Showing papers in "Journal of Geometry in 2016"

A generalization of translation surfaces with constant curvature in the isotropic space

Abstract: In this paper we study the translation surfaces generated by a space curve and a planar curve in the isotropic 3-space \({\mathbb{I}^{3}}\). We completely classify such surfaces in \({\mathbb{I}^{3}}\) with constant curvature. Several examples are also given by figures.

The characterization of Hermitian surfaces by the number of points

Abstract: The nonsingular Hermitian surface of degree \({\sqrt{q} +1}\) is characterized by its number of \({\mathbb{F}_q}\) -points among the surfaces over \({\mathbb{F}_q}\) of degree \({\sqrt{q} +1}\) in the projective 3-space without \({\mathbb{F}_q}\) -plane components.

Biharmonic hypersurfaces in Euclidean space E 5

Abstract: In this paper, we study biharmonic hypersurfaces in E 5. We prove that every biharmonic hypersurface in Euclidean space E 5 must be minimal.

An axiomatic foundation of Cayley-Klein geometries

Abstract: We present a common axiomatic characterization of Cayley-Klein geometries over fields of characteristic \({ eq 2}\). To this end the axiom system of Bachmann (Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd edn. Springer, Heidelberg 1973) for plane absolute geometry, which allows a common axiomatization of Euclidean, hyperbolic and elliptic geometry, is generalized. The notion of plane absolute geometry is broadened in several aspects. The most important one is that the principle of duality holds: the dual of a Cayley-Klein geometry is also a Cayley-Klein geometry. The various Cayley-Klein geometries are singled out by additional axioms like the Euclidean or hyperbolic parallel axiom or their dual statements.

Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search

Abstract: In the projective plane PG(2, q), upper bounds on the smallest size t 2(2, q) of a complete arc are considered. For a wide region of values of q, the results of computer search obtained and collected in the previous works of the authors and in the present paper are investigated. For q ≤ 301813, the search is complete in the sense that all prime powers are considered. This proves new upper bounds on t 2(2, q) valid in this region, in particular $$\begin{array}{ll}t_{2}(2, q) \;\; < 0.998 \sqrt{3q {\rm ln}\,q} \quad\;\;\,\,{\rm for} \;\; \qquad \quad \;\;7 \leq q \leq 160001;\\ t_{2}(2, q) \;\; < 1.05 \sqrt{3q {\rm ln}\, q}\qquad\,\,{\rm for}\;\; \qquad \quad \;\;7 \leq q \leq 301813;\\ t_{2}(2,q)\;\; < \sqrt{q}{\rm ln}^{0.7295}\,q \qquad \,\,\,\,\,{\rm for} \;\; \quad \quad \,\,\,109 \leq q \leq 160001;\\ t_{2}(2,q) \;\; < \sqrt{q}{\rm ln}^{0.7404}\,q \qquad \,\,\,\,\,{\rm for }\;\;\, \quad 160001 < q \leq 301813.\end{array}$$ The new upper bounds are obtained by finding new small complete arcs in PG(2,q) with the help of a computer search using randomized greedy algorithms and algorithms with fixed (lexicographical) order of points (FOP). Also, a number of sporadic q’s with q ≤ 430007 is considered. Our investigations and results allow to conjecture that the 2-nd and 3-rd bounds above hold for all q ≥ 109. Finally, random complete arcs in PG(2, q), q ≤ 46337, q prime, are considered. The random complete arcs and complete arcs obtained by the algorithm FOP have the same region of sizes; this says on the common nature of these arcs.

A constructive real projective plane

Abstract: The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. The topics include Desargues’s Theorem, harmonic conjugates, projectivities, involutions, conics, Pascal’s Theorem, poles and polars. The axioms used for the synthetic treatment are constructive versions of the traditional axioms. The analytic construction is used to verify the consistency of the axiom system; it is based on the usual model in three-dimensional Euclidean space, using only constructive properties of the real numbers. The methods of strict constructivism, following principles put forward by Errett Bishop, reveal the hidden constructive content of a portion of classical geometry. A number of open problems remain for future studies.

On Ricci flat warped products with a quarter-symmetric connection

Abstract: In this paper, we have computed the warping functions for a Ricci flat Einstein multiply warped product spaces M with a quarter-symmetric connection for different dimensions of M [i.e; (1). dimM = 2, (2). dimM = 3, (3). \({dim M \geq 4}\)] and all the fibers are Ricci flat.

On the existence of self-complementary and non-self-complementary strongly regular graphs with Paley parameters

Abstract: For p an odd prime, let $${{\mathcal A}_{p}}$$ be the complete classical affine association scheme whose associate classes correspond to parallel classes of lines in the classical affine plane AG(2, p). It is known that $${{\mathcal A}_{p}}$$ is an amorphic association scheme. We investigate rank 3 fusion schemes of $${{\mathcal A}_{p}}$$ whose basis graphs have the same parameters as the Paley graphs $${P(p^{2})}$$ . In contrast to the Paley graphs, the great majority of graphs we detect are non-self-complementary and non-Schurian. In particular, existence of non-self-complementary graphs with Paley parameters is established for $${p \ge 17}$$ , with an analogous existence result for non-Schurian such graphs when $${p \ge 11}$$ . We demonstrate that the number of self-complementary and non-self-complementary strongly regular graphs with Paley parameters grows rapidly as $${p \to \infty}$$ .

Rigidity results for spin manifolds with foliated boundary

Abstract: In this paper, we consider a compact Riemannian manifold whose boundary is endowed with a Riemannian flow. Under a suitable curvature assumption depending on the O’Neill tensor of the flow, we prove that any solution of the basic Dirac equation is the restriction of a parallel spinor field defined on the whole manifold. As a consequence, we show that the flow is a local product. In particular, in the case where solutions of the basic Dirac equation are given by basic Killing spinors, we characterize the geometry of the manifold and the flow.

On bi-null Cartan curves in semi-Euclidean 6-space with index 3

Abstract: In this paper, we study bi-null curves in semi-Euclidean 6-space with index 3, \({\mathbb{R}_{3}^{6}}\). We construct the Frenet frame and Cartan curvature functions of bi-null curves in \({\mathbb{R}_{3}^{6}}\). Also we discuss some properties of bi-null Cartan curves in terms of the Cartan curvatures.

Clifford parallelisms and external planes to the Klein quadric

Abstract: For any three-dimensional projective space $${\mathbb{P}(V)}$$ , where V is a vector space over a field F of arbitrary characteristic, we establish a one-one correspondence between the Clifford parallelisms of $${\mathbb{P}(V)}$$ and those planes of $${\mathbb{P} (V \wedge V)}$$ that are external to the Klein quadric representing the lines of $${\mathbb{P}(V)}$$ . We also give two characterisations of a Clifford parallelism of $${\mathbb{P}(V)}$$ , both of which avoid the ambient space of the Klein quadric.

Sets of type (q + 1, n) in PG(3, q)

Abstract: In this paper, a description of sets of points of PG(3, q) of type (q + 1, n) with respect to the planes is given.

Inversion with respect to a hypercycle of a hyperbolic plane of positive curvature

Abstract: Inversion with respect to a hypercycle of a hyperbolic plane $${\widehat{H}}$$ of positive curvature is investigated. The plane $${\widehat{H}}$$ is the projective Cayley–Klein model of two-dimensional de Sitter’s space. One of four analogs of a Euclidean circle on the plane $${\widehat{H}}$$ is a hypercycle. The formulae of inversion with respect to the hypercycle in a canonical frame of the first type are derived. The main properties of this inversion are proved.

Some new families of partial difference sets in finite fields

Abstract: In this paper we present some new cyclotomic families of partial difference sets. The argument rests on a general procedure for constructing cyclotomic difference sets or partial difference sets in Galois domains due to Ott (Des Codes Cryptogr, doi:10.1007/s10623-015-0082-6, 2015). Definitions and various properties of partial difference sets can be found for instance in Ma (Des Codes Cryptogr 4:221–261, 1994).

Screen conformal lightlike submanifolds of semi-Riemannian manifolds

Abstract: Since the induced objects on a lightlike submanifold depend on its screen distribution which, in general, is not unique and hence we can not use the classical submanifold theory on a lightlike submanifold in the usual way. Therefore, in present paper, we study screen conformal lightlike submanifolds of a semi-Riemannian manifold, which are essential for the existence of unique screen distribution. We obtain a characterization theorem for the existence of screen conformal lightlike submanifolds of a semi-Riemannian manifold. We prove that if the differential operator Ds is a metric Otsuki connection on transversal lightlike bundle for a screen conformal lightlike submanifold then semi-Riemannian manifold is a semi-Euclidean space. We also obtain some characterization theorems for a screen conformal totally umbilical lightlike submanifold of a semi-Riemannian space form. Further, we obtain a necessary and sufficient condition for a screen conformal lightlike submanifold of constant curvature to be a semi-Euclidean space. Finally, we prove that for an irrotational screen conformal lightlike submanifold of a semi-Riemannian space form, the induced Ricci tensor is symmetric and the null sectional curvature vanishes.

Convexity without convex combinations

Abstract: Separation theorems play a central role in the theory of Functional Inequalities. The importance of Convex Geometry has led to the study of convexity structures induced by Beckenbach families. The aim of the present note is to replace recent investigations into the context of an axiomatic setting, for which Beckenbach structures serve as models. Besides the alternative approach, some new results (whose classical correspondences are well-known in Convex Geometry) are also presented.

Reaching fixed points as limits in subspaces

Abstract: A strictly contracting mapping of a spherically complete ultrametric space has a unique fixed point. In general, the fixed point can be approached by a pseudo-convergent family. It will be discussed, when this family can be cut in such a way that the fixed point is the limit of the reduced family in an appropriate subspace. The more difficult part of the paper is the transfer of these results to valued fields.

Regular parallelisms from generalized line stars in \({P_{3} \mathbb{R}}\): a direct proof

Abstract: A generalized line star with respect to an elliptic quadric contained in the Klein quadric gives rise to a regular parallelism in real projective 3-space. This was shown by Betten and Riesinger (Results Math 47:226–241, 2005) using the Thas–Walker construction. They remark that the resulting description of the parallelism is equivalent to a much simpler one, but again the proof is hard. In practice, they always work with that simpler construction. We show that the Thas–Walker approach is not needed here. In fact, one can derive all relevant properties of the parallelism from the simple construction. We also give coordinate-free, short proofs of the topological properties of regular parallelisms in general and of those obtained from generalized line stars in particular.

On spherical four-point-spaces

Abstract: A four-point metric space X is called planar (resp. spherical) if it is isometric to a subspace of the Euclidean plane (resp. isometric to a subspace of a sphere with geodesic distance). We show, among other things, that a four-point subspace of the plane is spherical if and only if its convex hull is either a line segment or a convex quadrilateral. A way to determine whether a given four-point-space is spherical or not is also presented.

The principle of duality in Euclidean and in absolute geometry

Abstract: In Euclidean geometry and in absolute geometry fragments of the principle of duality hold. Bachmann (Aufbau der Geometrie aus dem Spiegelungsbegriff, 1973, §3.9) posed the problem to find a general theorem which describes the extent of an allowed dualization. It is the aim of this paper to solve this problem. To this end a first-order axiomatization of Euclidean (resp. absolute) geometry is provided which allows the application of Godel’s Completeness Theorem for first-order logic and the solution of Bachmann’s problem.

Desargues configurations: minors and ambient automorphisms

Abstract: We determine all possible minors of the Desargues configuration, their embeddings in projective spaces, and their ambient automorphism groups (i.e., the group of all projective collineations that leave the embedded configuration invariant) in Pappian projective spaces. Examples show that the latter problem becomes hard if the extra condition (Pappian) is dropped.

Wild nearfields of dimension 2 over rational function fields

Abstract: Let F be a field. We construct many subgroups of \({{\rm GL}_2(F(t))}\) that act regularly on \({F(t)^2{\setminus}\{0\}}\), and we show that the corresponding nearfields are not Dickson nearfields if chara \({F e 2}\).

Construction of hyperbolic Riemann surfaces with large systoles

Abstract: Let S be a compact hyperbolic Riemann surface of genus \({g \geq 2}\). We call a systole a shortest simple closed geodesic in S and denote by \({{\rm sys}(S)}\) its length. Let \({{\rm msys}(g)}\) be the maximal value that \({{\rm sys}(\cdot)}\) can attain among the compact Riemann surfaces of genus g. We call a (globally) maximal surface Smax a compact Riemann surface of genus g whose systole has length \({{\rm msys}(g)}\). In Section 2 we use cutting and pasting techniques to construct compact hyperbolic Riemann surfaces with large systoles from maximal surfaces. This enables us to prove several inequalities relating \({{\rm msys}(\cdot)}\) of different genera. In Section 3 we derive similar intersystolic inequalities for non-compact hyperbolic Riemann surfaces with cusps.

On $${\left(1,3 \right) }$$ 1 , 3 -Cartan null Bertrand curves in semi-Euclidean 4-space with index 2

Abstract: In this paper, we study generalized Cartan null Bertrand curves in semi-Euclidean 4-space $${\mathbb{E}_{2}^{4}}$$ with index 2

On a class of partial planes related to biplanes

Abstract: In this note we consider partial planes in which for each element x (point or line) there exists a unique opposite element or antipode x* which cannot be joined to x or has no intersection with x. We also require the existence of a triangle. Such partial planes will be called antipodal planes. We are mainly interested in the subclass of regular antipodal planes satisfying: p I L implies p* I L* for all points p and lines L. We shall provide a free construction of infinite regular antipodal planes. The objects thus constructed are not free objects in the usual sense since between antipodal planes there do not exist proper homomorphisms. On the other hand, regular antipodal planes do have a canonical homomorphic image which is a biplane (cf. Payne, J Comb Theory A 12:268–282, 1972). Regular antipodal planes can be coordinatized by certain algebraic systems in a similar way as projective planes are coordinatized by ternary rings. Again by a free construction, we shall provide examples satisfying a configuration theorem comparable to the Fano condition with fixed line at infinity.

Curvature for curves in semi-Euclidean spaces

Abstract: We show a formula for curvatures of curves in a semi-Euclidean space (or pseudo-sphere) with respect to Frenet–Serre type frame in terms of volumes. We also investigate versality of height unfolding and distance squared unfolding for a curve.

Continuous flattening of truncated tetrahedra

Abstract: Each Platonic polyhedron P can be folded using a continuous folding process into a face of P so that the resulting shape is flat and multilayered, while two of the faces are rigid during the motion. In previous works, explicit formulas of continuous functions for such motions were given and the same result as above was shown to hold for any tetrahedron. In this paper, we show that a truncated regular tetrahedron can be folded continuously via explicit continuous folding mappings into a flat (folded) state, such that two of the hexagonal faces are rigid. Furthermore, given any general tetrahedron P and any truncated tetrahedron Q of P, we show that if Q contains the largest inscribed sphere of P and satisfies some condition, then Q can be folded continuously into a flat folded state such that two of the hexagonal faces of Q are rigid during the motion.

Helicoidal surfaces satisfying $${\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)}$$

Abstract: In this paper, we study helicoidal surfaces without parabolic points in Euclidean 3-space \({\mathbb{R} ^{3}}\), satisfying the condition \({\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)}\), where \({\Delta ^{II}}\) is the Laplace operator with respect to the second fundamental form, f is a smooth function on the surface and C is a constant vector. Our main results state that helicoidal surfaces without parabolic points in \({ \mathbb{R} ^{3}}\) which satisfy the condition \({\Delta ^{II} \mathbf{G}=f(\mathbf{G}+C)}\), coincide with helicoidal surfaces with non-zero constant Gaussian curvature.

On collineation groups of finite projective spaces containing a Singer cycle

Abstract: By a result of Kantor, any subgroup of GL(n, q) containing a Singer cycle normalizes a field extension subgroup. This result has as a consequence a projective analogue, and this paper gives the details of this deduction, showing that any subgroup of PΓL(n, q) containing a projective Singer cycle normalizes the image of a field extension subgroup GL(n/s, qs) under the canonical homomorphism GL(n, q) → PGL(n, q), for some divisor s of n, and so is contained in the image of ΓL(n/s, qs) under the canonical homomorphism ΓL(n, q) → PΓL(n, q). The actions of field extension subgroups on V (n, q) are also investigated. In particular, we prove that any field extension subgroup GL(n/s, qs) of GL(n, q) has a unique orbit on s-dimensional subspaces of V (n, q) of length coprime to q. This orbit is a Desarguesian s-partition of V (n, q).

Lexicographic generation of projective spaces

Abstract: Lexicographic or first choice constructions of geometric objects sometimes lead to amazingly good results. Usually it is difficult to determine the precise identity of the resulting geometries. Here we find infinitely many cases where the identification actually can be accomplished.