projective translation planes
reviewed by: Markus Stroppel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 826781