Characterization of the Transformation Group of the Space of a Null System

In the space I_r of the invariant r-dimensional subspaces of a null system in (2r +1)-dimensional projective space, W.L. Chow characterized the basic group of transformations of I_r as all the transformations φ: I_r → I_r which are bijective and such that φ and φ^?1 preserve adjacency. In the present paper we examine arbitrary mappings φ of I_r which satisfy the two conditions: 1. φ preserves adjacency. 2. For any a ∈ I_r there exists b ∈ I_r such that a^φ ∩ b^φ = ø.     

Adjacency in generalized projective Veronese spaces

A bijection of strong subspaces of a generalized Veronese space preserving the adjacency need not to be determined by an automorphism of the underlying space. We give conditions which assure that the adjacency preserving bijection of points (or lines) of a generalized Veronese space is determined by an automorphism of this space. The results are applied to Veronese spaces associated with projective structures.     

Chebyshev polynomials on symmetric matrices

In this paper we evaluate Chebyshev polynomials of the second kind on a class of symmetric integer matrices, namely on adjacency matrices of simply laced Dynkin and extended Dynkin diagrams. As an application of these results we explicitly calculate minimal projective resolutions of simple modules of symmetric algebras with radical cube zero that are of finite and tame representation type.     

Distant-preserving epimorphisms between projective Klingenberg planes

This paper deals with epimorphisms between arbitrary projective Klingenberg planes which preserve the non-neighbor relation. Our main result is an algebraic characterization of such epimorphisms which generalizes a theorem of F. D. Veldkamp [7] for distant preserving epimorphisms between projective ring planes.     

射影平坦Finsler度量的解析构造

利用Hamel关于射影平坦的基本方程,我们导出了Randers度量的λ形变保持射影平坦的充分条件.特别,对一类具有特殊旗曲率性质的Randers度量我们证明了这类度量一定存在保持射影平坦性的λ形变.     

Geometry of free cyclic submodules over ternions

Given the algebra T of ternions (upper triangular 2×2 matrices) over a commutative field F we consider as set of points of a projective line over T the set of all free cyclic submodules of T ². This set of points can be represented as a set of planes in the projective space over F ⁶. We exhibit this model, its adjacency relation, and its automorphic collineations. Despite the fact that T admits an F-linear antiautomorphism, the plane model of our projective line does not admit any duality.     

Characterization of orthomaps on the Cayley plane

We classify maps which preserve orthogonality on the Cayley projective plane over octonions. In addition, we also classify orthogonality preserving maps on finite dimensional projective spaces over reals, complexes, or quaternions. Unlike similar results which extend Uhlhorns’s theorem we assume neither injectivity/surjectivity nor that orthogonality is preserved in both directions.     

Preserving levels of projective determinacy by tree forcings

We prove that various classical tree forcings—for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing—preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes are added to thin projective transitive relations by these forcings.     

Partitions of Three-Dimensional Pappian Projective Spaces by Ovoidal Quadrics

In his article Partitioning Projective Geometries Into Caps, Ebert showed inter alia that every finite three-dimensional projective space can be partitioned by ovoidal quadrics. The aim of the present note is to generalize this result by identifying a more comprehensive class of three-dimensional projective spaces which admit partitions of this kind. In addition, the group of all collineations which preserve such a partition is determined.     

On canonical almost geodesic mappings which preserve the weyl projective tensor

We study a partial case of canonical almost geodesic mappings of the first type of spaces with affine connection that preserve Weyl projective curvature tensor and certain other tensors. Main equations under consideration are reduced to a closed Cauchy system type in covariant derivatives. Therefore a general solution to these equations depends on a finite number of constants. We submit an example of above mappings between flat spaces. We establish that projective Euclidean and equiaffine spaces form closed classes of spaces with respect to these mappings.     

On Opposition in Spherical Buildings and Twin Buildings

In this paper, we prove a combinatorial property of twin apartments and opposition of chambers in twin buildings. We then characterize adjacency of chambers in twin buildings by meansof opposition of chambers. As an application, we study maps which satisfy certain conditions related to opposition of chambers, e.g., maps that preserve opposition. Applied to the special case of spherical buildings, all our main results as well as their corollaries are new.     

Projective holonomy I: principles and properties

The aim of this paper and its sequel is to introduce and classify the holonomy algebras of the projective Tractor connection. After a brief historical background, this paper presents and analyses the projective Cartan and Tractor connections, the various structures they can preserve, and their geometric interpretations. Preserved subbundles of the Tractor bundle generate foliations with Ricci-flat leaves. Contact- and Einstein-structures arise from other reductions of the Tractor holonomy, as do U(1) and bundles over a manifold of smaller dimension.     

Grassmann spaces of regular subspaces

The underlying metric affine geometry, or metric projective geometry, can be recovered from Grassmann spaces associated with the family of regular subspaces of respective space. In other words, automorphisms of such Grassmann spaces are collineations witch preserve orthogonality of the respective underlying space. This generalizes results of Pra?mowska et al. (Linear Algebra Appl 430:3066–3079, 2009) and Pra?mowska and ?ynel (Adv Geom, to appear).     

Algebraic characterization of lineations on subsets of desarguesian affine spaces

Lineations are mappings that preserve collinearity. It is shown that lineations, defined ou some subsets of a desarguesian affine spaceA with weak conditions of injectivity, are restrictions of injective lineations defined on the projective closure II(A) (see Theorem 2). This leads to a generalization of the fundamental theorem of affine geometry.Dedicated to Professor Helmut Mäurer on the occasion of his 60th birthday.     

On the genus of the groups PSL(2, q), PSL(3, q), and PSp(4, q)

SL(n, q) is the group of n×n matrices, over the Galois field GF(q), of determinate 1. PSL(n, q) is SL(n, q) modulo the scalar n×n matrices of determinate 1. PSL(n, q) acts on the Desarguesian projective space PG(n−1, q). Sp(4, q) is the group of 4 × 4 matrices of determinate 1 which preserve the symplectic bilinear form on the 4 × 1 matrices over GF(q). PSp(4, q) is Sp(4, q) modulo Z = {1,−1}. PSp(4, q) acts on the symplectic generalized quadrangle W(3, q), a subspace of the projective space PG(3, q), as a group of automorphisms. In this paper, bounds are given for the genus of these groups.     

Filtered colimits in the effective topos

It is shown that the “constant sheaves” functor ∇:Sets→Eff does not preserve ω₁-filtered colimits, and that as a consequence of this, the full subcategory of Eff on the countable projective objects is not dense.     

On the Z-eigenvalues of the adjacency tensors for uniform hypergraphs

The adjacency matrices for graphs are generalized to the adjacency tensors for uniform hypergraphs, and some fundamental properties for the adjacency tensor and its Z-eigenvalues of a uniform hypergraph are obtained. In particular, some bounds on the smallest and the largest Z-eigenvalues of the adjacency tensors for uniform hypergraphs are presented.     

Tensoring with Infinite-Dimensional Modules in $\mathcal {O}_0$

We show that the principal block O₀\mathcal {O}_0 of the BGG category O\mathcal {O} for a semisimple Lie algebra \frak g\frak g acts faithfully on itself via exact endofunctors which preserve tilting modules, via right exact endofunctors which preserve projective modules and via left exact endofunctors which preserve injective modules. The origin of all these functors is tensoring with arbitrary (not necessarily finite-dimensional) modules in the category O\mathcal {O}. We study such functors, describe their adjoints and show that they give rise to a natural (co)monad structure on O₀\mathcal {O}_0. Furthermore, all this generalises to parabolic subcategories of O₀\mathcal {O}_0. As an example, we present some explicit computations for the algebra \fraksl₃\frak{sl}_3.     

Adjacency posets of planar graphs

In this paper, we show that the dimension of the adjacency poset of a planar graph is at most 8. From below, we show that there is a planar graph whose adjacency poset has dimension 5. We then show that the dimension of the adjacency poset of an outerplanar graph is at most 5. From below, we show that there is an outerplanar graph whose adjacency poset has dimension 4. We also show that the dimension of the adjacency poset of a planar bipartite graph is at most 4. This result is best possible. More generally, the dimension of the adjacency poset of a graph is bounded as a function of its genus and so is the dimension of the vertex-face poset of such a graph.     

Cartan-Fubini type extension of holomorphic maps respecting varieties of minimal rational tangents

A fundamental result in the theory of minimal rational curves on projective manifolds is CartanFubini extension theorem proved by Hwang and Mok,which describes the extensibility of biholomorphisms between connected open subsets of two Fano manifolds of Picard number 1 which preserve varieties of minimal rational tangents(VMRT),under a mild geometric assumption on the second fundamental forms of VMRT’s.Hong and Mok have developed Cartan-Fubini extension for non-equidimensional holomorphic immersions from a connected open subset of a Fano manifold of Picard number 1 into a uniruled projective manifold,under the assumptions that the map sends VMRT’s onto linear sections of VMRT’s and it satisfies a mild geometric condition formulated in terms of second fundamental forms on VMRT’s.In the current paper,we give a generalization of Hong and Mok’s result,under the same condition on second fundamental forms,assuming only that the holomorphic immersions send VMRT’s to VMRT’s.Our argument is different from Hong and Mok’s and is based on the study of natural foliations on the total family of VMRT’s.This gives a substantially simpler proof than Hong and Mok’s argument.