Smooth Hughes planes are classical

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)We prove that the only compact projective Hughes planes which are smooth projective planes are the classical planes over the complex numbers C, the quaternions H, and the Caley numbers O. As a by-product this shows that an 8-dimensional smooth projective plane which admits a collineation group of dimension d≥17 is isomorphic to the quaternion projective plane P 2 H. For topological compact projective planes this is true if d≥19, and this bound is sharp

Smooth stable planes

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)This paper deals with smooth stable planes which generalize the notion of differentiable (affine or projective) planes. 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 (1.6)). Moreover, the flag space is a closed submanifold of the product manifold P×L (Theorem (1.14)), and the smooth structure on the set P of points and on the set 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 A p , see Theorem (2.5). Dually, we prove in Section 3 that for any line L∈L the tangent space T L L together with the set S L ={T L L p ∣p∈L} gives rise to some shear plane. It turned out that the translation planes A p are one of the most important tools in the investigation of smooth incidence geometries. The linearization theorems (3.9), (3.11), and (4.4) can be viewed as the main results of this paper. In the closing section we investigate some homogeneity properties of smooth projective planes

Automorphism groups of differentiable double loops

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)In this paper, we study local and global topological loops as well as topological double loops having a differentiable structure such that the loop operations are differentiable. The main result states that the group of differentiable automorphisms of a differentiable double loop is compact with respect to the compact-open topology

On the embedding of zero-dimensional double loops in locally euclidean double loops

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)It is shown that a closed (hence locally compact) zero-dimensional sub-double-loop of a locally Euclidean double loop is always tamely embedded and that its complement is simply connected

Collineations of smooth stable planes

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)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 Γ of all continuous collineations of a smooth stable plane is a Lie transformation group on both the set P of points and the set ℒ of lines. In particular, this shows that the point and line sets of a (topological) stable plane ℐ admit at most one smooth structure such that ℐ 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 (dim ℐ ≤ 4) are transferred to smooth stable planes of arbitrary finite dimension. As an application of these transfers we show that the stabilizers Γ[c,c] 1 and Γ[A,A] 1 (see (3.2) Notation) are closed, simply connected, solvable subgroups of Aut(ℐ) (Corollary (4.17)). Moreover, we show that Γ[c,c] is even abelian (Theorem (4.18)). In the closing section we investigate the behaviour of reflections

On the dimensions of automorphism groups of eight-dimensional double loops

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch

Automorphism groups of locally compact connected double loops are locally compact

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch

On the dimensions of automorphism groups of four-dimensional double loops

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch

On the dimensions of automorphism groups of eight-dimensional ternary fields II

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)Let T be an eight-dimensional, connected, locally compact ternary field and let Г denote a connected closed Lie subgroup of its automorphism group which is taken with the compact-open topology. It is proved that if the ternary fixed field F^Г of Г is connected, then Г is either isomorphic to one of the compact Lie groups G2 or SU3ℂ, or the (covering) dimension of Г is at most 7

16-dimensional smooth projective planes with large collineation groups

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)Smooth projective planes are projective planes defined on smooth manifolds (i.e. the set of points and the set of lines are smooth manifolds) such that the geometric operations of join and intersection are smooth. A systematic study of such planes and of their collineation groups can be found in previous works of the author. We prove in this paper that a 16-dimensional smooth projective plane which admits a collineation group of dimension d ≥ 39 is isomorphic to the octonion projective plane P2 O. For topological compact projective planes this is true if d ≥ 41. Note that there are nonclassical topological planes with a collineation group of dimension 40