Automorphism groups of compact projective planes

Compact connected projective planes have been investigated extensively in the last 30 years, mostly by studying their automorphism groups. It is our aim here to remove the connectedness assumption in some general results of Salzmann [31] and Hähl [14] on automorphism groups of compact projective planes. We show that the continuous collineations of every compact projective plane form a locally compact transformation group (Theorem 1), and that the continuous collineations fixing a quadrangle in a compact translation plane form a compact group (Corollary to Theorem 3). Furthermore we construct a metric for the topology of a quasifield belonging to a compact projective translation plane, using the modular function of its additive group (Theorem 2).     

4-dimensional compact projective planes with a 7-dimensional collineation group

The classification of 4-dimensional compact projective planes having a 7-dimensional collineation group is completed. Besides one single shift plane all such planes are either translation planes or dual translation planes.Dedicated to H. R. Salzmann on his 60th birthday     

Large semisimple groups on 16-dimensional compact projective planes are almost simple

The paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E₆(?26). A 16-dimensional, compact projective plane P admitting an automorphism group of dimension 41 or more is classical, [18] 87.5 and 87.7. For the special case of a semisimple group Δ acting on P the same result can be obtained if dim δ ≧ 37, see [16]. Our aim is to lower this bound. We show: if Δ is semisimple and dim δ ≧ 29, then P is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin₉ (?, r), r ∈ {0, 1 }. The underlying paper contains the first part of the proof showing that Δ is in fact almost simple.     

Polyanalytic forms on compact Riemann surfaces

A sheaf of differentials on a compact Riemann surface supplied with a projective structure is said to be n-analytic if, in local projective coordinates, sections of the sheaf satisfy the differential equation { } For the projective structure induced by a covering mapping from the disk, an explicit characterization of the space of cross sections and of the space of first cohomologies of an n-analytic sheaf is given in terms of known spaces of sections of certain holomorphic sheaves. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 15–25. Translated by S. V. Kislyakov.     

Four-dimensional compact projective planes with two fixed points

We determine all 4-dimensional compact projective planes with a solvable 6-dimensional collineation group fixing two distinct points, and acting transitively on the affine pencils through the fixed points. These planes form a 2-parameter family, and one exceptional member of this family is the dual of the exceptional translation plane with 8-dimensional collineation group.     

Smooth Stable Planes and the Moduli Spaces of Locally Compact Translation Planes

 Smooth stable planes have been introduced in [3]. At every point p of a smooth stable plane the tangent spaces of the lines through p form a compact spread (see the definition in Section 2) on the tangent space thus defining a locally compact topological affine translation plane . We introduce the moduli space of isomorphism classes of compact spreads, . We show that for the topology of is not by constructing a sequence of non-classical spreads in that converges to the classical spread in , where . Moreover, we prove that the isomorphism type of varies continuously with the point p. Finally, we give examples of smooth affine planes which have both classical and non-classical tangent translation planes. (Received 15 April 1999; in revised form 22 October 1999)     

A Geometric Relationship Between Equivalent Spreads

By Andrè theory, it is well known how to algebraically convert a spread in a projective space to an equivalent spread (representing the same translation plane) in a projective space of different dimension and of different order (corresponding to a subfield of the kernel). The goal of this paper is to establish a geometric connection between any two such equivalent spreads by embedding them as subspaces and subgeometries of an ambient projective spaces. The connection can be viewed as a generalization of a construction due to Hirschfeld and Thas.     

Polaritäten Ebener projektiver Ebenen

A projective plane is called flat if the spaces of points and lines are locally compact and 2-dimensional and the joining of points and the intersecting of lines are continuous. H. Salzmann studied planes of this type in [11]–[21]. Here polarities of such planes are considered. In II general properties of polarities of flat planes are discussed. For example, a polarity with absolute points has always an oval of absolute points. A flat projective plane with a cartesian ternary field K admits a polarity iff multiplication in K is commutative. In III the polarities of flat projective planes with a 3-dimensional collineation group are determined.     

On unitary equivalence of compact operator tuples

For a compact operator tuple A, if its projective spectrum P(A_*) is smooth, then there exists a natural Hermitian holomorphic line bundle E_A over P(A_*) which is a unitary invariant for A. This paper shows that under some additional spectral conditions, EAis a complete unitary invariant, i.e., EAcan determine the compact operator tuple up to unitary equivalence.     

The point and line space of a compact projective plane are homeomorphic

The local Hopf bundles at the points and lines of a compact projective plane with manifold lines are all weakly equivalent. In particular, the point space and the line space of such a projective plane are homeomorphic. This is a consequence of the following topological result. Theorem. Let $\xi,\xi'$ be orientable topological $\mathbb{R}^n$ -bundles over an n-dimensional CW-complex. If $\xi$ and $\xi'$ are fibre homotopy equivalent and stably equivalent, then $\xi$ and $\xi'$ are equivalent. Received: 10 December 1998 / Accepted: 18 May 1999     

Actions of almost simple groups on compact eight-dimensional projective planes are classical,almost

We show that the existence of an almost simple group of automorphisms of dimension greater than 10 characterizes the Hughes planes (including the quarternion plane) among the 8-dimensional compact projective planes.Dedicated to Prof. Helmut R. Salzmann on his 65th birthday     

Large Almost Simple Groups Acting on 16-Dimensional Compact Projective Planes

 This paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E₆(−26). A 16-dimensional, compact projective plane ? admitting an automorphism group of dimension 41 or more is clasical, [23] 87.5 and 87.7. For the special case of a semisimple group Δ acting on ? the same result can be obtained if dim , see [22]. Our aim is to lower this bound. We show: if Δ is semisimple and dim , then ? is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin₉ (ℝ, r), r∈{0, 1}. The proof consists of two parts. In [16] it has been shown that Δ is in fact almost simple or isomorphic to SL₃?ċSpin₃ℝ. In the underlying paper we can therefore restrict our considerations to the case that Δ is almost simple, and the corresponding planes are classified.     

Large Almost Simple Groups Acting on 16-Dimensional Compact Projective Planes

 This paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E₆(−26). A 16-dimensional, compact projective plane ? admitting an automorphism group of dimension 41 or more is clasical, [23] 87.5 and 87.7. For the special case of a semisimple group Δ acting on ? the same result can be obtained if dim , see [22]. Our aim is to lower this bound. We show: if Δ is semisimple and dim , then ? is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin₉ (ℝ, r), r∈{0, 1}. The proof consists of two parts. In [16] it has been shown that Δ is in fact almost simple or isomorphic to SL₃?ċSpin₃ℝ. In the underlying paper we can therefore restrict our considerations to the case that Δ is almost simple, and the corresponding planes are classified. Received 10 February 1997; in final form 19 December 1997     

Continuity of operators commuting with translations for compact groups

In this note we show for certain Frechet spacesF(G) of functions (distributions) on a compact groupG that if every translation invariant linear functional onF(G) is continuous then every linear operatorT:F(G)F(G) commuting with translations is continuous. This solves partially a problem in [7] ofG. H. Meisters and improves the result [5] ofC. J. Lester. An application for compact groups which do not have the mean zero weak containment property follows by the result [10] ofG. A. Willis.     

Nonexistence of disjoint compact Baer subplanes

We show that a certain fibration associated with a Baer subplane of a compact connected projective plane does not admit a cross section. This implies that, in marked contrast to the finite case, two Baer subplanes cannot be disjoint.     

A class of unitals of order q which can be embedded in two different planes of order q2

By deriving the desarguesian plane of order q² for every prime power q a unital of order q is constructed which can be embedded in both the Hall plane and the dual of the Hall plane of order q² which are non-isomorphic projective planes. The representation of translation planes in the fourdimensional projective space of J. André and F. Buekenhouts construction of unitals in these planes are used. It is shown that the full automorphism groups of these unitals are just the collineation groups inherited from the classical unitals.     

芬斯勒射影几何中的Ricci曲率

本文研究了保持Ricci曲率不变的Finsler射影变换。给定一个紧致无边的n维可微流形M，证明了：对于一个从M上的Berwald度量到Riemann度量的C-射影变换，如果Berwald度量的Ricci曲率关于Riemann度量的迹不超过Riemann度量的标量曲率，则该射影变换是平凡的。     

Die Oktavenebene als Translationsebene mit großer Kollineationsgruppe

It is shown that the affine plane over the Cayley numbers is the only 16-dimensional locally compact topological translation plane having a collineation group of dimension at least 41. This (hitherto unpublished) result is one of the ingredients of H. Salzmann's characterizations of the Cayley plane among general compact projective planes by the size of its collineation group.The proof involves various case studies of the possibilities for the structure and size of collineation groups of 16-dimensional locally compact translation planes. At the same time, these case studies are important steps for a classification program aiming at the explicit determination of all such translation planes having a collineation group of dimension at least 38.     

Coincidence of the upper Kuratowski topology with the co-compact topology on compact sets, and the Prohorov property

For a Hausdorff space X, let F be the hyperspace of all closed subsets of X and H a sublattice of F. Following Nogura and Shakhmatov, X is said to be H-trivial if the upper Kuratowski topology and the co-compact topology coincide on H. F-trivial spaces are the consonant spaces first introduced and studied by Dolecki, Greco and Lechicki. In this paper, we deal with K-trivial spaces and Fin-trivial space, where K and Fin are respectively the lattices of compact and of finite subsets of X. It is proved that if C_k(X) is a Baire space or more generally if X has ‘the moving off property’ of Gruenhage and Ma, then X is K-trivial. If X is countable, then C_p(X) is Baire if and only if X is Fin-trivial and all compact subsets of X are finite. As for consonant spaces, it turns out that every regular K-trivial space is a Prohorov space. This result remains true for any regular Fin-trivial space in which all compact subsets are scattered. It follows that every regular first countable space without isolated points, all compact subsets of which are countable, is Fin-nontrivial. Examples of K-trivial non-consonant spaces, of Fin-trivial K-nontrivial spaces and of countably compact Prohorov Fin-nontrivial spaces, are given. In particular, we show that all (generalized) Fréchet–Urysohn fans are K-trivial, answering a question by Nogura and Shakhmatov. Finally, we describe an example of a continuous open compact-covering mapping f :X→Y, where X is Prohorov and Y is not Prohorov, answering a long-standing question by Topsøe.