Showing papers in "Journal of Geometry in 2010"
TL;DR: In this article, Tachibana, Vishnevskii, and Yano-Ako operators are applied to pure tensor fields and their generalizations can be found in this context.
Abstract: The aim of this paper is to introduce some operators which are applied to pure tensor fields. In this context Tachibana, Vishnevskii, Yano–Ako operators and their generalizations can be found.
24 citations
TL;DR: In this paper, the Cayley-Klein geometry is generalized to projective spaces over arbitrary fields and of arbitrary dimensions, and a metric in a pappian projective space and the definition of substructures are defined.
Abstract: A. Cayley and F. Klein discovered in the nineteenth century that euclidean and non-euclidean geometries can be considered as mathematical structures living inside projective-metric spaces. They outlined this idea with respect to the real projective plane and established (“begrundeten”) in this way the hyperbolic and elliptic geometry. The generalization of this approach to projective spaces over arbitrary fields and of arbitrary dimensions requires two steps, the introduction of a metric in a pappian projective space and the definition of substructures as Cayley-Klein geometries. While the first step is taken in H. Struve and R. Struve (J Geom 81:155–167, 2004), the second step is made in this article. We show that the concept of a Cayley-Klein geometry leads to a unified description and classification of a wide range of non-euclidean geometries including the main geometries studied in the foundations of geometry by D. Hilbert, J. Hjelmslev, F. Bachmann, R. Lingenberg, H. Karzel et al.
21 citations
TL;DR: In this paper, it was shown that the Dehn invariants of any Bricard octahedron remain constant during the flex and that the Strong Bellows Conjecture holds true for the Steffen flexible polyhedron.
Abstract: We prove that the Dehn invariants of any Bricard octahedron remain constant during the flex and that the Strong Bellows Conjecture holds true for the Steffen flexible polyhedron.
18 citations
TL;DR: In this article, the authors introduced notions of orthogonality in terms of the 2−HH norm, and their properties are studied, including characterizations of inner product spaces and strictly convex spaces.
Abstract: Kikianty and Dragomir (Math Inequal Appl 13:1–32, 2010) introduced the p−HH norms on the Cartesian square of a normed space, which are equivalent, but are geometrically different, to the well-known p-norms. In this paper, notions of orthogonality in terms of the 2−HH norm are introduced; and their properties are studied. Some characterizations of inner product spaces are established, as well as a characterization of strictly convex spaces.
12 citations
TL;DR: The Schouten tensor as discussed by the authors is a tensor field of type (0, 2) arising in the remainder of the Weyl part in the standard decomposition of the curvature tensor of a Riemannian metric on a compact smooth manifold.
Abstract: Given a Riemannian metric on a compact smooth manifold, we consider its Schouten tensor, which is a tensor field of type (0, 2) arising in the remainder of the Weyl part in the standard decomposition of the curvature tensor of the metric. We study extremal properties of the Schouten functional, defined to be the scaling-invariant L2-norm of the Schouten tensor. It is proved, for instance, that space form metrics are characterized as critical points of the Schouten functional among conformally flat metrics.
11 citations
TL;DR: In this paper, it was shown that every convex quadrangle is self-affine, whereas only some, but not all convex pentagons are selfaffine.
Abstract: A polygon in $${{\mathbb R}^2}$$
is called self-affine if it can be dissected into k ≥ 2 affine images of itself. Self-affine convex polygons have at most five vertices. Triangles are trivially self-affine. It is shown that every convex quadrangle is self-affine, whereas only some, but not all convex pentagons are self-affine.
8 citations
Abstract: It has been conjectured that all non-desarguesian projective planes contain a Fano subplane. The Figueroa planes are a family of non-translation planes that are defined for both infinite orders and finite order q3 for q > 2 a prime power. We will show that there is an embedded Fano subplane in the Figueroa plane of order q3 for q any prime power.
7 citations
TL;DR: In this paper, it was shown that point sets of PG(n, q), q = p3h, p ≥ 7 prime, of size < 3(qn-1 + 1)/2 intersecting each line in 1 modulo \({\sqrt[3] q}\) points (these are always small minimal blocking sets with respect to lines) are linear blocking sets.
Abstract: The main result of this paper is that point sets of PG(n, q), q = p3h, p ≥ 7 prime, of size < 3(qn-1 + 1)/2 intersecting each line in 1 modulo \({\sqrt[3] q}\) points (these are always small minimal blocking sets with respect to lines) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p3), p ≥ 7 prime, of size < 3(p3(n-1) + 1)/2 with respect to lines are always linear.
6 citations
TL;DR: In this paper, the Euclidean plane is characterized via a relation between arc-length orthogonality and Birkhoff orthoghonality, and three characterizations are obtained by considering properties of certain points related to an exterior point of the unit disc and the two tangent segments corresponding to it.
Abstract: The following characterizations of the Euclidean plane are obtained: the two tangent segments of the unit circle of a normed plane from each point of a disc centered at the origin with sufficiently large diameter have equal lengths; the lengths of the tangent segments from each point of a fixed circle centered at the origin are determined only by the radius of this circle. Three further characterizations of the Euclidean plane are obtained by considering properties of certain points related to an exterior point of the unit disc and the two tangent segments corresponding to it. To obtain one of these characterizations, the notion of arc-length orthogonality is introduced, and the Euclidean plane is also characterized via a relation between arc-length orthogonality and Birkhoff orthogonality.
4 citations
TL;DR: In this article, the rank two BCDL2003-geometries of O'Nan were classified and the maximal rank of a 2-BCDL 2003-geometry for O’Nan was shown to be 4.
Abstract: We classify the rank two BCDL2003-geometries of O’Nan and show that the maximal rank of a BCDL2003-geometry for O’Nan is 4. This bound is sharp since it is satisfied by the rank four geometry given by Buekenhout (Contemp Math 45:1–32, 1985).
4 citations
TL;DR: In this article, a new axiom system for a continuous absolute plane was developed, which removed Side-Angle-Side as an axiom and replaced it with Side Angle Angle Angle (SAA) as a new Axiom.
Abstract: Following the approach of G.D. Birkhoff, we develop a new axiom system for a continuous absolute plane. In this new axiom system we remove Side-Angle-Side as an axiom and replace it with Side-Angle-Angle as a new axiom. We prove that the new axiom system is also a continuous absolute plane, and in particular, that Side-Angle-Side holds in the new axiom system. In addition, we give new proofs of well known results. These new proofs do not depend on Side-Angle-Side, but instead use Side-Angle-Angle.
TL;DR: In this paper, it was shown by a combination of theoretical argument and computer search that if a projective (75, 4, 12, 5) set in PG(3, 7) exists then its automorphism group must be trivial.
Abstract: We show by a combination of theoretical argument and computer search that if a projective (75, 4, 12, 5) set in PG(3, 7) exists then its automorphism group must be trivial. This corresponds to the smallest open case of a coding problem posed by H. Ward in 1998, concerning the possible existence of an infinite family of projective two-weight codes meeting the Griesmer bound.
TL;DR: In this paper, the cubic hypersurfaces in PG(5, 2) are classified into five large families and a description of the set of T-singular lines for a representative T of each of the five GL(6, 2)-orbits is given.
Abstract: The space Alt(×3V6) of alternating trilinear forms on V6 = V(6, 2) is naturally isomorphic to the space \({\wedge^{3}(V_{6} ^{\ast})}\) of trivectors based on the dual space \({V_{6}^{\ast}}\). Under the natural action of the group GL(6, 2) the nonzero elements of \({{\rm Alt}(\times^{3}V_{6})\cong\wedge^{3}(V_{6}^{\ast})}\) are shown to fall into five distinct orbits. In consequence, the cubic hypersurfaces in PG(5, 2) are classified into five large families. For \({T \in {\rm Alt}(\times^{3}V_{6})}\) let \({\mathcal{L}_{T}}\) denote the set of T-singular lines, consisting that is of those projective lines \({\langle a,b\rangle}\) in \({{\rm PG}(5,2)=\mathbb{P}V_{6}}\) such that T(a, b, x) = 0 for all \({x\in V_{6}}\). A description is given of the set \({\mathcal{L}_{T}}\) for a representative T of each of the five GL(6, 2)-orbits. In particular, for one of the orbits \({\mathcal{L}_{T}}\) is a Desarguesian line-spread in PG(5, 2).
TL;DR: In this paper, the defect function of an invariant reflection structure (P, I) is strictly connected to the precession maps of the corresponding K-loop, therefore it permits a classification of such structures with respect to the algebraic properties of their k-loop.
Abstract: The defect function [introduced in Karzel and Marchi (Results Math 47:305–326, 2005)] of an invariant reflection structure (P, I) is strictly connected to the precession maps of the corresponding K-loop (P, +), therefore it permits a classification of such structures with respect to the algebraic properties of their K-loop. In the ordinary case (i.e. when the K-loop is not a group) we define, by means of products of three involutions, four different families of blocks denoted, respectively, by \({\mathcal{L}_G, \mathcal{L}, \mathcal{B}_G, \mathcal{B}}\) (cf. Sect. 4) so that we can provide the reflection structure with some appropriate incidence structure. On the other hand we consider in (P, +) two types of centralizers and recognize a strong connection between them and the aforesaid blocks: actually we prove that all the blocks of (P, I) can be represented as left cosets of suitable centralizers of the loop (P, +) (Theorem 6.1). Finally we give necessary and sufficient conditions in order that the incidence structures \({(P, \mathcal{L}_G)}\) and \({(P,\mathcal{L})}\) become linear spaces (cf. Theorem 8.6).
TL;DR: In this article, the authors take the approach that the answer is no, that all the points in the intersection are somehow close to one another (neighbourly) and that two non-neighborly points determine a unique line.
Abstract: An old question regarding the world we live in concerns what is real regarding points and lines: if two distinct lines intersect, is their intersection a unique point? In this paper, we take the approach that the answer is no, that all the points in the intersection are somehow close to one another (neighbourly) and that two non-neighbourly points determine a unique line These are the Affine Klingenberg spaces (AK-spaces) How does one put a logical structure on points and lines that reflect the preceding view of reality? History has shown that such a structure is based upon the concept of coordinatization, which leads naturally to algebraic structures that allow a faithful representation of incidence, which in turn reflects the existence of relations between points and lines that recognise incidence The preceding view of reality is not new, and the history of this subject is of approaches that are too general (there are conditions on neighbourly points) Our approach is novel in that it is based upon a minimum number of assumptions that yield the existence of dilatations that are translations: the corner stones of coordinatization
TL;DR: In this paper, the underlying metric affine geometry can be recovered from Grassmann spaces associated with the family of regular subspaces of respective space, which are collineations witch preserve orthogonality of the respective underlying space.
Abstract: 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 Prazmowska et al. (Linear Algebra Appl 430:3066–3079, 2009) and Prazmowska and Żynel (Adv Geom, to appear).
TL;DR: In this paper, the authors present a solution for the largest regular m-gon contained in a regular n-gon, which depends critically on the coprimality of m and n. They show that the optimal polygons are concentric if and only if gcd(m, n) > 1.
Abstract: We present a solution for the largest regular m-gon contained in a regular n-gon. We find that the answer depends critically on the coprimality of m and n. We show that the optimal polygons are concentric if and only if gcd(m, n) > 1. Our principal result is a complete solution for the case where m and n share a common divisor. For the case of coprime m and n, we present partial results and a conjecture for the general solution. Our findings subsume some special cases which have previously been published on this problem.
TL;DR: Karzel et al. as mentioned in this paper considered an order concept which is based on the notion of separation for quadruples of concyclic points and established the connections between these two notions, and showed that these concepts are equivalent.
Abstract: In Karzel et al. (J. Geom. 99: 116–127, 2009) we introduced for a symmetric Minkowski plane $${ {\mathfrak M} := (P,\Lambda,{\mathfrak G}_1,{\mathfrak G}_2) }$$
an order concept by the notion of an orthogonal valuation for the circles of Λ and showed that there is a one to one correspondence between the valuations and the halforderings of the accompanying commutative field. Here we consider an order concept which is based on the notion of separation for quadruples of concyclic points and establish the connections between these two notions. Our main result (cf. Theorem 3.3) states that these concepts are equivalent.
TL;DR: This paper presents a survey of the structure of possible small weight vectors in the dual code of planes of even order, and discusses small sets of even type in the plane that could give words of low weight in theDual code.
Abstract: The problem of determining the minimum weight and the minimum-weight vectors of the dual code of a finite projective plane has attracted much attention over the years. In this paper, we present a survey of the structure of possible small weight vectors in the dual code of planes of even order. In particular, we discuss small sets of even type in the plane that could give words of low weight in the dual code.
TL;DR: In this paper, the authors study cyclic surfaces in E3 generated by spiral motions of a circle and find the representation of cyclic spiral surfaces which are envelopes of one-parametric set of spheres.
Abstract: In this paper, we study cyclic surfaces in E3 generated by spiral motions of a circle. We find the representation of cyclic spiral surfaces in E3 which are envelopes of one-parametric set of spheres. Finally, we give an example.
TL;DR: In this paper, the authors determine orbit representatives of all proper subplanes generated by quadrangles of a Veblen-Wedderburn (VW) plane and the Hughes plane under their full collineation groups.
Abstract: We determine orbit representatives of all proper subplanes generated by quadrangles of a Veblen-Wedderburn (VW) plane Π of order 112 and the Hughes plane Σ of order 112 under their full collineation groups. In Π, there are 13 orbits of Baer subplanes all of which are desarguesian and approximately 3000 orbits of Fano subplanes. In Σ, there are 8 orbits of Baer subplanes all of which are desarguesian, 2 orbits of subplanes of order 3 and at most 408,075 distinct Fano subplanes. This work was motivated by the well-known question: “Does there exist a non-desarguesian projective plane of prime order?” The question remains unsettled.
TL;DR: In this paper, closed conformal vector fields in a constant sectional curvature Riemannian manifold were used to study the geometry of its immersed submanifolds, and the authors obtained a characterization of sphere among compact submansifolds with positive Ricci curvature immersed in the manifold.
Abstract: We use closed conformal vector fields in a constant sectional curvature Riemannian manifold \({\mathbb{M}}\) to study the geometry of its immersed submanifolds. In this situation we obtain a characterization of sphere among compact submanifolds with positive Ricci curvature immersed in \({\mathbb{M}}\).
TL;DR: In this paper, the bilinear flocks of Cherowitzo are generalized to a large variety of bilinearly flocks on the cone of the cone, and net replacements of certain Hughes-Kleinfeld planes of order q4 are obtained that construct every Andre plane in PG(3, q).
Abstract: The bilinear flocks of Cherowitzo are generalized to a large variety of bilinear flocks of the cone \({\mathcal{C}_q}\) . Using these ideas, net replacements of certain Hughes-Kleinfeld planes of order q4 are obtained that construct every Andre plane in PG(3, q).
TL;DR: In this paper, it was shown that all four exchange properties and the minimal condition follow from the Existence Theorem for a basis, assuming finiteness condition or a weaker condition (called minimal condition).
Abstract: Four exchange properties, including the usual one, are discussed. Assuming the finiteness condition or a weaker condition (called minimal condition), all four are equivalent. But examples show that in general no two of the four properties are equivalent. Furthermore it is shown that all four properties and the minimal condition follow from the Existence Theorem for a basis.
TL;DR: In this paper, a new geometric characterization of the real absolute planes is presented, which is based upon few and simple axioms concerning properties of a congruence relation, and the ordering properties are developed from two axiomologies concerning triangles and circles.
Abstract: A new geometric characterization of the real absolute planes is presented, which is based upon few and simple axioms concerning properties of a congruence relation. The ordering properties are developed from two axioms concerning triangles and circles. We use essential results of reflection geometry in order to prove that the structures under consideration have the well-known representations.
TL;DR: In this article, necessary and sufficient conditions on m and n for inscribing a regular m-gon in a regular n-gon were given for a given m-approximation.
Abstract: We find necessary and sufficient conditions on m and n for inscribing a regular m-gon in a regular n-gon.
TL;DR: Algebraic varieties of projective spaces over the algebraic closure of a finite field, with two or few characters with respect to r-dimensional linear spaces, are studied in this paper.
Abstract: Algebraic varieties of projective spaces over the algebraic closure of a finite field, with two or few characters with respect to r-dimensional linear spaces are studied.
TL;DR: In this article, the concept of a monotone arc of convex sets was introduced, and it was shown that compact monotonous arcs have the Cebysev property in the hyperspace of compact strictly convex set.
Abstract: A set in a metric space is called a Cebysev set if it contains a unique “nearest neighbour” to each point of the space. In this paper we introduce the concept of a monotone arc of convex sets and show that compact monotone arcs have the Cebysev property in the hyperspace of compact strictly convex sets. In the hyperspace of compact convex sets only certain monotone arcs are Cebysev ; these are characterized. Results are also obtained for affine segments and for noncompact monotone arcs.