Showing papers in "Journal of Geometry in 2020"

A note on quasi-Yamabe solitons on contact metric manifolds

Abstract: We prove that a contact metric manifold does not admit a proper quasi-Yamabe soliton ($$M,\, g,\,\xi ,\,\lambda ,\,\mu $$). Next we prove that if a contact metric manifold admits a quasi-Yamabe soliton ($$M,\, g,\, V,\, \lambda ,\, \mu $$) whose soliton field is pointwise collinear with the Reeb vector field, then the scalar curvature is constant, and the quasi-Yamabe soliton reduces to Yamabe soliton. Finally, it is shown that if a contact metric manifold admits a quasi-Yamabe soliton whose soliton field is the V-Ric vector field, then the Ricci operator Q commutes with the (1, 1) tensor $$\phi $$. As a consequence of the main result we obtain several corollaries.

Pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces

Abstract: The aim of this work is to study homogeneous pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces. First, we deduce a formula for Ricci tensor of a homogeneous pseudo-Riemannian manifold with compact isotropy subgroup. Based on this formula, we establish a one-to-one correspondence between homogeneous pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces and homogeneous Riemannian Einstein metrics on compact homogeneous spaces. As an application, we prove that every noncompact connected simple Lie group except SL(2) admits at least two nonproportional left invariant pseudo-Riemannian Einstein metrics. Furthermore, we study left invariant pseudo-Riemannian Einstein metrics on solvable Lie groups. By showing that if a nilpotent Lie group admits a left invariant Riemannian Ricci soliton, then it admits a left invariant pseudo-Riemannian Ricci soliton as well, we construct a left invariant pseudo-Riemannian Einstein metric on any given Riemannian standard Einstein solvmanifold.

Complete spherical convex bodies

Abstract: Similarly to the classic notion in Euclidean space, we call a set on the sphere $$S^d$$ complete, provided adding any extra point increases its diameter. Complete sets are convex bodies on $$S^d$$ . Our main theorem says that on $$S^d$$ complete bodies of diameter $$\delta $$ coincide with bodies of constant width $$\delta $$ .

Automated exploration of inner isoptics of an ellipse

Abstract: For a given curve $${\mathcal {C}}$$ and a given angle $$\theta $$ , the $$\theta $$ -isoptic curve of $${\mathcal {C}}$$ is the geometric locus of points through which passes a pair of tangents to $${\mathcal {C}}$$ making an angle equal to $$\theta $$ If the curve $${\mathcal {C}}$$ is smooth and convex, isoptics exist for any angle, and through every point exterior to the curve, there is exactly one pair of tangents The isoptics of conics are well known In this paper, we explore the inner isoptics of ellipses, ie the envelopes of the lines joining the points of contact of the ellipse with the tangents through points on a given isoptic If $$\theta =90^{\circ }$$ , the isoptic is called orthoptic and the corresponding inner isoptic is called the inner orthoptic We show that the inner orthoptic of an ellipse is an ellipse, but in general the inner isoptics are more complicated

The Fermat–Torricelli theorem in convex geometry

Abstract: This paper studies a generalization of the Euclidean triangle, the generalized deltoid, which we believe to be the right one for convex geometry. To illustrate the process, our main result shows that the generalized deltoid satisfies a convex generalization of the Fermat–Torricelli theorem. A point that minimizes the sum of distances to the vertices of a triangle (Fermat–Torricelli point) is the same as one through which pass three equiangular affine diameters (Fermat–Ceder point). A generalized deltoid is a triangle whose sides are disjoint, outwardly-looking arcs of convex curves. The Fermat–Torricelli theorem in convex geometry extends the Fermat–Ceder point of a triangle to a Fermat–Ceder point of a generalized deltoid. As an application, we show that the Fermat–Ceder points for the continuous families of affine diameters, area-bisecting lines, and perimeter-bisecting lines are unique for every triangle, and non-unique for every pentagon. In the case of quadrilaterals, the uniqueness of the Fermat–Ceder point for affine diameters holds precisely for all non-trapezoids, the one for the Fermat–Ceder point for area-bisecting lines holds for all quadrilaterals, and the one for the Fermat–Ceder point for perimeter-bisecting lines is open.

Rotational Weingarten surfaces in hyperbolic 3-space

Abstract: We study rotational Weingarten surfaces in the hyperbolic space $$\mathbb {H}^3(-1)$$ with the principal curvatures $$\kappa $$ and $$\lambda $$ satisfying a certain functional relation $$\kappa = F(\lambda )$$ for a given continuous function F. We determine profile curves of such surfaces parameterized in terms of the principal curvature $$\lambda $$. Then we consider some special cases by taking $$F(\lambda ) = a\lambda + b$$ and $$F(\lambda ) = a\lambda ^m$$ for particular values of the constants a, b, and m.

On the Hadwiger covering problem in low dimensions

Abstract: Let $$H_n$$ be the minimal number of smaller homothetic copies of an n-dimensional convex body required to cover the whole body. Equivalently, $$H_n$$ can be defined via illumination of the boundary of a convex body by external light sources. The best known upper bound in three-dimensional case is $$H_3\le 16$$ and is due to Papadoperakis. We use Papadoperakis’ approach to show that $$H_4\le 96$$ , $$H_5\le 1091$$ and $$H_6\le 15373$$ which significantly improve the previously known upper bounds on $$H_n$$ in these dimensions.

A family of geometric operators on triangles with two complex variables

Abstract: Given a plane triangle $$\Delta $$, one can construct a new triangle $$\Delta '$$ whose vertices are intersections of two cevian triples of $$\Delta $$. We extend the family of operators $$\Delta \mapsto \Delta '$$ by complexifying the defining two cevian parameters and study its rich structure from arithmetic-geometric viewpoints. We also find another useful parametrization of the operator family via finite Fourier analysis and apply it to investigate area-preserving operators on triangles.

Nets of conics of rank one in $$\mathrm {PG}(2,q)$$ PG ( 2 , q ) , q odd

Abstract: We classify nets of conics of rank one in Desarguesian projective planes over finite fields of odd order, namely, two-dimensional linear systems of conics containing a repeated line. Our proof is geometric in the sense that we solve the equivalent problem of classifying the orbits of planes in $$\mathrm {PG}(5,q)$$ which meet the quadric Veronesean in at least one point, under the action of $$\mathrm {PGL}(3,q) \leqslant \mathrm {PGL}(6,q)$$ (for q odd). Our results complete a partial classification of nets of conics of rank one obtained by Wilson (Am J Math 36:187–210, 1914).

Optimization on slant submanifolds of golden Riemannian manifolds using generalized normalized $$\delta $$ δ -Casorati curvatures

Abstract: In the present paper, we derive optimal inequalities involving generalized normalized $$\delta $$ -Casorati curvatures for slant submanifolds in a golden Riemannian space form. We obtain these inequalities by analysing a suitable constrained extrememum problem on submanifold.

Torus cannot collapse to a segment

Abstract: In earlier work, we analyzed the impossibility of codimension-one collapse for surfaces of negative Euler characteristic under the condition of a lower bound for the Gaussian curvature. Here we show that, under similar conditions, the torus cannot collapse to a segment. Unlike the torus, the Klein bottle can collapse to a segment; we show that in such a situation, the loops in a short basis for homology must stay a uniform distance apart.

A new approach to Bertrand curves in Euclidean 3-space

Abstract: In this article, a new approach is given for Bertrand curves in 3-dimensional Euclidean space. According to this approach, the necessary and sufficient conditions including the known results have been obtained for a curve to be Bertrand curve in $${\mathbb {E}}^{3}$$ . In addition, the related examples and graphs are given by showing that general helices and anti-Salkowski curves can be Bertrand curves or their mates, which is their new characterization.

Inequalities for generalized normalized $$\delta $$ δ -Casorati curvatures of slant submanifolds in metallic Riemannian space forms

Abstract: In the present paper, we establish sharp inequalities involving generalized normalized $$\delta $$ -Casorati curvatures for invariant, anti-invariant and slant submanifolds in metallic Riemannian space forms and characterize the submanifolds for which the equality holds.

Recognition of plane paths and plane curves under linear pseudo-similarity transformations

Abstract: $$E^{2}_{1}$$ be the 2-dimensional pseudo-Euclidean space of index 1, $$G=Sim_{L}(E^{2}_{1})$$ be the group of all linear pseudo-similarities of $$E^{2}_{1}$$ and $$G=Sim_{L}^{+}(E^{2}_{1})$$ be the group of all orientation-preserving linear pseudo-similarities of $$E^{2}_{1}$$ . In this paper, groups $$Sim_{GL}^{+}(E^{2}_{1})$$ and $$Sim_{GL}(E^{2}_{1})$$ are defined. For the groups $$G=Sim_{GL}^{+}(E^{2}_{1}),Sim_{GL}(E^{2}_{1})$$ , $$Sim_{L}(E^{2}_{1}),$$ $$Sim_{L}^{+}(E^{2}_{1})$$ , G-invariants of paths in $$E^{2}_{1}$$ are investigated. Using hyperbolic numbers, a method to detect G-similarities of paths and curves is presented. We give an evident form of a path and a curve in terms of their G-invariants. For two paths and two curves with the common G-invariants, evident forms of all linear pseudo-similarity transformations, carrying the paths and the curves, are obtained.

Curvature types of planar curves for gauges

Abstract: In this paper results from the differential geometry of curves are extended from normed planes to gauge planes which are obtained by neglecting the symmetry axiom. Based on the gauge analogue of the notion of Birkhoff orthogonality from Banach space theory, we study all curvature types of curves in gauge planes, thus generalizing their complete classification for normed planes. We show that (as in the subcase of normed planes) there are four such types, and we call them analogously Minkowski, normal, circular, and arc-length curvature. We study relations between them and extend, based on this, also the notions of evolutes and involutes to gauge planes.

Polynomial triangle centers on the line at infinity

Abstract: Points on the line at infinity in the extended plane of a triangle ABC are discussed in terms of barycentric coordinates that are polynomials in the sidelengths a, b, c. Various properties of the line at infinity are discussed, including two theorems, with related conjectures, on polynomial representations of triangle centers that are at opposite ends of a diameter of the circumcircle—along with their isogonal conjugates on the line at infinity. Also considered are an equal-areas locus, symbolic substitution, and historical comments.

Linearly dependent subsets of Segre varieties

Abstract: We study the linear algebra of finite subsets S of a Segre variety X. In particular we classify the pairs (S, X) with S linear dependent and $$\#(S)\le 5$$ . We consider an additional condition for linear dependent sets, i.e. that no two of their points are contained in a line of X, and get far better lower bounds for $$\#(S)$$ in term of the dimension and number of the factors of X. In this discussion and in the classification of the case $$\#(S)= 5$$ , $$X\cong {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$ we use the rational normal curves contained in X.

Critical point equation on a class of almost Kenmotsu manifolds

Abstract: In the present paper, we characterize $$(k,\mu )'$$-almost Kenmotsu manifolds admitting $$*$$-critical point equation. It is shown that if $$(g, \lambda )$$ is a non-constant solution of the $$*$$-critical point equation of a connected non-compact $$(k,\mu )'$$-almost Kenmotsu manifold, then (1) the manifold M is locally isometric to $$\mathbb {H}^{n+1}(-4)$$$$\times $$$$\mathbb {R}^n$$, (2) the manifold M is $$*$$-Ricci flat and (3) the function $$\lambda $$ is harmonic. Finally an illustrative example is presented.

On the extremal compatible linear connection of a Randers space

Abstract: A linear connection on a Finsler manifold is called compatible to the metric if its parallel transports preserve the Finslerian length of tangent vectors. Generalized Berwald manifolds are Finsler manifolds equipped with a compatible linear connection. Since the compatibility to the Finslerian metric does not imply the unicity of the linear connection in general, the first step of checking the existence of compatible linear connections on a Finsler manifold is to choose the best one to look for. A reasonable choice is introduced in Vincze (J Differ Geom Appl, 2019. arXiv:1909.03096 ) called the extremal compatible linear connection, which has torsion of minimal norm at each point. Randers metrics are special Finsler metrics that can be written as the sum of a Riemannian metric and a 1-form (they are “translates” of Riemannian metrics). In this paper, we investigate the compatibility equations for a linear connection to a Randers metric. Since a compatible linear connection is uniquely determined by its torsion, we transform the compatibility equations by taking the torsion components as variables. We determine when these equations have solutions, i.e. when the Randers space becomes a generalized Berwald space admitting a compatible linear connection. Describing all of them, we can select the extremal connection with the norm minimizing property. As a consequence, we obtain the characterization theorem in Vincze (Indag Math 26(2):363–379, 2014): a Randers space is a non-Riemannian generalized Berwald space if and only if the norm of the perturbating term with respect to the Riemannian part of the metric is a positive constant.

Almost hypercomplex manifolds with Hermitian–Norden metrics and 4-dimensional indecomposable real Lie algebras depending on one parameter

Abstract: We study almost hypercomplex structure with Hermitian–Norden metrics on 4-dimensional Lie groups considered as smooth manifolds. All the basic classes of a classification of 4-dimensional indecomposable real Lie algebras depending on one parameter are investigated. There are studied some geometrical characteristics of the respective almost hypercomplex manifolds with Hermitian–Norden metrics.

Uniqueness of the translating bowl in $$\mathbb {H}^2\times \mathbb {R}$$ H 2 × R

Abstract: In this paper we extend the proof of Theorem 5.2 in [1], which characterizes the translating bowl $$\mathcal {B}$$ in $${\mathbb {H}}^2\times {\mathbb {R}}$$ as the only properly immersed translating soliton with finite topology and one end that is smoothly asymptotic to $$\mathcal {B}$$ . We exploit the asymptotic behavior at infinity of $$\mathcal {B}$$ and the possibility of applying Alexandrov’s reflection technique with respect to vertical planes.

Two remarks on ordering in absolute planes

Abstract: As well known, the underlying field K of an Euclidean plane E is Euclidean if and only if E fulfills the Circle Axiom. In this paper we consider an apparently weaker form of the Circle Axiom which leads to weaker properties of K. It will be shown that these weaker properties still characterize Euclidean fields. In the second remark we complete the construction of a certain kind of Hilbert plane considered in Pambuccian and Schacht (Beitr Algebra Geom, 2019. https://doi.org/10.1007/s13366-019-00445-y).

Four-dimensional almost Einstein manifolds with skew-circulant stuctures

Abstract: We consider a four-dimensional Riemannian manifold M with an additional structure S, whose fourth power is minus identity. In a local coordinate system the components of the metric g and the structure S form skew-circulant matrices. Both structures S and g are compatible, such that an isometry is induced in every tangent space of M. By a special identity for the curvature tensor, generated by the Riemannian connection of g, we determine classes of Einstein and almost Einstein manifolds. For such manifolds we obtain propositions for the sectional curvatures of some characteristic 2-planes in a tangent space of M. We consider a Hermitian manifold associated with the studied manifold and find conditions for g, under which it is a Kahler manifold. We construct some examples of the considered manifolds on Lie groups.

Sasakian manifolds satisfying certain conditions on Q tensor

Abstract: The object of the present work is to investigate Sasakian manifolds satisfying certain conditions on Q tensor whose trace is well-known Z tensor.

2-Special normal rational curves in $$\mathrm{PG}(4,q)$$PG(4,q)

Abstract: This article looks at subconics of order q of $$\mathrm{PG}(2,q^2)$$ in the Bruck–Bose representation in $$\mathrm{PG}(4,q)$$. A unified characterisation is given, namely a normal rational curve $${{{\mathcal {N}}}}$$ in $$\mathrm{PG}(4,q)$$ corresponds to an $${\mathbb {F}}_{\!q\!\;}$$-conic of $$\mathrm{PG}(2,q^2)$$ if and only if $${{{\mathcal {N}}}}$$ is 2-special - a notion related to how $${{{\mathcal {N}}}}$$ interacts with the regular 1-spread of the Bruck–Bose representation.

A combinatorial characterization of the Baer and the unital cone in $$PG(3,q^2)$$ P G ( 3 , q 2 )

Abstract: Bruen, see [5], and Bruen and Thas, see [7], proved that in $$PG(2,q^2)$$ a blocking set of type $$(1,q+1)_1$$ is either a Baer subplane or a unital. In this paper a cone-generalization of this result in $$PG(3,q^2)$$ is provided.

Concurrence of tetrahedral cevians associated with triangle centers

Abstract: It is well known that the center of mass of a tetrahedron is the intersection of the line segments that join its vertices to the centers of mass of the opposite faces, and that a similar statement holds for simplices in higher dimensions. This paper addresses the question whether this nice inductive property of the center of mass also holds for other centers. It proves that the center of mass is the only center that has this property, and investigates in detail the situation for the other well known centers, namely, the Gergonne center, the Nagel center, the Lemoine center, the incenter, the orthocenter, the Fermat-Torricelli center, and the circumcenter. Some questions for further research are raised.

On Holditch’s theorem

Abstract: In this paper we present a certain modification of the Holditch construction. This construction allows to consider a geometric family of pairs of ring domains. It is proved that the ratio of areas of ring domains of each pair belonging to this family is constant. Problems on extremal chords of constant length sliding around a given oval with both endpoints on it are also considered.

The cylinder theorem in $${{\mathcal {H}}}^2\times R$$ H 2 × R

Abstract: We consider cylinders in $${{\mathcal {H}}}^2\times R$$ (see definitions in the introduction) and prove that a complete and connected surface in $${{\mathcal {H}}}^2\times R$$ with the vanishing of the Gauss and extrinsic curvatures is a cylinder.