Collineations of smooth stable planes

Richard Bödi
1998 Forum mathematicum  
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.
more » ... th 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.
doi:10.1515/form.10.6.751 fatcat:car4kuvhybb45pe5ouxyflo6xu