Showing papers in "Journal of Geometry in 2019"

A condition for scattered linearized polynomials involving Dickson matrices

Abstract: A linearized polynomial over $${{\mathbb {F}}}_{q^n}$$ is called scattered when for any $$t,x\in {{\mathbb {F}}}_{q^n}$$, the condition $$xf(t)-tf(x)=0$$ holds if and only if x and t are $${\mathbb {F}}_q$$-linearly dependent. General conditions for linearized polynomials over $${{\mathbb {F}}}_{q^n}$$ to be scattered can be deduced from the recent results in Csajbok (Scalar q-subresultants and Dickson matrices, 2018), Csajbok et al. (Finite Fields Appl 56:109–130, 2019), McGuire and Sheekey (Finite Fields Appl 57:68–91, 2019), Polverino and Zullo (On the number of roots of some linearized polynomials, 2019). Some of them are based on the Dickson matrix associated with a linearized polynomial. Here a new condition involving Dickson matrices is stated. This condition is then applied to the Lunardon–Polverino binomial $$x^{q^s}+\delta x^{q^{n-s}}$$, allowing to prove that for any n and s, if $${{\,\mathrm{N}\,}}_{q^n/q}(\delta )=1$$, then the binomial is not scattered. Also, a necessary and sufficient condition for $$x^{q^s}+bx^{q^{2s}}$$ to be scattered is shown which is stated in terms of a special plane algebraic curve.

Lie groups as 3-dimensional almost paracontact almost paracomplex Riemannian manifolds

Abstract: Almost paracontact almost paracomplex Riemannian manifolds of the lowest dimension 3 are considered. Such structures are constructed on a family of Lie groups and the obtained manifolds are studied. Curvature properties of these manifolds are investigated. An example is commented as support of the obtained results.

Classification of large partial plane spreads in $PG(6,2)$ and related combinatorial objects

Abstract: The partial plane spreads in $${{\,\mathrm{PG}\,}}(6,2)$$ of maximum possible size 17 and of size 16 are classified. Based on this result, we obtain the classification of the following closely related combinatorial objects: vector space partitions of $${{\,\mathrm{PG}\,}}(6,2)$$ of type $$(3^{16} 4^1)$$ , binary $$3\times 4$$ MRD codes of minimum rank distance 3, and subspace codes with the optimal parameters $$(7,17,6)_2$$ and $$(7,34,5)_2$$ .

On constructions of surfaces using a geodesic in Lie group

Abstract: In this study, we investigate the problem how to characterize a surface family from a given common geodesic curve in a 3-dimensional Lie group G. We obtain a parametric representation of the surface as a linear combination of the Frenet frame in G. Also, we give the relation about developability along the common geodesic of the members of surface family. Finally, we illustrate some examples to verify this method.

$$\eta $$ η -Ricci solitons in $$\epsilon $$ ϵ -Kenmotsu manifolds

Abstract: The object of the present paper is to characterize $$\epsilon $$ -Kenmotsu manifolds admitting $$\eta $$ -Ricci solitons Finally, the existence of $$\eta $$ -Ricci soliton in an $$\epsilon $$ -Kenmotsu manifold has been proved by a concrete example

On the existence of a closed, embedded, rotational $$\lambda $$ λ -hypersurface

Abstract: In this paper we show the existence of a closed, embedded $$\lambda $$ -hypersurfaces $$\Sigma \subset \mathbb {R}^{2n}$$ . The hypersurface $$\Sigma $$ is diffeomorphic to $$\mathbb {S}^{n-1} \times \mathbb {S}^{n-1} \times \mathbb {S}^1$$ and exhibits $$SO(n) \times SO(n)$$ symmetry. Our approach uses a “shooting method” similar to the approach used by McGrath in constructing a generalized self-shrinking “torus” solution to mean curvature flow. The result generalizes the $$\lambda $$ torus found by Cheng and Wei.

Envelopes of one-parameter families of framed curves in the Euclidean space

Abstract: As a one-parameter family of singular space curves, we consider a one-parameter family of framed curves in the Euclidean space. Then we define an envelope for a one-parameter family of framed curves and investigate properties of envelopes. Especially, we concentrate on one-parameter families of framed curves in the Euclidean 3-space. As applications, we give relations among envelopes of one-parameter families of framed space curves, one-parameter families of Legendre curves and one-parameter families of spherical Legendre curves, respectively.

The geometry of Gauss map and shape operator in simply isotropic and pseudo-isotropic spaces

Abstract: In this work, we are interested in the differential geometry of surfaces in simply isotropic $${\mathbb {I}}^3$$ and pseudo-isotropic $${\mathbb {I}}_{\mathrm {p}}^3$$ spaces, which consists of the study of $${\mathbb {R}}^3$$ equipped with a degenerate metric such as $$\mathrm {d}s^2=\mathrm {d}x^2\pm \mathrm {d}y^2$$ . The investigation is based on previous results in the simply isotropic space (Pavkovic in Glas Mat Ser III 15:149–152, 1980; Rad JAZU 450:129–137, 1990), which point to the possibility of introducing an isotropic Gauss map taking values on a unit sphere of parabolic type and of defining a shape operator from it, whose determinant and trace give the known relative Gaussian and mean curvatures, respectively. Based on the isotropic Gauss map, a new notion of connection is also introduced, the relative connection (r-connection, for short). We show that the new curvature tensor in both $${\mathbb {I}}^3$$ and $${\mathbb {I}}_{\mathrm {p}}^3$$ does not vanish identically and is directly related to the relative Gaussian curvature. We also compute the Gauss and Codazzi–Mainardi equations for the r-connection and show that r-geodesics on spheres of parabolic type are obtained via intersections with planes passing through their center (focus). Finally, we show that admissible pseudo-isotropic surfaces are timelike and that their shape operator may fail to be diagonalizable, in analogy to Lorentzian geometry. We also prove that the only totally umbilical surfaces in $${\mathbb {I}}_{\mathrm {p}}^3$$ are planes and spheres of parabolic type and that, in contrast to the r-connection, the curvature tensor associated with the isotropic Levi-Civita connection vanishes identically for any pseudo-isotropic surface, as also happens in simply isotropic space.

On the model flexibility of Siamese dipyramids

Abstract: Polyhedra called Siamese dipyramids are known to be non-flexible, however their physical models behave like physical models of flexible polyhedra. We discuss a simple mathematical method for explaining the model flexibility of the Siamese dipyramids.

Non-existence of $$*$$ ∗ -Ricci solitons on $$(\kappa ,\mu )$$ ( κ , μ ) -almost cosymplectic manifolds

Abstract: In this short note, we prove a non-existence result for $$*$$ -Ricci solitons on non-cosymplectic $$(\kappa ,\mu )$$ -almost cosymplectic manifolds.

Translation surfaces with non-zero constant mean curvature in Euclidean space

Abstract: We prove that the only surface in 3-dimensional Euclidean space $${\mathbb {R}}^3$$ with constant and non-zero mean curvature H, constructed by the sum of a planar curve and a space curve, is the circular cylinder of radius $$\frac{1}{2|H|}$$ .

The Fischer–Marsden conjecture on non-Kenmotsu $$(\kappa , \mu )^\prime $$ ( κ , μ ) ′ -almost Kenmotsu manifolds

Abstract: The purpose of this paper is to study the Fischer–Marsden conjecture on a class of almost Kenmotsu manifolds. We characterize non-Kenmotsu $$(\kappa , \mu )^\prime $$ -almost Kenmotsu manifolds satisfying the Fischer–Marsden equation.

Certain results on almost contact pseudo-metric manifolds

Abstract: We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields $$h:=\frac{1}{2}\pounds _\xi \varphi $$ and $$\ell := R(\cdot ,\xi )\xi $$ , emphasizing analogies and differences with respect to the contact metric case. Certain identities involving $$\xi $$ -sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost CR structure $$(\mathcal {H}(M), J, \theta )$$ corresponding to almost contact pseudo-metric manifold M to be CR manifold. Finally, we prove that a contact pseudo-metric manifold $$(M, \varphi ,\xi ,\eta ,g)$$ is Sasakian pseudo-metric if and only if the corresponding nondegenerate almost CR structure $$(\mathcal {H}(M), J)$$ is integrable and J is parallel along $$\xi $$ with respect to the Bott partial connection.

Developable surfaces along frontal curves on embedded surfaces

Abstract: We consider two types of developable surfaces along a frontal curve on an embedded surface in the Euclidean 3-space. One is called the osculating developable surface, and the other is called the normal developable surface. The frontal curve may have singular points. We give new invariants of the frontal curve which characterize singularities of the developable surfaces. Moreover, a frontal curve is a contour generator with respect to an orthogonal projection or a central projection if and only if one of these invariants is constantly equal to zero.

Algebraic CMC hypersurfaces of order 3 in Euclidean spaces

Abstract: Understanding and finding of general algebraic constant mean curvature surfaces in the Euclidean spaces is a hard open problem. The basic examples are the standard spheres and the round cylinders, all defined by a polynomial of degree 2. In this paper, we prove that there are no algebraic hypersurfaces of degree 3 in $$\mathbb {R}^n$$ , $$n\ge 3$$ , with nonzero constant mean curvature.

Clifford-like parallelisms

Abstract: Given two parallelisms of a projective space we describe a construction, called blending, that yields a (possibly new) parallelism of this space. For a projective double space $$({\mathbb P},{\mathrel {\parallel _{\ell }}},{\mathrel {\parallel _{r}}})$$ over a quaternion skew field we characterise the “Clifford-like” parallelisms, i.e. the blends of the Clifford parallelisms $$\mathrel {\parallel _{\ell }}$$ and $$\mathrel {\parallel _{r}}$$ , in a geometric and an algebraic way. Finally, we establish necessary and sufficient conditions for the existence of Clifford-like parallelisms that are not Clifford.

$$\mathcal {D}$$ D -Homothetic deformation on almost contact B-metric manifolds

Abstract: In this paper, the notion of $$\mathcal {D}$$ -homothetic deformation of an almost contact B-metric structure is introduced. We give an example of $$\mathcal {D}$$ -homothetic deformation of a 3-dimensional Sasaki-like accR manifold. Finally, the notion of $$\mathcal {D}$$ -homothetic warping is defined.

Ceva’s and Menelaus’ theorems in projective-metric spaces

Abstract: We prove that Ceva’s and Menelaus’ theorems are valid in a projective-metric space if and only if the space is any of the elliptic geometry, the hyperbolic geometry, or the Minkowski geometries.

On the isomorphism types of Moebius–Kantor complexes

Abstract: We study the isomorphism types of simply connected complexes with Moebius–Kantor links using a local invariant called the parity. We show that the parity can be computed explicitly in certain constructions arising from surgery.

Two characterizations of the family of secant lines of an ovoid of PG(3, $${\varvec{q}}$$ q )

Abstract: Two new characterizations of the family of secant lines to an ovoid of $$\mathrm {PG}(3, q)$$ are given.

One-factorisations of complete graphs arising from hyperbolae in the Desarguesian affine plane

Abstract: In a recent paper Korchmaros et al. (J Combin Theory Ser A 160:62–83, 2018) the geometry of finite planes is exploited for the construction of one-factorisations of the complete graph $$K_n$$ from configurations of points in $$\mathrm {PG}(2,q)$$ . Here we provide an alternative procedure where the vertices of $$K_n$$ correspond to the points of a hyperbola in $$\mathrm {AG}(2,q)$$ . In this way, we obtain one-factorisations for parameters which are either not covered by the constructions in Korchmaros et al. (J Combin Theory Ser A 160:62–83, 2018), or isomorphic to known examples but arising from different geometric configurations.

The Thomsen–Bachmann correspondence in metric geometry II

Abstract: We continue the investigations of the Thomsen–Bachmann correspondence between metric geometries and groups, which is often summarized by the phrase ‘Geometry can be formulated in the group of motions’. In the first part (H. Struve and R. Struve in J Geom, 2019. https://doi.org/10.1007/s00022-018-0465-8 ) of this paper it was shown that the Thomsen–Bachmann correspondence can be precisely stated in a framework of first-order logic. We now prove that the correspondence, which was established by Thomsen and Bachmann for Euclidean and for plane absolute geometry, holds also for Hjelmslev geometries, Cayley–Klein geometries, isotropic and equiform geometries, and that these geometries and the theory of their group of motions are mutually faithfully interpretable (and bi-interpretable, but not definitionally equivalent). Hence a reflection-geometric axiomatization of a class of motion groups corresponds to an elementary axiomatization of the underlying geometry and provides with the calculus of reflections a powerful proof method.

A characterization of the set of internal points of a conic in PG(2,q), q odd

Abstract: A point P not on a non-degenerate conic C in PG(2, q), q odd, is called internal to C if no tangent line to C contains P, external otherwise. The set of internal points of C is a $$\frac{q(q-1)}{2}$$ -set of type $$(0,\frac{q-1}{2},\frac{q+1}{2})$$ . In this paper, we classify all $$\frac{q(q-1)}{2}$$ -sets of class [0, m, n] having exactly two kinds of outer points.

A sufficient condition for a polyhedron to be rigid

Abstract: We study oriented connected closed polyhedral surfaces with non-degenerate triangular faces in three-dimensional Euclidean space, calling them polyhedra for short. A polyhedron is called flexible if its spatial shape can be changed continuously by changing its dihedral angles only. We prove that for every flexible polyhedron some integer combination of its dihedral angles remains constant during the flex. The proof is based on a recent result of A. A. Gaifullin and L. S. Ignashchenko.

An algebraic characterization of properly congruent-like quadrilaterals

Abstract: In this paper, we will provide a characterization of pairs of non-congruent quadrilaterals for which all elements are pairwise congruent (‘properly congruent-like quadrilaterals’). As a consequence of this main result, we demonstrate a method to establish, given a generic quadrilateral, whether some quadrilaterals that are properly congruent-like to it exist and, if so, how to determine the values of their elements. In particular, this approach allows us to provide examples of quadrilaterals that are not congruent-like to any other quadrilateral and to show constructive examples of pairs of properly congruent-like quadrilaterals.

Radial expansion preserves hyperbolic convexity and radial contraction preserves spherical convexity

Abstract: On a flat plane, convexity of a set is preserved by both radial expansion and contraction of the set about any point inside it. Using the Poincare disk model of hyperbolic geometry, we prove that radial expansion of a hyperbolic convex set about a point inside it always preserves hyperbolic convexity. Using stereographic projection of a sphere, we prove that radial contraction of a spherical convex set about a point inside it, such that the initial set is contained in the closed hemisphere centred at that point, always preserves spherical convexity.

Algebraic affine rotation surfaces of parabolic type

Abstract: Affine rotation surfaces, which appear in the context of affine differential geometry, are generalizations of surfaces of revolution. These affine rotation surfaces can be classified into three different families: elliptic, hyperbolic and parabolic. In this paper we investigate some properties of algebraic parabolic affine rotation surfaces, i.e. parabolic affine rotation surfaces that are algebraic, generalizing some previous results on algebraic affine rotation surfaces of elliptic type (classical surfaces of revolution) and hyperbolic type (hyperbolic surfaces of revolution). In particular, we characterize these surfaces in terms of the structure of their implicit equation, we describe the structure of the form of highest degree defining an algebraic parabolic affine rotation surface, and we prove that these surfaces can have either one, or two, or infinitely many axes of affine rotation. Additionally, we characterize the surfaces with more than one parabolic axis.

Duality of isosceles tetrahedra

Abstract: In this paper we define a so-called dual simplex of an n-simplex and prove that the dual of each simplex contains its circumcenter, which means that it is well-centered. For triangles and tetrahedra S we investigate when the dual of S, or the dual of the dual of S, is similar to S, respectively. This investigation encompasses the study of the iterative application of taking the dual. For triangles, this iteration converges to an equilateral triangle for any starting triangle. For tetrahedra we study the limit points of period two, which are known as isosceles or equifacetal tetrahedra.

Affine semipolar spaces

Abstract: Deleting a hyperplane from a polar space associated with a symplectic polarity we get a specific, symplectic, affine polar space. Similar geometry, called an affine semipolar space arises as a result of generalization of the notion of an alternating form to a semiform. Some properties of these two geometries are given and their automorphism groups are characterized.