Showing papers in "Journal of Geometry in 2008"
TL;DR: In this article, Boyer and Galicki showed that a complete K-contact gradient soliton is a Jacobi vector field along the geodesics of the Reeb vector field.
Abstract: Inspired by a result of Boyer and Galicki, we prove that a complete K-contact gradient soliton is compact Einstein and Sasakian. For the non-gradient case we show that the soliton vector field is a Jacobi vector field along the geodesics of the Reeb vector field. Next we show that among all complete and simply connected K-contact manifolds only the unit sphere admits a non-Killing holomorphically planar conformal vector field (HPCV). Finally we show that, if a (k, μ)-contact manifold admits a non-zero HPCV, then it is either Sasakian or locally isometric to E3 or En+1 × Sn (4).
157 citations
TL;DR: In this paper, the necessary and sufficient conditions for the Mannheim partner curves in Euclidean space and Minkowski space were obtained for three-dimensional space, respectively, and some examples are also given.
Abstract: In this paper, we study Mannheim partner curves in three dimensional space. We obtain the necessary and sufficient conditions for the Mannheim partner curves in Euclidean space $${\mathbb{E}}^{3}$$
and Minkowski space $${\mathbb{E}}^{3}_{1}$$
, respectively. Some examples are also given.
156 citations
TL;DR: In this article, an enumeration of all semifields of order at most 256 is presented, for the order 81, where the order is ordered by the order of the order in which the semidefend is given.
Abstract: In [17] Kantor states “...it is surprising that there has not yet been an enumeration of all semifields of order at most 256 ...”. In this note we present such an enumeration for the order 81.
31 citations
TL;DR: In this paper, the sufficiency condition employed to show the existence and uniqueness of solutions to the discrete, planar L0-Minkowski problem with data containing, at least, a pair of opposite vectors is also a necessary condition.
Abstract: We prove that the sufficiency condition employed to show the existence and, in certain cases the uniqueness, of solutions to the discrete, planar L0-Minkowski problem with data containing, at least, a pair of opposite vectors is also a necessary condition
30 citations
TL;DR: In this article, a 4-parametric family of 4-dimensional pseudo-Hermitian manifolds is constructed on a Lie group and characterized geometrically, and the condition for a 4manifold to be isotropic hyper-Kahler is given.
Abstract: Almost hypercomplex pseudo-Hermitian manifolds are considered. Isotropic hyper-Kahler manifolds are introduced. A 4-parametric family of 4-dimensional manifolds of this type is constructed on a Lie group. This family is characterized geometrically. The condition a 4-manifold to be isotropic hyper-Kahler is given.
25 citations
TL;DR: In this paper, the class of Riemannian almost product manifolds with nonintegrable structure is considered and some identities for curvature tensors as invariant tensors and quantities are obtained.
Abstract: The class of the Riemannian almost product manifolds with nonintegrable structure is considered. Some identities for curvature tensor as certain invariant tensors and quantities are obtained.
24 citations
TL;DR: In this article, the Schouten tensor of a compact Riemannian manifold (M, g) with metric g and dimension n ≥ 3 was considered, and the authors gave an isometric classification for compact locally conformally flat manifolds which satisfy the conditions: Ag is semi-positive definite and σk(Ag) is a nonzero constant for some k ∈ {2,..., n}.
Abstract: Consider a compact Riemannian manifold (M, g) with metric g and dimension n ≥ 3. The Schouten tensor Ag associated with g is a symmetric (0, 2)-tensor field describing the non-conformally-invariant part of the curvature tensor of g. In this paper, we consider the elementary symmetric functions {σk(Ag), 1 ≤ k ≤ n} of the eigenvalues of Ag with respect to g; we call σk(Ag) the k-th Schouten curvature function. We give an isometric classification for compact locally conformally flat manifolds which satisfy the conditions: Ag is semi-positive definite and σk(Ag) is a nonzero constant for some k ∈ {2, ... , n}. If k = 2, we obtain a classification result under the weaker conditions that σ2(Ag) is a non-negative constant and (Mn, g) has nonnegative Ricci curvature. The corresponding result for the case k = 1 is well known. We also give an isometric classification for complete locally conformally flat manifolds with constant scalar curvature and non-negative Ricci curvature.
22 citations
TL;DR: In this article, the singularities of light-like surfaces of spacelike curves in de Sitter 3-space have been investigated and the geometric meaning of the singularity of such surfaces has been investigated.
Abstract: The Lorentzian space form with the positive curvature is called de Sitter space which is an important subject in the theory of relativity. In this paper we consider spacelike curves in de Sitter 3-space. We define the notion of lightlike surfaces of spacelike curves in de Sitter 3-space. We investigate the geometric meanings of the singularities of such surfaces.
21 citations
TL;DR: In this paper, the curvature of generalized symmetric spaces is described and the curvatures of four-dimensional generalized symmetrized spaces are classified in the Lorentzian case and in pseudo-Riemannian case.
Abstract: The curvature of four-dimensional generalized symmetric spaces is completely described. In particular, Einstein-like metrics on these spaces are classified. Interesting behaviours are found in the Lorentzian case and in one of the pseudo-Riemannian ones.
20 citations
TL;DR: In this article, it was shown that the Erdős-Mordell inequality is equivalent to the statement that the sum of the angles of a triangle is less than or equal to two right angles.
Abstract: We prove that, in plane absolute geometry, the Erdős-Mordell inequality is equivalent to the statement that the sum of the angles of a triangle is less than or equal to two right angles.
13 citations
TL;DR: In this article, the authors studied the properties of trajectories for canonical magnetic fields induced by structure tensors on real hypersurfaces of types A0 and A1 in a complex space form.
Abstract: In this paper we study some basic properties of trajectories for canonical magnetic fields induced by structure tensor on real hypersurfaces of types A0 and A1 in a complex space form. On each such real hypersurface, there are infinitely many canonical magnetic fields whose trajectories with null structure torsion are closed, and also infinitely many canonical magnetic fields whose trajectories with null structure torsion are open. We give a condition dividing canonical magnetic fields into these two classes and investigate lengths of closed trajectories. Our study also provide us information on the moduli space of Killing helices of proper order 4 on complex space forms.
TL;DR: In this paper, the authors construct n-dimensional hypersurfaces in (n + 1)-dimensional Euclidean space that satisfy a higher dimensional generalization of the equal area zones property of the sphere.
Abstract: We construct and study n-dimensional hypersurfaces in (n + 1)-dimensional Euclidean space that satisfy a higher dimensional generalization of the equal area zones property of the sphere (that the surface area of a zone between two parallel planes depends only on the distance between the planes). These new hypersurfaces are hypersurfaces of revolution. The relative simplicity of their construction allows us to describe them in great detail, revealing some interesting curiosities and motivating further questions.
TL;DR: In this article, the authors characterize hyperbolic geometry among Hilbert geometry by the property that three medians of any Hyperbolic triangle all pass through one point in one point.
Abstract: In this paper we characterize hyperbolic geometry among Hilbert geometry by the property that three medians of any hyperbolic triangle all pass through one point.
TL;DR: In this paper, a pseudo-Riemannian 4-manifold with a field of parallel null 2-planes is defined, which admits an orthogonal almost complex structure.
Abstract: We call a pseudo-Riemannian 4-manifold, which admits a field of parallel null 2-planes, a Walker 4-manifold A pseudo-Riemannian metric of a Walker 4-manifold is necessarily of neutral signature, and it admits an orthogonal almost complex structure We show that such a Walker 4-manifold can carry various structures with respect to a certain kind of almost complex structure, eg, symplectic structures, Kahler structures, Hermitian structures, according as the properties of certain functions which define the canonical form of the metric The combination of these structures are also analyzed
TL;DR: A set C in the Euclidean space is called isometrically m-divisible if there exists a disjoint decomposition of C into m subsets Ci pairwise congruent with respect to the group of isometries as discussed by the authors.
Abstract: A set C in the Euclidean space \({\mathbb{R}}^d\) is called isometrically m-divisible if there exists a disjoint decomposition of C into m subsets Ci pairwise congruent with respect to the group of Euclidean isometries. We present a necessary condition for the isometric m-divisibility, which shows in particular that most convex bodies – in the sense of Baire category – are not isometrically m-divisible for any choice of m\(m \in \{2, 3, \ldots, \aleph_0 \}\).
TL;DR: A new class of translation planes of order qn that admit very small affine homology groups is constructed by the replacement of a new hyper-regulus that cannot be an Andre or generalized Andre.
Abstract: A new class of translation planes of order qn that admit very small affine homology groups is constructed These planes are constructed by the replacement of a new hyper-regulus that cannot be an Andre hyper-regulus Hence, the planes cannot be Andre or generalized Andre
TL;DR: In this paper, the following question is considered: given any point in an n-dimensional simplex in Euclidean space, does there exist a reflection in one of the facets of the simplex such that it can be found in a closed ball such that the point is contained in the closed ball?
Abstract: Let Δ be an n-dimensional simplex in Euclidean space $${\mathbb{E}}^n$$
contained in an n-dimensional closed ball B. The following question is considered. Given any point $$x \,{\in}\,\triangle$$
, does there exist a reflection $$r : {\mathbb{E}}^n \rightarrow {\mathbb{E}}^n$$
in one of the facets of Δ such that $$r(x)\in B$$
?
TL;DR: A semioval in a projective plane is a nonempty subset S of points with the property that for every point P ∈ S there exists a unique line l such that S \cap \le = \{P\}\) as mentioned in this paper.
Abstract: A semioval in a projective plane \(\prod\) is a nonempty subset S of points with the property that for every point P ∈ S there exists a unique line l such that \(S \cap \le = \{P\}\). It is known that \(q +1 \leq \|S\| \leq q\sqrt{q} + 1\) and both bounds are sharp. We say that S is a small semioval in \(\prod\) if \(\|S\| \leq 3(q + 1)\).
TL;DR: In this article, the celebrated Erdos, Faber and Lovasz Conjecture may be expressed as follows: any linear hypergraph on ν points has chromatic index at most ν, where ν denotes the linear intersection number and χ denotes the chromatic number of the graph.
Abstract: The celebrated Erdos, Faber and Lovasz Conjecture may be stated as follows: Any linear hypergraph on ν points has chromatic index at most ν We show that the conjecture is equivalent to the following assumption: For any graph \(\chi(G) \leq
u(G)\), where ν(G) denotes the linear intersection number and χ(G) denotes the chromatic number of G As we will see \(\chi(G) + \chi(\bar{G}) \leq
u(G)+
u(\overline{G})\) for any graph G = (V, E), where \((\overline{G})\) denotes the complement of G Hence, at least G or \(\overline{G}\) fulfills the conjecture
TL;DR: In this article, the authors studied locally strongly convex, hyperbolic centroaffine Tchebychev hypersurfaces with complete centro affine metric and gave a classification in case that the Ricci tensor is semi-positive and the scalar curvature is constant.
Abstract: We study locally strongly convex, hyperbolic centroaffine Tchebychev hypersurfaces with complete centroaffine metric. We give a classification in case that the centroaffine Ricci tensor is semi-positive and the scalar curvature is constant. The classification extends known global results for hyperbolic affine hyperspheres and for complete centroaffine extremal hypersurfaces.
TL;DR: In this article, a 6-dimensional example of non-compact complete Einstein non-Kahler almost Kahler manifold G with negative scalar curvature was constructed by Apostolov-Draghici-Moroianu.
Abstract: In this paper, we introduce a 6-dimensional example of non-compact complete Einstein non-Kahler almost-Kahler manifold G with negative scalar curvature which was constructed by Apostolov-Draghici- Moroianu ([5]) and discuss the geometric structures.
TL;DR: In this paper, the notion of quasi-constant holomorphic sectional curvature is introduced in the Kahlerian framework and necessary and sufficient conditions for these manifolds to be locally symmetric are obtained.
Abstract: In correspondence with the manifolds of quasi-constant sectional curvature defined (cf [5], [9]) in the Riemannian context, we introduce in the Kahlerian framework the geometric notion of quasi-constant holomorphic sectional curvature. Some characterizations and properties are given. We obtain necessary and sufficient conditions for these manifolds to be locally symmetric, Ricci or Bochner flat, Kahler η-Einstein or Kahler-Einstein, etc. The characteristic classes are studied at the end and some examples are provided throughout.
TL;DR: In this paper, basic properties of the affine slit spaces and axiomatic characterizations of reducts of affine, Desarguesian and Pappian affine planes are discussed.
Abstract: We discuss basic properties of the affine slit spaces and give elementary axiomatic characterizations of reducts of affine, Desarguesian affine, and Pappian affine planes.
TL;DR: In this paper, it was shown that the sum of the angles under which we see a diameter from two points of the set separated by the diameter is never smaller than 5π/6.
Abstract: This paper is about diameters of compact sets. These are chords of maximal length. On one hand we see that the sum of the angles under which we see a diameter from two points of the set separated by the diameter is never smaller than 5π/6. On the other hand we describe cases in which the diameter of a point-symmetric set must join symmetric points.
TL;DR: A classification of projective translation planes of order q 2 admits a collineation group G admitting a two-transitive orbit of q + 1 points is given in this article.
Abstract: A classification is given of all projective translation planes of order q2 that admit a collineation group G admitting a two-transitive orbit of q + 1 points.
TL;DR: In this paper, the authors determine all metric and periodic lines in de Sitter's world over a real pre-Hilbert space, and show that all lines in this space are periodic.
Abstract: We determine all metric and periodic lines in de Sitter’s world over a real pre-Hilbert space.
TL;DR: In this article, it was shown that for every odd integer s ≥ 1, there is a corresponding set of mutually disjoint reguli in any Desarguesian plane, such that the line intersection properties of the reguli of S ≥ 1 are inherited from those of K ≥ 2.
Abstract: Let S
k
denote a set of k reguli in a Desarguesian affine plane $$\sum_{q^2}$$
of order q
2. It is shown that, for every odd integer s > 1, there is a corresponding set S
s
k
of k reguli in any Desarguesian plane $$\sum_{q^{2s}}$$
of order q
2s
such the line intersection properties of the reguli of S
s
k
are inherited from those of S
k
. Hence, sets of mutually disjoint reguli in $$\sum_{q^2}$$
‘lift’ to sets of mutually disjoint reguli in $$\sum_{q^{2s}}$$
. Thus, the existence of a subregular spread in PG(3, q) produces an infinite class of subregular spreads in spaces PG(3, q
s
).
TL;DR: In this article, a two-parameter family of affinely connected surfaces which admit the cylinder group as a collineation group of their geodesics is presented, and the Moulton Planes in the radial model of Betten, the circular cone, as well as the real affine plane are part of this family.
Abstract: We present a two-parameter family of affinely connected surfaces which admit the cylinder group as a collineation group of their geodesics. The Moulton Planes in the radial model of Betten, the circular cone, as well as the real affine plane, are part of this family. The Moulton Planes occur in this family in the same way as the real affine plane is contained in a family of cones with decreasing steepness.
TL;DR: In this article, two examples of 4-dimensional complex manifolds with Norden metric are constructed by means of Lie groups and Lie algebras, and the form of the curvature tensor for each of the examples is obtained.
Abstract: Two examples of 4-dimensional complex manifolds with Norden metric are constructed by means of Lie groups and Lie algebras. Both manifolds are characterized geometrically. The form of the curvature tensor for each of the examples is obtained. Conditions these manifolds to be isotropic-Kahlerian are given.
TL;DR: In this paper, a Lie group as a 4-dimensional pseudo-Riemannian manifold is considered, which is equipped with an almost product structure and a Killing metric in two ways.
Abstract: A Lie group as a 4-dimensional pseudo-Riemannian manifold is considered. This manifold is equipped with an almost product structure and a Killing metric in two ways. In the first case Riemannian almost product manifold with nonintegrable structure is obtained, and in the second case – a pseudo-Riemannian one. Each belongs to a 4-parametric family of manifolds, which are characterized geometrically.