Showing papers in "Journal of Geometry in 2009"

Almost Kenmotsu Manifolds and Nullity Distributions

Abstract: We characterize almost contact metric manifolds which are CR-integrable almost Kenmotsu, through the existence of a suitable linear connection. We classify almost Kenmotsu manifolds satisfying a certain nullity condition, we give examples and completely describe the three dimensional case.

On sizes of complete caps in projective spaces PG( n , q ) and arcs in planes PG(2, q )

Abstract: More than thirty new upper bounds on the smallest size t 2(2, q) of a complete arc in the plane PG(2, q) are obtained for (169 ≤ q ≤ 839. New upper bounds on the smallest size t 2(n, q) of the complete cap in the space PG(n, q) are given for n = 3 and 25 ≤ q ≤ 97, q odd; n = 4 and q = 7, 8, 11, 13, 17; n = 5 and q = 5, 7, 8, 9; n = 6 and q = 4, 8. The bounds are obtained by computer search for new small complete arcs and caps. New upper bounds on the largest size m 2(n, q) of a complete cap in PG(n, q) are given for q = 4, n = 5, 6, and q = 3, n = 7, 8, 9. The new lower bound 534 ≤ m 2(8, 3) is obtained by finding a complete 534-cap in PG(8, 3). Many new sizes of complete arcs and caps are obtained. The updated tables of upper bounds for t 2(n, q), n ≥ 2, and of the spectrum of known sizes for complete caps are given. Interesting complete caps in PG(3, q) of large size are described. A proof of the construction of complete caps in PG(3, 2 h ) announced in previous papers is given; this is modified from a construction of Segre. In PG(2, q), for q = 17, δ = 4, and q = 19, 27, δ = 3, we give complete $${(\frac{1}{2}(q + 3) + \delta)}$$ -arcs other than conics that share $${\frac{1}{2}(q + 3)}$$ points with an irreducible conic. It is shown that they are unique up to collineation. In PG(2, q), $${{q \equiv 2}}$$ (mod 3) odd, we propose new constructions of $${\frac{1}{2} (q + 7)}$$ -arcs and show that they are complete for q ≤ 3701.

Stability for some extremal properties of the simplex

Abstract: Some geometric inequalities for convex bodies, where the equality cases characterize simplices, are improved in the form of stability estimates. The inequalities all deal with covering by homothetic copies.

Projective metric realizations of cone-manifolds with singularities along 2-bridge knots and links

Abstract: Using projective metric geometry we develop a technique to describe homogeneous 3-dimensional metrics on cone-manifolds generated by two rotations In particular, for some cone-manifolds with singularities along 2-bridge knots and links we give explicit descriptions of all possible geometries (\({\mathbb S^3}\), \({\widetilde{SL_2(\mathbb R)}}\), and Nil)

Singularities of lightcone Gauss images of spacelike hypersurfaces in de Sitter space

Abstract: We define the notions of lightcone Gauss images of spacelike hypersurfaces in de Sitter space. We investigate the relationships between singularities of these maps and geometric properties of spacelike hypersurfaces as an application of the theory of Legendrian singularities. We classify the singularities and give some examples in the generic case in de Sitter 3-space.

On semi-parallel lightlike hypersurfaces of indefinite Kenmotsu manifolds

Abstract: We study semi-parallel lightlike hypersurfaces of an indefinite Kenmotsu manifold, tangent to the structure vector field. Some Theorems on parallel and semi-parallel vector field, geodesibility of lightlike hypersurfaces are obtained. The geometrical configuration of such lightlike hypersurfaces is established. We prove that, in totally contact umbilical lightlike hypersurfaces of an indefinite Kenmotsu manifold which has constant \({\overline{\phi}}\)-holomorphic sectional curvature c, tangent to the structure vector field and such that its distribution is parallel, the parallelism and semi-parallelism notions are equivalent.

Circular trajectories on real hypersurfaces in a nonflat complex space form

Abstract: In this paper we study which trajectories for Sasakian magnetic fields are circles on certain standard real hypersurfaces which are called hypersurfaces of type A in a nonflat complex space form. We also give a characterization of these real hypersurfaces by such a circular property of trajectories for Sasakian magnetic fields.

Vectors, Cyclic Submodules, and Projective Spaces Linked with Ternions

Abstract: Given a ring of ternions R, i. e., a ring isomorphic to that of upper triangular 2×2 matrices with entries from an arbitrary commutative field F, a complete classification is performed of the vectors from the free left R-module Rn+1, n ≥ 1, and of the cyclic submodules generated by these vectors. The vectors fall into 5 + |F| and the submodules into 6 distinct orbits under the action of the general linear group GLn+1(R).

Generalized Line Stars and Topological Parallelisms of the Real Projective 3-Space

Abstract: Let Q be an elliptic quadric of the real projective 3-space PG(3,\({\mathbb{R}}\)) =: \(\prod_{3}\) and denote by Q¬i the set of non-interior points with respect to Q. A simple covering \(\mathfrak{G}\) of Q¬i by 2-secants of Q is called generalized line star with respect to Q. The generalized line star \(\mathfrak{G}\) is called continuous, if the determination of the unique line of \(\mathfrak{G}\) through a given point of Q¬i is continuous. A parallelismis a family P of spreads such that each line of \(\prod_{3}\) is contained in exactly one spread of P; two lines of \(\prod_{3}\) are P-parallel if, and only if, they are members of the same spread of P. The parallelism P is called topological, if the operation of drawing a line P-parallel to a given line through a given point is continuous. In [1] the authors give a construction \({\mathbb{P}}\) such that \({\mathbb{P}}(\mathfrak{G})\) is a parallelism of \(\prod_{3}\) ; cf. Theorem 1.4 below.

Differential Geometry of Submanifolds of Warped Product Manifolds I × f Sm−1(k)

Abstract: We provide a general study of submanifolds in Rm(k, f) := I × f S, which is the warped product of an open interval I and a Riemannian manifold S of constant sectional curvature k. Fundamental properties of submanifolds in Rm(k, f) are obtained. Several classification theorems on parallel, curvature-invariant and totally umbilical submanifolds in Rm(k, f) are proved. Moreover, hypersurfaces of constant curvature in Rm(k, f) are also classified.

Embedding Finite Partial Linear Spaces in Finite Translation Nets

Abstract: In the 1970’s Paul Erdős and Dominic Welsh independently posed the problem of whether all finite partial linear spaces \({\mathbb{L}}\) are embeddable in finite projective planes. Except for the case when \(\mathbb{L}\) has a unique embedding in a projective plane with few additional points, very little has been done which is directly applicable to this problem. In this paper it is proved that every finite partial linear space \(\mathbb{L}\) is embeddable in a finite translation net generated by a partial spread of a vector space of even dimension. The question of whether every finite partial linear space is embedded in a finite Andre net is also explored. It is shown that for each positive integer n there exist finite partial linear spaces which do not embed in any Andre net of dimension less than or equal to n over its kernel.

A Missing Prime Configuration in the Hausdorff Metric Geometry

Abstract: The Hausdoff metric h was introduced by Felix Hausdoff in the early 20th century as a way to measure the distance between elements in the hyperspace \({\mathcal{H}}({\mathbb{R}}^{n})\) of non-empty compact subsets of \({\mathbb{R}}^{n}\). The geometry this metric imposes on \({\mathcal{H}}({\mathbb{R}}^{n})\) has many interesting properties: lines in this geometry can have endpoints; there can be many elements at a given location between two sets in \({\mathcal{H}}({\mathbb{R}}^{n})\); the Fibonacci and Lucas numbers arise in a natural way in this geometry; and for infinitely many different values of the positive integer k, there is a configuration [A, B] in \({\mathbb{R}}^{n}\) so that there exist exactly k elements at each location between A and B. In this paper we will show that if k is between 1 and 18, there is always a configuration having exactly k elements at each location between the end elements and prove the surprising result that no such configuration exists if k equals 19.

Duality between hyperbolic and de Sitter geometry

Abstract: We study the trigonometry on the de Sitter surface. Since this surface carries a metric of Lorentzian signature, care has to be taken when defining lengths and angles. We provide trigonometric formulae for triangles of all causality types. This is basically achieved by transferring the concept of polar triangles from spherical geometry into the Minkowski space. As a byproduct, we obtain a new simple proof of the hyperbolic law of cosines for angles.

On Blocking Sets of External Lines to a Hyperbolic Quadric in PG(3, q), q Even

Abstract: Minimum size blocking sets with respect to the external lines to a hyperbolic quadric of PG(3, q), q > 4 even, are characterized.

The Pons Asinorum for Tetrahedra

Abstract: Proposition 5 of Book I of Euclid’s Elements, better known as the Pons Asinorum or the Asses’ Bridge, and its converse, Proposition 6, state that two sides of a triangle are equal if and only if the opposite angles are equal. A generalization of this statement to higher dimensional d-simplices is considered in [14], where it is shown that such a generalization holds only if the underlying d-simplex is orthocentric. In this paper, it is shown that a reasonably satisfactory generalization holds for all, not necessarily orthocentric, tetrahedra. It is also shown that a stronger statement than that given in [14] holds for orthocentric tetrahedra.

Invariant submanifolds of Sasakian space forms

Abstract: In the present study, we consider isometric immersions \({f : M \rightarrow \tilde{M}(c)}\) of (2n + 1)-dimensional invariant submanifold M2n+1 of (2m + 1) dimensional Sasakian space form \({\tilde{M}^{2m+1}}\) of constant \({ \varphi}\)-sectional curvature c. We have shown that if f satisfies the curvature condition \({\overset{\_}{R}(X, Y) \cdot \sigma =Q(g, \sigma)}\) then either M2n+1 is totally geodesic, or \({||\sigma||^{2}=\frac{1}{3}(2c+n(c+1)),}\) or \({||\sigma||^{2}(x) > \frac{1}{3}(2c+n(c+1)}\) at some point x of M2n+1. We also prove that \({\overset{\_ }{R}(X, Y)\cdot \sigma = \frac{1}{2n}Q(S, \sigma)}\) then either M2n+1 is totally geodesic, or \({||\sigma||^{2}=-\frac{2}{3}(\frac{1}{2n}\tau -\frac{1}{2}(n+2)(c+3)+3)}\), or \({||\sigma||^{2}(x) > -\frac{2}{3}(\frac{1}{2n} \tau (x)-\frac{1}{2} (n+2)(c+3)+3)}\) at some point x of M2n+1.

On M. T. Calapso’s Characterization of the Metric of an Absolute Plane

Abstract: We show that the equivalence of the rectangle existence axiom to an axiom used by Maria Teresa Calapso to characterize the Euclidean metric can be proved inside rather weak, purely metric axiom systems for absolute planes.

Non-existence of stable currents in submanifolds of the Euclidean spaces

Abstract: We prove the non-existence theorems of stable integral currents for certain classes of hypersurfaces or higher codimensional submanifolds in the Euclidean spaces.

A Characterisation of the Lines External to an Oval Cone in PG(3, q), q Even

Abstract: In this article, the lines not meeting an oval cone in PG(3, q) (q even) are characterised by their intersection properties with points and planes.

The sequence of generalized median triangles and a new shape function

Abstract: Starting with a triangle ABC and a real number s, we let AA s , BB s , CC s be the cevians that divide the sides BC, CA, AB, respectively, in the ratio s : 1 − s, and we let $${\mathcal{H}_s(ABC)}$$ be the triangle whose side lengths are equal to those of AA s , BB s , CC s . We investigate the sequence of (the shapes of) triangles $${\mathcal{H}_s^n(ABC)}$$ , n = 1, 2, ... by introducing a new shape function that suits this sequence. We also use this shape function to prove a theorem of C. F. Parry concerning automedian triangles.

Two non existence results for the self-similar equation in Euclidean 3-space

Abstract: We prove that the only self-similar surfaces of Euclidean 3-space which are foliated by circles are the self-similar surfaces of revolution discovered by S. Angenent and that the only ruled, self-similar surfaces are the cylinders over planar self-similar curves.

On the equivalence of Lagrange’s axiom to the Lotschnittaxiom

Abstract: We prove that, in Hilbert’s plane absolute geometry, an axiom used by Lagrange in a proof of the Euclidean parallel postulate in a paper read on 3 February 1806 at the Institut de France, which states that “If a and b are two parallels from P to g, then the reflection of a in b is parallel to g as well”, is equivalent to F. Bachmann’s Lotschnittaxiom, which states that “The perpendiculars on the sides a right angle always intersect.”

A new method for finding the center of gravity of polygons

Abstract: A direct and easy general method is described for finding the center of gravity of all types of convex and concave polygons such as quadrilaterals, pentagons, hexagons and n-sided polygons.

Propellers in Affine Cayley–Klein Planes

Abstract: In the present paper the generalized Propeller theorem from planar Euclidean geometry is extended to all planar affine Cayley–Klein geometries. Since there are no equilateral triangles in affine Cayley–Klein planes (except for the Euclidean case), there is no direct extension of the Propeller theorem. In order to find the respective non-Euclidean analogues of it, we introduce the notion of Ωk-equilateral triangle. Some properties of such triangles are given, too. Finally, we prove a Propeller theorem related to isocentric triangles in all affine Cayley–Klein planes.

Complete Lifts of Tensor Fields on a Pure Cross-section in the Tensor Bundle

Abstract: The main purpose of the present paper is to study the behaviour on the pure cross-section of the complete lifts of tensor fields in M n to its tensor bundle T0 q (M n ), q > 0. The results obtained are to some extent similar to results previously established for cotangent T01 (M n ) [11] and tensor bundles T1q (M n ) [7]. However there are various important differences and it appears that the problem of lifting tensor fields to the tensor bundle T0 q (M n ), q > 1 the cross-section presents difficulties connected with purity of the cross-section and γ-operator which are not encountered in the case of the cotangent bundle and tensor bundle T1 q (M n ), respectively. These difficulties were solved with applications of the Tachibana operator.

On non-conjugate braids with the same closure link

Abstract: We use a refinement of an argument by Shinjo, and some study of the 3-strand Burau representation, to extend from knots to links her previous construction of infinite sequences of pairwise non-conjugate braids with the same closure of a non-minimal number of (and at least 4) strands.

Minimum Words of Codes from Affine Planes

Abstract: We show that there are non-desarguesian affine planes of order 16 for which the binary codes have vectors of minimum weight that are not the incidence vectors of lines. This is in contrast to the desarguesian case and answers an open question as to the nature of the minimum words of the code of a non-desarguesian affine plane. Further, we show that all the nontranslation planes of order 16 have hull of minimum weight smaller than 32, in fact containing words of weight 24. Most of these words of weight 24 yield words of weight 16 in the binary code of some affine plane of order 16 that are not the incidence vectors of affine lines. The search has also shown that all the non-desarguesian planes of order at most 16 are not tame. These results are also in contrast to what is known in the desarguesian case. The results are mainly by computer, using Magma.

On Polar Spaces of Infinite Rank

Abstract: Many properties of polar spaces of finite rank fail to hold in polar spaces of infinite rank. For instance, in a polar space of infinite rank it can happen that maximal singular subspaces have different dimensions; every polar space of infinite rank contains singular subspaces that cannot be obtained as intersections of any family of maximal singular subspaces, whereas in a polar space of finite rank every singular subspace is the intersection of a finite number of maximal singular subspaces. In this paper we shall examine peculiar properties of polar spaces of infinite rank, to test how far them can drug us from the familiar world of polar spaces of finite rank.

Total Angle around a Point in Minkowski Plane

Abstract: A new angular measure in Minkowski plane was introduced recently. It is additive, invariant under invertible linear transformations, and has a clear interpretation as an "amount of rotation" from one direction to another one. The total angle τ around a point depends on the unit ball. It is known that \(4.443 \approx \sqrt{2}\pi \leq \tau \leq 8\). We show that \(4.985 \approx 4 \sqrt{2}\, {\rm ln}\,{\rm tan} \frac{3\pi}{8} \leq \tau \leq 8\, {\rm ln}\,{\rm tan} \frac{3\pi}{8}\, \approx 7.051\).

On Existence of Semifields with Free Automorphism Groups

Abstract: The aim of this paper is to prove the existence of a semifield A of order q4 where q is a power of 2, having the properties, A contains GF(q2) in its left-nucleus and admits a free four-group E ≅ Z2 × Z2 of automorphisms.