# Showing papers in "Journal of Geometry in 2019"

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TL;DR: In this paper, the Lunardon-Polverino binomial is shown to not be a binomial binomial if and only if x and t are linear dependent on the Dickson matrix.

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.

19 citations

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TL;DR: In this paper, almost paracontact Riemannian manifolds of the lowest dimension 3 were constructed on a family of Lie groups and the obtained manifolds were studied. And the Curvature properties of these manifolds are investigated.

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.

13 citations

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TL;DR: In this paper, the partial plane spread in the Gaussian space is classified as a combinatorial object, and the classification of the following closely related objects are obtained: vector space partitions of the Gaussian space partitions with the optimal parameters of the maximum possible size 17 and of size 16.

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 $$(7,34,5)_2$$
.

11 citations

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TL;DR: In this paper, a simple mathematical method for explaining the model flexibility of the Siamese dipyramids is presented. But their physical models behave like physical models of flexible polyhedra.

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.

7 citations

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TL;DR: Pavkovic et al. as discussed by the authors showed that the curvature tensor associated with the isotropic Levi-Civita connection vanishes identically for any pseudo-isotropic surface.

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.

7 citations

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6 citations

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TL;DR: In this paper, the authors show the existence of a closed, embedded hypersurface with the same symmetry as the self-shrinking torus, which is known as the shot-based torus.

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.

6 citations

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TL;DR: The only surface in 3D Euclidean space 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 H as discussed by the authors.

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 $$\frac{1}{2|H|}$$
.

5 citations

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TL;DR: This work characterises the “Clifford-like” parallelisms, i.e. the blends of the Clifford parallelisms and establishes necessary and sufficient conditions for the existence of Clifford-like parallelisms that are not Clifford.

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.

4 citations

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TL;DR: In this article, it was shown that there are no algebraic hypersurfaces of degree 3 in Euclidean spaces with nonzero constant mean curvature, except for the standard spheres and the round cylinders.

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.

4 citations

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TL;DR: In this paper, the authors studied the geometry of almost contact pseudo-metric manifold in terms of tensor fields, emphasizing analogies and differences with respect to the contact metric case.

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.

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TL;DR: In this paper, the authors prove a non-existence result for Ricci solitons on non-cosymplectic manifolds, and prove the same result for almost cosympelous manifolds.

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

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TL;DR: In this paper, two characterizations of the family of secant lines to an ovoid of polygonal O(3, q) are given, where q is the length of the secant line.

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

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TL;DR: In this paper, the Fischer-Marsden conjecture on almost Kenmotsu manifolds was studied and the authors characterized non-kappa, \mu, \mu )^\prime -almost-kempe manifolds satisfying the Fischer−Marsden equation.

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.

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TL;DR: In this paper, a one-parameter family of singular space curves is considered and an envelope is defined for it, and relations among envelopes of oneparameter families of framed space curves are investigated.

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.

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TL;DR: In this article, it was shown that Ceva's and Menelaus' theorem is 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.

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.

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TL;DR: In this article, the vertices of the complete graph K_n are represented as points of a hyperbola in the plane and the points of the hyperbolic graph are represented by points of points in a polygonal graph.

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.

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TL;DR: Struve and R. Struve as mentioned in this paper showed that the Thomsen-Bachmann correspondence between metric geometries and groups can be precisely stated in a framework of first-order logic.

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.

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TL;DR: In this article, the outer points of a non-degenerate conic C in PG(2, q), q odd, are classified into two kinds of outer points.

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.

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TL;DR: In this article, the authors studied oriented connected closed polyhedral surfaces with non-degenerate triangular faces in three-dimensional Euclidean space and proved that for every flexible polyhedron some integer combination of its dihedral angles remains constant during the flex.

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.

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TL;DR: In this paper, the authors investigate the problem of characterizing a surface family from a given common geodesic curve in a 3-dimensional Lie group G. They obtain a parametric representation of the surface as a linear combination of the Frenet frame in G.

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.

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TL;DR: Using the Poincare disk model of hyperbolic geometry, it was shown in this paper that radial expansion of a spherical convex set about a point inside a sphere, such that the initial set is contained in the closed hemisphere centred at that point, always 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.

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TL;DR: In this paper, the algebraic parabolic affine rotation surfaces of elliptic, hyperbolic and parabolic surfaces of revolution are characterized in terms of the structure of their implicit equation.

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.

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TL;DR: In this article, the authors 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.

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.

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TL;DR: In this paper, a hyperplane from a polar space associated with a symplectic polarity is deletion, resulting in a specific, symplectic, affine polar space.

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.

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TL;DR: In this article, two types of developable surfaces along a frontal curve on an embedded surface in the Euclidean 3-space are considered. But the frontal curve may have singular points.

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.

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TL;DR: In this paper, it was shown that the absolute planes in which the elementary Archimedean axiom holds satisfy Bachmann's Lotschnittaxiom are Euclidean.

Abstract: Absolute planes in which the elementary Archimedean axiom holds satisfy Aristotle’s axiom. Absolute planes satisfying both the elementary Archimedean axiom and Bachmann’s Lotschnittaxiom are Euclidean. The Corollary to Aristotle’s axiom is equivalent to Aristotle’s axiom.

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TL;DR: In this paper, it was shown that the most homogeneous parallelisms of oriented lines other than the Clifford parallelism do not necessarily arise in this way, and that there are far more oriented parallelism of this kind than ordinary parallelisms.

Abstract: We introduce topological parallelisms of oriented lines (briefly called oriented parallelisms). Every topological parallelism (of lines) on $${\mathrm{PG}(3,{\mathbb {R}})}$$
gives rise to a parallelism of oriented lines, but we show that even the most homogeneous parallelisms of oriented lines other than the Clifford parallelism do not necessarily arise in this way. In fact we determine all parallelisms of both types that admit a reducible $${\mathrm{SO}_3 {\mathbb {R}}}$$
-action (only the Clifford parallelism admits a larger group (Lowen in Innov Incid Geom. arXiv:1702.03328
), and it turns out surprisingly that there are far more oriented parallelisms of this kind than ordinary parallelisms. More specifically, Betten and Riesinger (Aequ Math 81:227–250, 2011) construct ordinary parallelisms by applying $${\mathrm{SO}_3 {\mathbb {R}}}$$
to rotational Betten spreads. We show that these are the only ordinary parallelisms compatible with this group action, but also the ‘acentric’ rotational spreads considered by them yield oriented parallelisms. The automorphism group of the resulting (oriented or non-oriented) parallelisms is always $${\mathrm{SO}_3 {\mathbb {R}}}$$
, no matter how large the automorphism group of the non-regular spread is. The isomorphism type of the parallelism depends not only on the isomorphism type of the spread used, but also on the rotation group applied to it. We also study the rotational Betten spreads used in this construction and their automorphisms.

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TL;DR: In this paper, the authors consider a supporting hyperplane of a regular simplex and its reflected image at this hyperplane and show that the volume of the convex hull of these two simplices is maximal when the hyperplane goes through on a vertex and is orthogonal to the height of the simplex at this vertex.

Abstract: The new result of this paper is connected with the following problem: consider a supporting hyperplane of a regular simplex and its reflected image at this hyperplane. When will the volume of the convex hull of these two simplices be maximal? We prove that in the case when the dimension is less or equal to 4, the maximal volume attained in the case when the hyperplane goes through on a vertex and is orthogonal to the height of the simplex at this vertex. More interesting that in the higher dimensional cases this position is not optimal. We also determine an optimal position of the hyperplane in the 5-dimensional case. This corrects an erroneous statement in my paper (Horvath in Beitr Geom Algebra 55(2):415–428, 2014).

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TL;DR: In this paper, a characterization of pairs of non-congruent quadrilaterals for which all elements are pairwise congruent (properly conformal) is provided.

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.