Showing papers in "Journal of Geometry in 2005"
TL;DR: In this article, it was shown that the Hilbert geometry associated to a bounded convex domain is isometric to a normed vector space if and only if D is an open n-simplex.
Abstract: It is shown that the Hilbert geometry (D, h
D
) associated to a bounded convex domain
$$D \subset \mathbb{E}^{n} $$
is isometric to a normed vector space
$${\left( {{\text{V}},{\left\| {\, \cdot \,} \right\|}} \right)}$$
if and only if D is an open n-simplex. One further result on the asymptotic geometry of Hilbert’s metric is obtained with corollaries for the behavior of geodesics. Finally we prove that every geodesic ray in a Hilbert geometry converges to a point of the boundary.
49 citations
TL;DR: In this paper, the authors studied ruled Weingarten surfaces in Minkowski 3-space on which there is a nontrivial functional relation between a pair of elements of the set {K, K, K I. II.
Abstract: In this paper, we study ruled Weingarten surfaces M : x (s, t) = α(s) + tβ (s) in Minkowski 3-space on which there is a nontrivial functional relation between a pair of elements of the set {K, K
II
, H, H
II
}, where K is the Gaussian curvature, K
II
is the second Gaussian curvature, H is the mean curvature, and H
II
is the second mean curvature. We also study ruled linear Weingarten surfaces in Minkowski 3-space such that the linear combination aK
II
+ bH + cH
II
+ dK is constant along each ruling for some constants a, b, c, d with a
2 + b
2 + c
2 ≠ 0.
49 citations
TL;DR: In this paper, the authors give a closed formula for volumes of generic hyperbolic tetrahedra in terms of edge lengths, based on the volume conjecture for the Turaev-Viro invariant of closed 3-manifolds, defined from the quantum 6j -symbols.
Abstract: We give a closed formula for volumes of generic hyperbolic tetrahedra in terms of edge lengths. The cue of our formula is by the volume conjecture for the Turaev-Viro invariant of closed 3-manifolds, which is defined from the quantum 6j -symbols. This formula contains the dilogarithm functions, and we specify the adequate branch to get the actual value of the volumes.
45 citations
TL;DR: In this paper, the spectrum of possible sizes k of complete k-arcs in finite projective planes PG(2,q) is investigated by computer search and backtracking algorithms that try to construct complete arcs joining the orbits of some subgroup of collineation groups are applied.
Abstract: The spectrum of possible sizes k of complete k-arcs in finite projective planes PG(2 ,q) is investigated by computer search. Backtracking algorithms that try to construct complete arcs joining the orbits of some subgroup of collineation group PL( 3 ,q) and randomized greedy algorithms are applied. New upper bounds on the smallest size of a complete arc are given for q = 41, 43, 47, 49, 53, 59, 64, 71 ≤ q ≤ 809 ,q � 529, 625, 729, and q = 821. New lower bounds on the second largest size of a complete arc are given for q = 31, 41, 43, 47, 53, 125. Also, many new sizes of complete arcs are obtained for 31 ≤ q ≤ 167.
39 citations
TL;DR: In this article, it was shown that there exists a unique (up to an isometry) convex cyclic polygon with edge lengths a 1,..., a fixme n ≥ 0.
Abstract: Let a1, ..., a
n
be positive numbers satisfying the condition that each of the a
i
’s is less than the sum of the rest of them; this condition is necessary for the a
i
’s to be the edge lengths of a (closed) polygon. It is proved that then there exists a unique (up to an isometry) convex cyclic polygon with edge lengths a1, ..., a
n
. On the other hand, it is shown that, without the convexity condition, there is no uniqueness—even if the signs of all central angles and the winding number are fixed, in addition to the edge lengths.
25 citations
TL;DR: In this paper, it was shown that every non-flat complex space form of complex dimension greater than one does not admit a warped product representation and that warped product immersions cannot be represented by warped products in real space forms.
Abstract: Warped product immersions appeared naturally in several recent studies. In this article we study fundamental geometric properties of such immersions. In addition we prove that every non-flat complex space form of complex dimension greater than one does not admit a warped product representation. As an application we obtain an improvement of an earlier result in [3] concerning warped products in real space forms.
23 citations
TL;DR: In this article, flat translation invariant surfaces in 3D Heisenberg group are classified and shown to be invariant to 3D translations. But they are not invariant against 3D collision.
Abstract: Abstract.Flat translation invariant surfaces in 3-dimensional Heisenberg group are classified.
13 citations
TL;DR: In this paper, the authors consider the problem of partitioning a convex body into equal-area parts by means of chords and prove two basic results that hold with some specific exceptions: (a) When chords are pairwise non-crossing, the dual tree of the partition has to be a path, and (b) A convex ngon admits no equipartition produced by more than n chords having a common interior point.
Abstract: In this paper we consider the problem of partitioning a plane compact convex body into equal-area parts, i.e., an equipartition, by means of chords. We prove two basic results that hold with some specific exceptions: (a) When chords are pairwise non-crossing, the dual tree of the partition has to be a path, (b) A convex n-gon admits no equipartition produced by more than n chords having a common interior point.
11 citations
TL;DR: In this article, the problem of classifying finite projective planes of order n with an automorphism group G and a point orbit on which G acts two-transitively is investigated in considerable detail, under the assumption that G has length at last n.
Abstract: The problem of classifying finite projective planes \(\Pi \) of order n with an automorphism group G and a point orbit \(\mathcal{O}\) on which G acts two-transitively is investigated in considerable detail, under the assumption that \(\mathcal{O}\) has length at last n. Combining old and new results a rather satisfying classification is obtained, even though some cases for orbit lengths n and n + 1 remain unsolved.
10 citations
TL;DR: In this article, small holes through which regular 3-, 4-, and 5-dimensional simplices can pass were studied for regular 3, 4, and 5 dimensional simplices, respectively.
Abstract: Abstract.We study small holes through which regular 3-, 4-, and 5-dimensional simplices can pass.
9 citations
TL;DR: In this article, a simple classification of triples of Lie cycles is given, based on 2-dimensional Lie contact geometry in the form of two of its subgeometries, Laguerre geometry and oriented Mobius geometry.
Abstract: A simple classification of triples of Lie cycles is given. The class of each triad determines the number of solutions to the associated oriented Apollonius contact problem. The classification is derived via 2-dimensional Lie contact geometry in the form of two of its subgeometries—Laguerre geometry and oriented Mobius geometry. The method of proof illustrates interactions between the two subgeometries of Lie geometry. Two models of Laguerre geometry are used: the classic model and the 3-dimensional affine Minkowski space model.
TL;DR: In this paper, it was shown that conformal Anosov flow is conformally anosov on a compact contact metric 3-manifolds with critical metric for the Chern-Hamilton functional.
Abstract: We prove that on a compact (non Sasakian) contact metric 3-manifold with critical metric for the Chern-Hamilton functional, the characteristic vector field ξ is conformally Anosov and there exists a smooth curve in the contact distribution of conformally Anosov flows As a consequence, we show that negativity of the ξ-sectional curvature is not a necessary condition for conformal Anosovicity of ξ (this completes a result of [4]) Moreover, we study contact metric 3-manifolds with constant ξ-sectional curvature and, in particular, correct a result of [13]
TL;DR: In this paper, the authors discuss the singular case in which the square of a correlation does not fix any non-absolute point of that correlation and show that there is, up to isomorphisms, just one such correlation, and that it possesses q 2n+1 + 1 absolute points.
Abstract: This paper is the last in a series of four articles devoted to the classification of correlations of finite Desarguesian planes of odd nonsquare order. The first three papers (see references [7], [8], [9]) were devoted to correlations whose squares (which are, of course, collineations) leave invariant at least one nonabsolute point of the respective correlation. The present article discusses the “singular” situation in which the square of a correlation does not fix any nonabsolute point of that correlation. It turns out that there is, up to isomorphisms, just one such correlation, and that it possesses q
2n+1 + 1 absolute points. Its square leaves invariant one absolute point.
TL;DR: In this article, the relation between distance-regular graphs and (α, β)-geometries was studied in two different ways: the first is the necessary and sufficient conditions for the neighbourhood geometry of a distance regular graph to be an ε, β-geometry.
Abstract: We study the relation between distance-regular graphs and (α, β)-geometries in two different ways. We give necessary and sufficient conditions for the neighbourhood geometry of a distance-regular graph to be an (α, β)-geometry, and describe some (classes of) examples. On the other hand, properties of certain regular two-graphs allow us to construct (0, α)-geometries on the corresponding Taylor graphs.
TL;DR: In this article, the Gergonne and Nagel centers of a tetrahedron were extended to simplices of any dimension, and it was shown that both centers exist and are unique for all simplices.
Abstract: In contrast with the analogous situation for a triangle, the cevians that join the vertices of a tetrahedron to the points where the faces touch the insphere (or the exspheres) are not concurrent in general. This observation led the present author and P. Walker in [4] to devise alternative definitions of the Gergonne and Nagel centers of a tetrahedron that do not assume the concurrence of such cevians and that coincide with the ordinary definitions in the case of a triangle. They then proved that the Gergonne center exists and is unique for all tetrahedra and that the Nagel center, though unique, exists only for tetrahedra that satisfy certain conditions. In this article, we extend these definitions to simplices of any dimension. By keeping the requirement that the Gergonne center be interior and relaxing such a condition for the Nagel center, we prove that both centers exist and are unique for all simplices, thus polishing the definitions and generalizing the results of the above-mentioned article.
TL;DR: In this article, a (2m + 3)-dimensional Riemannian manifold with a vertical skew symmetric almost contact 3-structure is considered, and the fundamental 2-form Ω associated with φ is a presymplectic form.
Abstract: We consider a (2m + 3)-dimensional Riemannian manifold M(ξ r
, η
r
, g ) endowed with a vertical skew symmetric almost contact 3-structure. Such manifold is foliated by 3-dimensional submanifolds
$$M^ \bot $$
of constant curvature tangent to the vertical distribution and the square of the length of the vertical structure vector field is an isoparametric function. If, in addition, M(ξ r
, η
r
, g ) is endowed with an f -structure φ, M, turns out to be a framed f−CR-manifold. The fundamental 2-form Ω associated with φ is a presymplectic form. Locally, M is the Riemannian product
$$M = M^ \top \times M^ \bot $$
of two totally geodesic submanifolds, where
$$M^ \top$$
is a 2m-dimensional Kaehlerian submanifold and
$$M^ \bot $$
is a 3-dimensional submanifold of constant curvature. If M is not compact, a class of local Hamiltonians of Ω is obtained.
TL;DR: In this article, the stability of certain harmonic sections of the normal bundles for compact submanifolds in the spheres is studied and the stability properties of these sections are analyzed. But the authors focus on the stability in terms of the harmonic properties of the submannifolds.
Abstract: We study harmonic sections of the normal bundles for submanifolds. Especially, the stability of certain harmonic sections of the normal bundles for compact submanifolds in the spheres are considered.
TL;DR: In this article, a characterisation of the Rosati oval in the regular nearfield plane of dimension 2 over its centre is given, based on a 3D model of the sphere.
Abstract: We give a characterisation of the Rosati oval in the regular nearfield plane of dimension 2 over its centre.
TL;DR: In this paper, the simple connectedness condition was introduced for a simply connected orthogonal polygon in the plane, where the set S is a union of two sets which are starshaped via staircase paths if and only if for every three points of S, at least two points see (via staircase paths) a common point of S.
Abstract: Let S be a simply connected orthogonal polygon in the plane. The set S is a union of two sets which are starshaped via staircase paths (i.e., orthogonally starshaped) if and only if for every three points of S, at least two of these points see (via staircase paths) a common point of S. Moreover, the simple connectedness condition cannot be deleted.
TL;DR: In this paper, it was shown that every mapping that preserves the distances 1 and 2 is an isometry, provided d ≥ 5, where d ≥ is the number of isometries.
Abstract: Benz proved that every mapping \(f:\mathbb{Q}^d \to \mathbb{Q}^d \) that preserves the distances 1 and 2 is an isometry, provided d ≥ 5. We prove that every mapping \(f:\mathbb{Q}^d \to \mathbb{Q}^d \) that preserves the distances 1 and \(\sqrt 2 \) is an isometry, provided d ≥ 5.
TL;DR: In this paper, the authors describe the contact of higher order and the dual contact for Cayley's cubics and show that there are three exceptional cases for higher order cubics on the Cayley surface.
Abstract: Cayley’s (ruled cubic) surface carries a three-parameter family of twisted cubics. We describe the contact of higher order and the dual contact of higher order for these curves and show that there are three exceptional cases.
TL;DR: In this article, the number of non-isomorphic semifield planes of order p4 and kernel GF(p2) for p prime, 3 ≤ p ≤ 11, and the class of p-primitive planes is the largest.
Abstract: In this article we determine the number of non-isomorphic semifield planes of order p4 and kernel GF(p2) for p prime, 3 ≤ p ≤ 11. We show that for each of these values of p, the plane is either desarguesian, p-primitive, or a generalized twisted field plane. We also show that the class of p-primitive planes is the largest. We also discuss the autotopism group of the semifields under study.
TL;DR: In this paper, a variety of partitions (flocks) of Segre varieties by caps are obtained, and the partitions arise from semifield planes and are thus called "semifield flat flocks".
Abstract: Using ‘fusion’ methods on finite semifields, a variety of partitions (flocks) of Segre varieties by caps are obtained. The partitions arise from semifield planes and are thus called “semifield flat flocks”. Furthermore, the finite transitive semifield flat flocks are completely determined.
TL;DR: In this paper, the foundations of the construction of the direct product of affine partial linear spaces, as defined by Johnson and Ostrom, are analyzed and fundamental preservation theorems are proved, and the role of frequently used affine axioms in the context of this theory is discussed.
Abstract: We analyze foundations of the construction of the direct product of affine partial linear spaces, as defined by Johnson and Ostrom. Fundamental preservation theorems are proved, and the role of some of frequently used affine axioms in the context of this theory is discussed.
TL;DR: In this article, the authors studied the projective space of univariate rational parameterized equations of degree d or less in real projective spaces and gave a geometric characterization of the automorphism group of projective kinematics.
Abstract: We study the projective space
$$\mathbb{S}_n^d $$
of univariate rational parameterized equations of degree d or less in real projective space
$$\mathbb{P}^n $$
The parameterized equations of degree less than d form a special algebraic variety
$$\mathcal{K}_1 $$
We investigate the subspaces on
$$\mathcal{K}_1 $$
and their relation to rational curves in
$$\mathbb{P}^n, $$
give a geometric characterization of the automorphism group of
$$\mathcal{K}_1 $$
and outline applications of the theory to projective kinematics
TL;DR: In this paper, the number of hamiltonian circuits of a graph through which a graph can be characterized as having a dominant dominant circuit of minimal length is given, and a formula for the total number of such circuits is given.
Abstract: We get a formula for the number of hamiltonian circuits of a graph through which we characterize the special hamiltonian graphs, that is containing a dominant circuit of minimal length
TL;DR: In this paper, a local condition on a planar space is given which is sufficient for its points, lines and planes to be the points, the lines and some subspaces of a projective space.
Abstract: A local condition on a planar space is given which is sufficient for its points, lines and planes to be the points, the lines and some subspaces of a projective space.
TL;DR: Congruent classes of Frenet curves of order 2 in the complex quadric are studied in this article, obtaining that each congruence class is a level set of a family of smooth functions.
Abstract: Congruent classes of Frenet curves of order 2 in the complex quadric are studied, obtaining that each congruence class is a level set of a family of certain smooth functions, that are generalizations of isoparametric functions on the unit sphere in the tangent space of the complex quadric.
TL;DR: In order to identify multipliers of abelian (υ, k, λ)-difference sets the First and Second Multiplier Theorem of Hall, Ryser and Chowla, resp. of Hall and Menon, need a divisor m of n = k − λ that is coprime to υ as mentioned in this paper.
Abstract: In order to identify multipliers of abelian (υ, k, λ)-difference sets the First and the Second Multiplier Theorem of Hall, Ryser and Chowla, resp. of Hall and Menon, need a divisor m of n = k − λ that is coprime to υ. Moreover, both theorems require that m > λ. The famous Multiplier Conjecture asserts that the restriction m > λ is not necessary.