Showing papers in "Journal of Geometry in 2014"
TL;DR: A tensor is presented by combining Riemann–Christoffel curvature Tensor, Ricci tensor, the metric tensor and scalar curvature which describe various curvature tensors as its particular cases and is proved to have equivalency of different geometric structures.
Abstract: In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann–Christoffel curvature tensor, the same type structures given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various geometric structures obtained by same restriction imposing on different curvature tensors. In this purpose we present a tensor by combining Riemann–Christoffel curvature tensor, Ricci tensor, the metric tensor and scalar curvature which describe various curvature tensors as its particular cases. Then with the help of this generalized tensor and using algebraic classification we prove the equivalency of different geometric structures (see Theorems 6.3, 6.4, 6.5, 6.6 and 6.7; Tables 1 and 2).
56 citations
TL;DR: In this article, it was shown that there are no null Mannheim curves in Minkowski 3-space, except for pseudo null straight lines and pseudo null circles with Mannheim partner curves.
Abstract: In this paper, we prove that there are no null Mannheim curves in Minkowski 3-space. We also prove that the only pseudo null Mannheim curves in Minkowski 3-space are pseudo null straight lines and pseudo null circles whose Mannheim partner curves are pseudo null straight lines. Finally, we give some examples of pseudo null Mannheim curves in \({E^3_1}\).
13 citations
TL;DR: In this article, the Chern and the Stiefel-Whitney characteristic classes of the manifold TM and the manifold SrM are computed. But the Chern characteristic is not applicable to manifold TM.
Abstract: We prove a Theorem on homotheties between two given tangent sphere bundles SrM of a Riemannian manifold M, g of \({{\rm dim}\geq3}\), assuming different variable radius functions r and weighted Sasaki metrics induced by the conformal class of g. New examples are shown of manifolds with constant positive or with constant negative scalar curvature which are not Einstein. Recalling results on the associated almost complex structure IG and symplectic structure \({\omega^G}\) on the manifold TM, generalizing the well-known structure of Sasaki by admitting weights and connections with torsion, we compute the Chern and the Stiefel–Whitney characteristic classes of the manifoldsTM and SrM.
12 citations
TL;DR: In this paper, it was shown that for the measure arising from what is known as the polar sine, the sum of measures of the corner angles of an orthocentric tetrahedron is constant and equals π.
Abstract: Focusing on the fact that the sum of the angles of any Euclidean triangle is constant and equals π for all triangles, Hajja and Martini raised, in [Math Intell 35(3):16–28, 2013, Problem 9], the question whether an analogous statement holds for higher dimensional d-simplices. An interesting answer was given by Hajja and Hammoudeh in (Beit Algebra Geom (to appear), 2014), where they proved that for the measure arising from what is known as the polar sine, the sum of measures of the corner angles of an orthocentric tetrahedron is constant and equals π. A crucial ingredient in that treatment is the fact that orthocentric d-simplices are pure, in the sense that the planar subangles of every corner angle are all acute, all obtuse, or all right. In this article, it is shown that this property is not shared by any of the three other special families of d-simplices that appear in the literature, namely, the families of circumscriptible, isodynamic, and isogonic (or rather tetra-isogonic) d-simplices, thus answering Problem 3 of (Hajja and Martini in Math Intell 35(3):16–28, 2013). Specifically, it is proved that there are d-simplices in each of these families in which one of the corner angles has an acute, an obtuse, and a right planar subangle. The tools used are expected to be useful in various other contexts. These tools include formulas for the volumes of d-simplices in these families in terms of the parameters in their standard parameterizations, simple characterizations of the Cayley–Menger determinants of such d-simplices, embeddability of a given d-simplex belonging to any of these families in a (d + 1)-simplex in the same family, formulas for some special determinants, and a nice property of a certain class of quadratic forms.
10 citations
TL;DR: In this article, a characterization of cones in PG(3, q) as sets of points of a plane of size q 2 + q + 1 projecting from a point V, with three intersection numbers with respect to the planes is given.
Abstract: A characterization of cones in PG(3, q) as sets of points of PG(3, q) of size q2 + q + 1 projecting from a point V a set of q + 1 points of a plane of PG(3, q) and with three intersection numbers with respect to the planes is given.
9 citations
TL;DR: In this article, it was shown that Ricci and Yamabe solitons are gradient only in the steady case in a proper second-order symmetric spacetime and four-dimensional Lorentzian plane wave manifolds.
Abstract: We show that a proper second-order symmetric spacetime, and four-dimensional Lorentzian plane wave manifolds admit different vector fields resulting in expanding, steady and shrinking Ricci and Yamabe solitons. Moreover, it is proved that those Ricci and Yamabe solitons are gradient only in the steady case.
9 citations
TL;DR: In this article, a series of 2-generator Sol-manifolds depending on a positive integer n, introduced by Molnar and Szirmai, were studied and constructed as tetrahedron manifolds and showed that they are twofold coverings of the 3-sphere branched over specified links.
Abstract: We study a series of 2-generator Sol-manifolds depending on a positive integer n, introduced by Molnar and Szirmai. We construct them as tetrahedron manifolds and show that they are twofold coverings of the 3-sphere branched over specified links. Finally, we give a surgery description of the considered 3-manifolds; indeed, they can be obtained by n−2 and 0 Dehn surgeries along the components of the Whitehead link.
9 citations
TL;DR: In this paper, the set of T-singular lines in an alternating trilinear form is defined, and a few of them are notable by virtue of the special nature of their set.
Abstract: Given an alternating trilinear form \({T\in\text{Alt}(\times^{3}V_{n})}\) on \({V_{n}=V(n,\mathbb{F})}\) let \({\mathcal{L}_{T}}\) denote the set of T-singular lines in \({\text{PG}(n-1)=\mathbb{P}V_{n},}\) consisting that is of those lines \({\langle a,b\rangle}\) of \({\text{PG}(n-1)}\) such that T(a, b, x) = 0 for all \({x\in V_{n}.}\) Amongst the immense profusion of different kinds of T we single out a few which we deem noteworthy by virtue of the special nature of their set \({\mathcal{L}_{T}}\).
8 citations
TL;DR: New bounds for the maximum size of a set of generators of H(2d − 1, q2) which pairwise intersect in codimension i are provided by applying a multiplicity bound by C. D. Godsil.
Abstract: We provide new bounds for the maximum size of a set of generators of H(2d − 1, q2) which pairwise intersect in codimension i by applying a multiplicity bound by C. D. Godsil. This implies a new bound on the maximum size of partial spreads of H(2d − 1, q2), d even.
8 citations
TL;DR: The largest class of Riemannian almost product P-manifolds, which is closed with respect to the group of conformal transformations of the RiemANNian metric, is the class of the conformal P-Manifolds as discussed by the authors, whose curvature tensors have similar properties as the Kahler tensors.
Abstract: The largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric, is the class of the conformal Riemannian P-manifolds This class is an analogue of the class of the conformal Kahler manifolds in almost Hermitian geometry The main aim of this work is to obtain properties of manifolds of this class with connections, whose curvature tensors have similar properties as the Kahler tensors in Hermitian geometry
6 citations
TL;DR: In this paper, translation balls and their packing in the hyperbolic 3-space were discussed and an infinite family of packings generated by discrete groups of isometries were presented.
Abstract: For one of Thurston model spaces, \({\widetilde{{\rm SL}_2({\mathbb{R}})}}\), we discuss translation balls and packing that space by such balls in contrast to the packing by standard (geodesic) balls. We present an infinite family of packings generated by discrete groups of isometries, and observe numerical results on their densities. In particular, we found packings whose densities are close to the upper bound density for ball packings in the hyperbolic 3-space.
TL;DR: The relationship between ordered groups and ordered geometries was studied in this article, where it was shown that the geometric concept of an ordered plane corresponds on the group-theoretical side to an order structure of the group of translations.
Abstract: The article studies the relationship between ordered groups and ordered geometries. To this end we consider metric planes of a very general kind (without any assumptions about order, continuity, free mobility and the existence or uniqueness of a joining line) which are singular (the set of translations forms a group) and show that the geometric concept of an ordered plane corresponds on the group-theoretical side to an order structure of the group of translations. This correspondence shows that not only Euclidean planes but also Minkowskian and Galilean planes are orderable if and only if the associated coordinate field is orderable. The article closes with some implications for the foundations of ordered geometry which include an axiomatic analysis of the Pasch axiom and some remarks on the relationship of the notions of incidence and order.
TL;DR: In this paper, it was shown that any horizontally minimal smooth surface embedded in the Heisenberg group is locally a straight ruled surface, i.e., it consists of straight lines tangent to a horizontal vector field along a smooth curve.
Abstract: We generalise a result of Garofalo and Pauls: a horizontally minimal smooth surface embedded in the Heisenberg group is locally a straight ruled surface, i.e., it consists of straight lines tangent to a horizontal vector field along a smooth curve. We show additionally that any horizontally minimal surface is locally contactomorphic to the complex plane.
TL;DR: In this article, an n-dimensional manifold Mn evolving under the Ricci flow was considered and gradient estimates for positive solutions of porous medium equations on Mn were derived for Harnack type inequalities.
Abstract: In this paper we consider an n-dimensional manifold Mn evolving under the Ricci flow and establish gradient estimates for positive solutions of porous medium equations on Mn. As applications, we derive Harnack type inequalities. In particular, our results generalize gradient estimates for positive solutions of the heat equations in Liu (Pacific J Math 243:165–180 [18]).
TL;DR: In this article, it was shown that for any positive integer k, if q ≥ 2k − 1, then there are partitions of PG(n − 1, q) in normal rational curves of degree 2k−1.
Abstract: Let \({L = \mathbb{F}_{q^n}}\) be a finite field and let \({F=\mathbb{F}_q}\) be a subfield of L. Consider L as a vector space over F and the associated projective space that is isomorphic to PG(n − 1, q). The properties of the projective mapping induced by \({x \mapsto x^{-1}}\) have been studied in Csajbok (Finite Fields Appl. 19:55–66, 2013), Faina et al. (Eur. J. Comb. 23:31–35, 2002), Havlicek (Abh. Math. Sem. Univ. Hamburg 53:266–275, 1983), Herzer (Abh. Math. Sem. Univ. Hamburg 55:211–228 1985, Handbook of Incidence Geometry. Buildings and Foundations. Elsevier, Amsterdam, 1995). The image of any line is a normal rational curve in some subspace. In this note a more detailed geometric description is achieved. Consequences are found related to mixed partitions of the projective spaces; in particular, it is proved that for any positive integer k, if q ≥ 2k − 1, then there are partitions of PG(2k − 1, q) in normal rational curves of degree 2k − 1. For smaller q the same construction gives partitions in (q + 1)-tuples of independent points.
TL;DR: In this paper, a compact and semisimple Lie group G is considered endowed with a 2-plectic structure ω, induced by the Killing form, and it is shown that ω induces a left-invariant valued 2-form which is proportional to dΘ, where Θ is the Cartan-Maurer 1-form on G.
Abstract: In this paper a compact and semisimple Lie group G is considered endowed with a 2-plectic structure ω, induced by the Killing form. We show that the Lie group of 2-plectomorphisms of G is finite dimensional and compact, and hence the Darboux’s theorem fails to be true for this 2-plectic structure. Also it is shown that ω induces a left-invariant $${\mathfrak{g}^{*}}$$
valued 2-form which is proportional to dΘ, where Θ is the Cartan–Maurer 1-form on G. At last we consider the action of G on its tangent bundle which is furnished with the 2-plectic structure ω
c
, the complete lift of ω, and calculate covariant momentum map of this action.
TL;DR: In this article, the authors constructed invariant hypercomplex structures with hyper-Hermitian and Norden metrics on Lie groups of dimension 4, where the signature of the basic pseudo-Riemannian metric is considered.
Abstract: Integrable hypercomplex structures with Hermitian and Norden metrics on Lie groups of dimension 4 are considered. The corresponding five types of invariant hypercomplex structures with hyper-Hermitian metric, studied by Barberis, are constructed here. The different cases regarding the signature of the basic pseudo-Riemannian metric are considered.
TL;DR: In this paper, a higher fundamental form of a hypersurface is defined as a form obtained by inserting a power of the shape operator into the first fundamental form, and the hypersurfaces in a Riemannian space form or in a three-dimensional Bianchi-Cartan-Vranceanu space for which the covariant derivative of a high fundamental form vanishes are classified.
Abstract: By a “higher fundamental form” of a hypersurface is understood a form obtained by inserting a power of the shape operator into the first fundamental form. The hypersurfaces in a Riemannian space form or in a three-dimensional Bianchi–Cartan–Vranceanu space for which the covariant derivative of a higher fundamental form vanishes are classified.
TL;DR: In this paper, the authors considered centroaffine surfaces as projective surfaces and studied the projective minimality of centro-affine minimal surfaces with one-parameter family of deformation known as associated surfaces.
Abstract: Centroaffine minimal surfaces are considered as an interesting class of surfaces from the viewpoint of not only variational problems in centroaffine differential geometry but also integrable systems. Typical examples of centroaffine minimal surfaces are proper affine spheres centered at the origin when we regard them as centroaffine surfaces. On the other hand, the study of projective minimal surfaces has a long history in projective differential geometry. Typical examples of projective minimal surfaces are proper affine spheres again, and so-called Demoulin surfaces or Godeaux-Rozet surfaces. In this paper, we shall regard centroaffine surfaces as projective surfaces and study projective minimality of centroaffine minimal surfaces. Using the fact that any centroaffine minimal surfaces have a one-parameter family of deformation known as associated surfaces, we shall give a classification of indefinite centroaffine minimal surfaces whose associated surfaces are all projective minimal, which includes centroaffine surfaces with vanishing Tchebychev operator and those found by the first author before. We shall also show that any indefinite centroaffine minimal surface whose associated surfaces are all Godeaux-Rozet surfaces is a proper affine sphere.
TL;DR: In this article, the authors provide characterizations of euclidean or hyperbolic spaces based on intrinsic four point properties which are related to known four point embedding properties.
Abstract: Many characterizations of euclidean spaces (real inner product spaces) among metric spaces have been based on euclidean four point embeddability properties. Related “intrinsic” four point properties have also been used to characterize euclidean or hyperbolic spaces among a suitable class of metric spaces. The present paper provides new characterizations of euclidean or hyperbolic spaces based on intrinsic four point properties which are related to known four point embedding properties.
TL;DR: In this paper, it was shown that the binary code C of the projective Hall plane Hq2 of even order q 2 where q = 2 t, for t 2, has words of weight 2q 2 in its hull that are not the dierence of the incidence vectors of two lines of Hq 2.
Abstract: We show that the binary code C of the projective Hall plane Hq2 of even order q 2 where q = 2 t , for t 2 has words of weight 2q 2 in its hull that are not the dierence of the incidence vectors of two lines of Hq2; together with an earlier result for the dual Hall planes of even order, this shows that for all t 2 the Hall plane and its dual are not tame. We also deduce that dim(C) > 3 2t +1, the dimension of the binary code of the desarguesian projective plane of order 2 2t , thus supporting the Hamada-Sachar conjecture for this innite class of planes.
TL;DR: In this article, a three-dimensional Riemannian manifold equipped with two circulant structures is considered and a condition for the structure q to be parallel with respect to the connection of g and q is given.
Abstract: We consider a three-dimensional Riemannian manifold equipped with two circulant structures—a metric g and a structure q, which is an isometry with respect to g and the third power of q is minus identity. We discuss some curvature properties of this manifold, we give an example of such a manifold and find a condition for q to be parallel with respect to the Riemannian connection of g.
TL;DR: In this paper, it was shown that a quaternary linear code is FH-free if there is no plane meeting the code in a Fano plane or in a non-singular Hermitian curve.
Abstract: A set \({\fancyscript{K}}\) in PG(r, 4), r ≥ 2, is odd if every line meets \({\fancyscript{K}}\) in an odd number of points. An odd set \({\fancyscript{K}}\) in PG(r, 4), r ≥ 3, is FH-free if there is no plane meeting \({\fancyscript{K}}\) in a Fano plane or in a non-singular Hermitian curve. We prove that an odd set \({\fancyscript{K}}\) contains a hyperplane of PG(r, 4) if and only if \({\fancyscript{K}}\) is FH-free. As an application to coding theory, a new extension theorem for quaternary linear codes is given.
TL;DR: In this article, the notion of k-polyharmonic morphisms is introduced, which preserves polyharmonic functions of order k. The morphisms are defined as maps between Riemannian manifolds that preserve harmonic functions and biharmonic function respectively.
Abstract: Polyharmonic functions have been studied in various fields. There are maps between Riemannian manifolds called harmonic morphisms and biharmonic morphisms that preserve harmonic functions and biharmonic functions respectively. In this paper, we introduce the notion of k-polyharmonic morphisms as maps that preserves polyharmonic functions of order k. For k = 3, we obtain several characterizations of triharmonic morphisms. We also give some relationships among harmonic, biharmonic, and triharmonic morphisms, and a relationship between triharmonic morphisms and p-harmonic morphisms.
TL;DR: In this article, the authors used tactical decompositions by automorphism group (TDA) to study symmetric configurations in detail and gave models which display the structure of the group.
Abstract: As is known, there are 229 symmetric configurations 123, (Daublebsky von Sterneck in Monatshefte Math Phys 5:223–255, 1895; Gropp in J Comb Inf Syst Sci 15:34–48, 1990). We use tactical decompositions by automorphism group (TDA) to study these configurations in detail. In (Daublebsky von Sterneck in Monatshefte Math Phys 14, 254–260, 1903) the automorphism groups of the configurations were determined. We find some errors there and correct them. For the configurations with a rather big automorphism group, we give models which display the structure of the group.
TL;DR: In this paper, the connection between Fuss' relation for bicentric n-gons where conics are two nested circles and Fuss's relation for BICentric ngons with conics being two separated circles is discussed.
Abstract: The article primarily deals with the connection between Fuss’ relation for bicentric n-gons where conics are two nested circles and Fuss’ relation for bicentric n-gons where conics are two separated circles. It is proved that one of them is quite determined by other. Also, the article deals with solutions of some old and difficult problems where characteristic points of two corresponding circles are defined and often used. Several interesting properties are established and some difficult problems are solved.
TL;DR: In this article, it was shown that every possible k-cycle can be embedded into PG(n, q), for all n ≥ 3, where q is the number of classes.
Abstract: We prove that every possible k-cycle can be embedded into PG(n, q), for all n ≥ 3. Mathematics Subject Classification (1991). 05C38, 05B25, 51E20.
TL;DR: In this paper, the side lengths of a non-degenerate cyclic quadrilateral are given, but not necessarily in cyclic order, and a formula for the sum of the lengths of the three diagonals of the triangle is given.
Abstract: If the side lengths of a non-degenerate cyclic quadrilateral are given, but not necessarily in cyclic order, then three diagonal lengths arise in the resulting three cyclic quadrilaterals, just as three possible pairs of supplementary angles arise as opposite vertices, and where the diagonals intersect, in each of the three configurations. We obtain a formula for the sum of the lengths of the three diagonals minus the sum of the four sides which enables us to deduce the geometric inequality that the sum of the side lengths is less than the sum of the lengths of the three diagonals. We obtain another formula when these lengths are replaced by their squares, and this yields a similar inequality. A proof of both formulas is given which uses algebraic geometry, but which proceeds by analysis of degenerate situations. Two alternative proofs of the linear version of the inequality (which implies the quadratic version) are supplied which use trigonometry and Lagrange multipliers respectively. An unusual feature of these results is that they refer not to one configuration, but rather concern three possible configurations.
TL;DR: In this article, the authors estimate volumes of trajectory-balls for normal magnetic Jacobi fields under some assumptions on sectional curvatures or Ricci curvatures of the underlying Kahler manifold.
Abstract: In this paper, applying comparison theorems on normal magnetic Jacobi fields we estimate volumes of trajectory-balls for Kahler magnetic fields under some assumptions on sectional curvatures or Ricci curvatures of the underlying Kahler manifold.
TL;DR: In this paper, it was shown that a cubic curve is invariant under the isogonal transformation of the projective plane determined by some triangle, which is the case for all cubic curves.
Abstract: We develop techniques to prove that a cubic curve is invariant under the isogonal transformation of the projective plane determined by some triangle.