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Collineations of Smooth Stable Planes 7 5 3 Planes Forum Math. 10 (1998), 751-773 Forum Mathematicum © de Gruyter 1998 C o l l i n e a t i o n s o f s m o o t h s t a b l e p l a n e s Richard Bödi ( C o m m u n i c a t e d b y K a r l H. H o f m a n n ) A b s t r a c t . Smooth stable planes have been introduced in [4]. We show that every continuous collineation between two smooth stable planes is in fact a smooth collineation. This implies that the group $ of all continuous collineations of a smooth stable plane is a Lie transformation group on both the set P of points and the set $ of lines. In particular, this shows that the point and line sets of a (topological) stable plane $ admit at most one smooth structure such that $ becomes a smooth stable plane. The investigation of central and axial collineations in the case of (topological) stable planes due to R. Löwen ([25], [26], [27]) is continued for smooth stable planes. Many results of [26] which are only proved for low dimensional planes $ are transferred to smooth stable planes of arbitrary finite dimension. As an application of these transfers we show that the stabilizers $ and $ (see (3.2) Notation) are closed, simply connected, solvable subgroups of $ (Corollary (4.17)). Moreover, we show that $ is even abelian (Theorem (4.18)). In the closing section we investigate the behaviour of reflections. 1 9 9 1 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n : 5 1 H 2 5 , 5 1 A 4 0 . 1. Introduction I n the e a r l y 3 0 ’ s K o l m o g o r o f f w a s the first w h o c o m b i n e d p r o j e c t i v e s p a c e s w i t h topological structures. In the m i d - f i f t i e s , Skornyakov ( 1 9 5 4 ) and Freudenthal ( 1 9 5 7 ) p u b l i s h e d t w o papers o n topological planes, and i t w a s Helmut S a l z m a n n w h o started the s y s t e m a t i c investigation o f topological plane g e o m e t r i e s i n 1 9 5 5 . R e ¬ m e m b e r i n g the initial task o f E u c l i d and Hilbert to characterize the e u c l i d e a n plane $, i t i s natural to c o n s i d e r not o n l y the topology o f $ but a l s o its differentiate structure. This l e a d s f r o m topological g e o m e t r y to smooth geometry, i . e . i n c i d e n c e g e o m e t r y o n smooth m a n i f o l d s s u c h that the g e o m e t r i c operations are smooth. In sharp contrast to the case of (compact) topological planes, only a few papers on smooth geometries have been published up to now. The pioneer of smooth geometry, S. Breitsprecher, wrote three papers in the late 60's, [7], [8], [9]. In 1972, Bereitgestellt von | ZHAW Zuercher Hochschule für Angewandte Wissenschaften Angemeldet Heruntergeladen am | 23.02.18 10:00 7 5 2 Richard Bodi Dieter Betten published a paper [2] on differentiable two-dimensional projective planes, in which he answered a question of Hellmuth Kneser by showing that the proper Moulton planes cannot be turned into smooth projective planes. After a period of 14 years where nothing was published about smooth geometry, Hermann Hahl revived thisfieldwith two papers, [18] and [14], (one of them is a joint paper with Theo Grundhofer) about spherical fibrations which lead to smooth translation planes defined over division algebras. The main result of these two papers is that every smooth translation plane defined over some division algebra is isomorphic to one of the classical projective planes $, i.e. the projective planes defined over the real, complex, quaternion, or octonion numbers, which we abbreviate by $, $, a n d $, r e s p e c t i v e l y . Joachim Otte g e n e r a l i z e d this r e s u l t i n 1 9 9 2 to arbitrary translation planes, s e e [33] and [34]. W e introduced the concept o f a smooth s t a b l e plane i n [4]. I n this paper w e w i l l s t u d y collineations and collineation groups o f smooth s t a b l e planes. W e r e c a l l the most b a s i c d e f i n i t i o n s o f [4]. (1.1) Definition. A linear space is a triple $ of sets $ and $, where P denotes the set of points, $ is the set of lines and $ is the set of flags, such that for every pair of distinct points p, q there is exactly one joining line $, i . e . $. I f $, we will say that p and L are incident, or that p lies on L, or th a t L passes through p. (1.2) Definition. A st a b l e pl a n e $ is a li n e a r sp a c e $ which satisfies the following three axioms: ( S 1 ) T h e r e are H a u s d o r f f t o p o l o g i e s o n both P and $ that are neither d i s c r e t e nor anti-discrete s u c h that the join map $ and the i n t e r s e c t i o n map $ are c o n ¬ tinuous. Moreover, the domain $ of the intersection map is an open subset of $. ( S 2 ) The topology o n P i s l o c a l l y compact and has p o s i t i v e finite c o v e r i n g d i m e n ¬ s i o n . ( 5 3 ) $ contains a quadrangle. A smooth stable plane $ is a stable plane $ s u c h t h a t P and $ are smooth manifolds and such that the join and intersection mappings are smooth on their (respective) domains. W e w i l l u s e the words smooth and C ∞-differentiable s y n o n y m o u s l y . Note, h o w e v e r , that the r e s u l t s o f [4] a s w e l l a s o f the present paper are s t i l l v a l i d f o r C2- d i f f e r e n t i a b l e stable planes. (1.3) Examples. Every subgeometry defined on some nonempty open subset of the point set of one of the classical projective planes (see the introduction for the definition of a classical plane ) is a smooth stable plane. This includes the hyperbolic Bereitgestellt von | ZHAW Zuercher Hochschule für Angewandte Wissenschaften Angemeldet Heruntergeladen am | 23.02.18 10:00 Collineations of smooth stable planes 7 5 3 planes. Otte, [33], has shown that there is a plethora of non-classical smooth a f f i n e and projective planes. He describes these examples algebraically in terms of a coordinatizing ternary field. However, his examples are in general rather unaccessible from a geometric point of view. More explicit examples of smooth a f f i n e and stable planes w i l l be given in forthcoming papers. For the readers’ convenience the following theorem summarizes some properties of stable planes. Proofs can be found in [25] and [28]. (1.4) Theorem. Le t $ be a stable plane and let $. a) The point row L is a closed subset of the set P of points, the line pencil $is compact, and the flag space $ is a closed subset of $. b) The set $ of lines and the line pencils$are connected. c) $ with $, and $, $, where dim denotes the covering dimension. The stable plane $ is said to be 2l-dimensional. d) dim U = dim X for every open subset U of $. Throughout this paper, let $ denote a smooth stable plane of dimen¬ sion n = 2l. 2. Collineations The main topic of this chapter is the investigation of smooth collineations (i.e. automorphisms) of smooth stable planes. We show that every continuous collinea- tion of such a plane i s in fact a smooth collineation. This implies that the group $ of all continuous collineations is a Lie group with respect to the compact-open topology. In particular, the smooth structures of a smooth stable plane are uniquely determined by the underlying (topological) stable plane.
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