Showing papers in "Journal of Geometry in 2017"

{\eta}-Ricci solitons on para-Sasakian manifolds

Abstract: The object of this paper is to study a special type of metrics called $${\eta}$$ -Ricci solitons on para-Sasakian manifolds. We give the existence of para-Sasakian $${\eta}$$ -Ricci solitons in our settings. In addition, the non-existence of certain geometric characteristics of para-Sasakian $${\eta}$$ -Ricci solitons are studied. Finally, we discuss 3-dimensional and conformally flat para-Sasakian $${\eta}$$ -Ricci solitons.

Kenmotsu statistical manifolds and warped product

Abstract: A notion of a Kenmotsu statistical manifold is introduced, which is locally obtained as the warped product of a holomorphic statistical manifold and a line A statistical manifold is a Riemannian manifold equipped with a torsion-free affine connection satisfying the Codazzi equation It can be considered as being in information geometry, Hessian geometry and various submanifold theory On the other hand, a Kenmotsu manifold is in a meaningful class of almost contact metric manifolds In this paper, we construct a suitable statistical structure on it Although the notion of the warped product of Riemannian manifolds is well known, the one for statistical manifolds is not established We consider it in general, and study the statistical sectional curvature of the warped product of two statistical manifolds We show that a Kenmotsu statistical manifold of constant $$\phi $$ -sectional curvature is constructed from a special Kahler manifold, which is an important example of holomorphic statistical manifold A Sasakian statistical manifold is also studied from the viewpoint of the warped product of statistical manifolds

On curvature properties of Som–Raychaudhuri spacetime

Abstract: Som–Raychaudhuri (Proc R Soc Lond A 304:81–86, 1968) spacetime is a stationary cylindrical symmetric solution of Einstein field equation corresponding to a charged dust distribution in rigid rotation. The main object of the present paper is to investigate the curvature restricted geometric structures admitting by the Som–Raychaudhuri spacetime and it is shown that such a spacetime is a 2-quasi-Einstein, generalized Roter type, Ein(3) manifold satisfying \(R\cdot R = Q(S,R)\), \(C\cdot C = \frac{2a^2}{3} Q(g,C)\), and its Ricci tensor is cyclic parallel and Riemann compatible. Finally, we make a comparison between Godel spacetime and Som–Raychaudhuri spacetime.

Moving frames and the characterization of curves that lie on a surface

Abstract: In this work we are interested in the characterization of curves that belong to a given surface. To the best of our knowledge, there is no known general solution to this problem. Indeed, a solution is only available for a few examples: planes, spheres, or cylinders. Generally, the characterization of such curves, both in Euclidean ( $$E^3$$ ) and in Lorentz–Minkowski ( $$E_1^3$$ ) spaces, involves an ODE relating curvature and torsion. However, by equipping a curve with a relatively parallel moving frame, Bishop was able to characterize spherical curves in $$E^3$$ through a linear equation relating the coefficients which dictate the frame motion. Here we apply these ideas to surfaces that are implicitly defined by a smooth function, $$\varSigma =F^{-1}(c)$$ , by reinterpreting the problem in the context of the metric given by the Hessian of F, which is not always positive definite. So, we are naturally led to the study of curves in $$E_1^3$$ . We develop a systematic approach to the construction of Bishop frames by exploiting the structure of the normal planes induced by the casual character of the curve, present a complete characterization of spherical curves in $$E_1^3$$ , and apply it to characterize curves that belong to a non-degenerate Euclidean quadric. We also interpret the casual character that a curve may assume when we pass from $$E^3$$ to $$E_1^3$$ and finally establish a criterion for a curve to lie on a level surface of a smooth function, which reduces to a linear equation when the Hessian is constant.

Existence of global Chebyshev nets on surfaces of absolute Gaussian curvature less than {{2\pi}}

Abstract: We prove the existence of a global smooth Chebyshev net on complete, simply connected surfaces when the total absolute curvature is bounded by $${2\pi}$$ . Following Samelson and Dayawansa, we look at Chebyshev nets given by a dual curve, splitting the surface into two connected half-surfaces, and a distribution of angles along it. An analogue to the Hazzidakis’ formula is used to control the angles of the net on each half-surface with the integral of the Gaussian curvature of this half-surface and the Cauchy boundary conditions. We can then prove the main result using a theorem about splitting the Gaussian curvature with a geodesic, obtained by Bonk and Lang.

Certain results on K-paracontact and paraSasakian manifolds

Abstract: The purpose of this paper is to study 3-dimensional paraSasakian manifold and conformally flat K-paracontact manifold. Moreover, we show that in a K-paracontact manifold the conditions Einstein, conformally flat, semi-symmetric and Ricci semi-symmetric are all equivalent. Finally, compact regular 3-dimensional paraSasakian and conformally flat K-paracontact manifolds are studied.

Existence conditions of framed curves for smooth curves

Abstract: A framed curve is a smooth curve in the Euclidean space with a moving frame. We call the smooth curve in the Euclidean space the framed base curve. In this paper, we give an existence condition of framed curves. Actually, we construct a framed curve such that the image of the framed base curve coincides with the image of a given smooth curve under a condition. As a consequence, polygons in the Euclidean plane can be realised as not only a smooth curve but also a framed base curve.

Affine translation surfaces with constant mean curvature in Euclidean 3-space

Abstract: Using classical methods and solving certain differential equations we classify a kind of new surface, affine translation surfaces, which has the non-zero constant mean curvature in three dimensional Euclidean space $${{\bf E}^3}$$ . Therefore a kind of solutions for the constant mean curvature equation is given.

The Critical Point Equation And Contact Geometry

Abstract: In this paper, we consider the CPE conjecture in the frame-work of \({K}\)-contact manifold and \({(\kappa, \mu)}\)-contact manifold. First, we prove that a complete \({K}\)-contact metric satisfying the CPE is Einstein and is isometric to a unit sphere \({S^{2n+1}}\). Next, we prove that if a non-Sasakian \({ (\kappa, \mu) }\)-contact metric satisfies the CPE, then \({ M^{3} }\) is flat and for \({ n > 1 }\), \({ M^{2n+1} }\) is locally isometric to \({ E^{n+1} \times S^{n}(4)}\).

Geometry of {\mathcal{P}\mathcal{R}}-semi-invariant warped product submanifolds in paracosymplectic manifold

Abstract: The purpose of this paper is to study $${\mathcal{P}\mathcal{R}}$$ -semi-invariant warped product submanifolds of a paracosymplectic manifold $${\widetilde{M}}$$ . We prove that the distributions associated with the definition of $${\mathcal{P}\mathcal{R}}$$ -semi-invariant warped product submanifold M are always integrable. A necessary and sufficient condition for an isometrically immersed $${\mathcal{P}\mathcal{R}}$$ -semi-invariant submanifold of $${\widetilde{M}}$$ to be a $${\mathcal{P}\mathcal{R}}$$ -semi-invariant warped product submanifold is obtained in terms of the shape operator.

Persistent rigid-body motions and study’s “Ribaucour” problem

Abstract: In this work we show that the concept of a one-parameter persistent rigid-body motion is a slight generalisation of a class of motions called Ribaucour motions by Study. This allows a simple description of these motions in terms of their axode surfaces. We then investigate other special rigid-body motions, and ask if these can be persistent. The special motions studied are line-symmetric motions and motions generated by the moving frame adapted to a smooth curve. We are able to find geometric conditions for the special motions to be persistent and, in most cases, we can describe the axode surfaces in some detail. In particular, this work reveals some subtle connections between persistent rigid-body motions and the classical differential geometry of curves and ruled surfaces.

The perimeter sixpartite center of a triangle

Abstract: We show that the perimeter of a triangle can be divided into six equal parts by three concurrent lines in only one way. The perimeter sixpartite point is presented exactly, by its coordinates, and proved to be a non-constructible point and a new center of the triangle.

Differential inequalities on homogeneous toric bundles

Abstract: We study the generalized Abreu equation in n-dimensional polytopes and derive some differential inequalities for homogeneous toric bundles.

A generalisation of Sylvester’s problem to higher dimensions

Abstract: In this article we consider S to be a set of points in d-space with the property that any d points of S span a hyperplane and not all the points of S are contained in a hyperplane. The aim of this article is to introduce the function \(e_d(n)\), which denotes the minimal number of hyperplanes meeting S in precisely d points, minimising over all such sets of points S with \(|S|=n\).

Loxodromes and geodesics on rotational surfaces in a simply isotropic space

Abstract: A loxodrome is a curve on a parametrized surface that intersects one family of parametric lines at a constant angle. In this paper, we investigate loxodromes on rotational surfaces in the 3-dimensional simply isotropic space \({I^3}\) and give an example of loxodromes. Also, we completely classify geodesics on rotational surfaces in \({I^3}\).

A construction of small complete caps in projective spaces

Abstract: In this work complete caps in PG(N, q) of size $${O(q^{\frac{N-1}{2}} \log^{300} q)}$$ are obtained by probabilistic methods. This gives an upper bound asymptotically very close to the trivial lower bound $${\sqrt{2}q^{\frac{N-1}{2}}}$$ and it improves the best known bound in the literature for small complete caps in projective spaces of any dimension. The result obtained in the paper also gives a new upper bound for l(m, 2, q)4, that is the minimal length n for which there exists an [n, n−m, 4] q 2 covering code with given m and q.

Evolutes of plane curves and null curves in Minkowski 3-space

Abstract: We use the isotropic projection of Laguerre geometry in order to establish a correspondence between plane curves and null curves in the Minkowski 3-space. We describe the geometry of null curves (Cartan frame, pseudo-arc parameter, pseudo-torsion, pairs of associated curves) in terms of the curvature of the corresponding plane curves. This leads to an alternative description of all plane curves which are Laguerre congruent to a given one.

Generalized Weierstrass–Enneper representations of Euclidean, spacelike, and timelike surfaces: a unified Lie-algebraic formulation

Abstract: The complex-analytic approach to constructing minimal surfaces carried out by Weierstrass and Enneper has since been extended to create conformal parametrizations of minimal and more general surfaces, including Euclidean, spacelike, and timelike surfaces in three-dimensional Euclidean and Lorentz spaces. In this work, we present a Lie-algebraic formulation that unites these representations into one coherent framework and completes the formulas in the general timelike case. Integrable moving frames for surfaces are created inside of three-dimensional Lie algebras by applying a scaling, together with an isometric action of the appropriate Lie group, to a standard constant orthonormal frame. Using complex coordinates for Euclidean and spacelike surfaces and hyperbolic coordinates for timelike surfaces allows for simple characterizations of geometric properties of the surfaces in terms of the complex and hyperbolic structures. We give expressions in this framework of the first and second fundamental forms and of mean and Gaussian curvatures on Euclidean, spacelike, and timelike surfaces. The expression of the Gaussian curvature affirms in an efficient and elegant way that Gauss’ Theorem Egregium holds for surfaces of all three types. We also provide examples and illustrations of special surfaces and deformations of surfaces arising from the Weierstrass–Enneper representations.

On sets of type $${\varvec{(m,m+q)_2}}$$ ( m , m + q ) 2 in $${\varvec{PG(3,q)}}$$ P G ( 3 , q )

Abstract: A set K of type $$(m,m+q)_2$$ in the projective space PG(3, q) is a set of points such that every plane meets K in either m or $$m+q$$ points and, furthermore, there are both planes meeting K in m points and planes meeting K in $$m+q$$ points. The classical example of such a set is a partial spread of size m; however, several other families are known. In this paper we study some properties of these sets and their classification for small parameters.

On anti-invariant Riemannian submersions whose total manifolds are locally product Riemannian

Abstract: In this paper, we study Riemannian, anti-invariant and Lagrangian submersions from locally product Riemannian manifolds onto Riemannian manifolds. We first give a characterization theorem for Riemannian submersions. It is proved that the fibers of a Lagrangian submersion are always totally geodesic. We also consider the first variational formula of anti-invariant Riemannian submersions and give a new condition for the harmonicity of such submersions.

Deformation of minimal surfaces with planar curvature lines

Abstract: Minimal surfaces with planar curvature lines are classical geometric objects, having been studied since the late nineteenth century. In this paper, we revisit the subject from a different point of view. After calculating their metric functions using an analytical method, we recover the Weierstrass data, and give clean parametrizations for these surfaces. Then, we show that there exists a single continuous deformation between all minimal surfaces with planar curvature lines. In the process, we establish the existence of axial directions for these surfaces.

Counting arcs in projective planes via Glynn’s algorithm

Abstract: An n-arc in a projective plane is a collection of n distinct points in the plane, no three of which lie on a line. Formulas counting the number of n-arcs in any finite projective plane of order q are known for $$n \le 8$$ . In 1995, Iampolskaia, Skorobogatov, and Sorokin counted 9-arcs in the projective plane over a finite field of order q and showed that this count is a quasipolynomial function of q. We present a formula for the number of 9-arcs in any projective plane of order q, even those that are non-Desarguesian, deriving Iampolskaia, Skorobogatov, and Sorokin’s formula as a special case. We obtain our formula from a new implementation of an algorithm due to Glynn; we give details of our implementation and discuss its consequences for larger arcs.

General position subsets and independent hyperplanes in d-space

Abstract: Erdős asked what is the maximum number \({\alpha(n)}\) such that every set of \({n}\) points in the plane with no four on a line contains \({\alpha(n)}\) points in general position. We consider variants of this question for \({d}\)-dimensional point sets and generalize previously known bounds. In particular, we prove the following two results for fixed \({d}\): Every set \({\mathcal{H}}\) of \({n}\) hyperplanes in \({\mathbb{R}^d}\) contains a subset \({S\subseteq \mathcal{H}}\) of size at least \({c \left(n \log n\right)^{1/d}}\), for some constant \({c=c(d)> 0}\), such that no cell of the arrangement of \({\mathcal{H}}\) is bounded by hyperplanes of \({S}\) only. Every set of \({cq^d\log q}\) points in \({\mathbb{R}^d}\), for some constant \({c=c(d)> 0}\), contains a subset of \({q}\) cohyperplanar points or \({q}\) points in general position. Two-dimensional versions of the above results were respectively proved by Ackerman et al. [Electronic J. Combinatorics, 2014] and by Payne and Wood [SIAM J. Discrete Math., 2013].

On bipartite graphs whose interval space is a closed join space

Abstract: We characterize the bipartite graphs whose geodesic interval spaces are (closed) join spaces, i.e. which share a number of geometrical properties with Euclidean spaces. We prove that the geodesic interval space \((V(G),I_G)\) of a bipartite graph G is a join space if and only if G is a partial cube all of whose finite convex subgraphs have a pre-hull number which is at most 1. Such a partial cube is a called a Peano graph. Also we study several fundamental notions related to the interval spaces of connected graphs viewed as commutative quasihypergroups, namely, homomorphisms, subhypergroupoids, direct and weak direct products, and retracts. We show that the interval space of an interval monotone bipartite graph is a join space if and only if all its cosets are subhypergroupoids, and we prove that the class of interval spaces of a Peano graph is closed under subhypergroups, weak direct products and retracts, but generally not for homomorphic images.

Synthetic foundations of cevian geometry, I: fixed points of affine maps

Abstract: We give synthetic proofs of new results in triangle geometry, focusing especially on fixed points of certain affine maps which are defined in terms of the cevian triangles of a point P and its isotomic conjugate P′, with respect to a given triangle ABC. We give a synthetic proof of Grinberg’s formula for the cyclocevian map in terms of the isotomic and isogonal maps, and show that the complement Q of the isotomic conjugate P′ has many interesting properties. If T P is the affine map taking ABC to the cevian triangle DEF for P, it is shown that Q is the unique ordinary fixed point of T P when P does not lie on the sides of triangle ABC, its anticomplementary triangle, or the Steiner circumellipse of ABC. This paper forms the foundation for several more papers to follow, in which the conic on the 5 points A, B, C, P, Q is studied and its center is characterized as a fixed point of the map $${\lambda = T_{P'} \circ T_P^{-1}}$$ .

Classification of subspaces in $${{\mathbb{F}}^2\otimes {\mathbb{F}}^3}$$ F 2 ⊗ F 3 and orbits in $${{\mathbb{F}}^2\otimes{\mathbb{F}}^3 \otimes {\mathbb{F}}^r}$$ F 2 ⊗ F 3 ⊗ F r

Abstract: This paper contains the classification of the orbits of elements of the tensor product spaces $${{\mathbb{F}}^2\otimes{\mathbb{F}}^3 \otimes {\mathbb{F}}^r}$$ , $${r\geq 1}$$ , under the action of two natural groups, for all finite; real; and algebraically closed fields. For each of the orbits we determine: a canonical form; the tensor rank; the rank distribution of the contraction spaces; and a geometric description. The proof is based on the study of the contraction spaces in $${{\mathrm{PG}}({\mathbb{F}}^2\otimes {\mathbb{F}}^3)}$$ and is geometric in nature. Although the main focus is on finite fields, the techniques are mostly field independent.

In $${}$$ any $${q^2}$$q2-set of class $${[0,m,n]_2}$$[0,m,n]2 containing a line is a cylinder

Abstract: In this short note we characterize cylinders of \({AG(3,q)}\) as \({q^2}\)-sets of class \({[0,m,n]_2}\) containing at least one line.

Curvature properties of $$\varvec{}$$-dimensional Riemannian manifolds with a circulant structure

Abstract: We consider a 4-dimensional Riemannian manifold M equipped with a circulant structure q, which is an isometry with respect to the metric g and \(q^{4}=\mathrm{id}\), \(q^{2} e \pm \mathrm{id}\). For such a manifold (M, g, q) we obtain some assertions for the sectional curvatures of 2-planes. We construct an example of such a manifold on a Lie group and we find some of its geometric characteristics.

Certain results on generalized $${\varvec{(k,\,\mu )}}$$-contact metric manifolds

Abstract: The object of the present paper is to study second order symmetric parallel tensors in generalized \((k,\,\mu )\)-contact metric manifolds and its applications to Ricci solitons. Next, we prove that a generalized \((k,\,\mu )\)-contact metric manifold M admits a Ricci soliton whose potential vector field is the Reeb vector field \(\xi \) if and only if M is a Sasaki–Einstein manifold. Finally, we give some examples of generalized \((k,\,\mu )\)-contact metric manifold.

Projective linear groups as automorphism groups of chiral polytopes

Abstract: It is already known that the automorphism group of a chiral polyhedron is never isomorphic to \( PSL \left( 2,q\right) \) or \( PGL \left( 2,q\right) \) for any prime power q. In this paper, we show that \( PSL \left( 2,q\right) \) and \( PGL \left( 2,q\right) \) are never automorphism groups of chiral polytopes of rank at least 5. Moreover, we show that \( PGL \left( 2,q\right) \) is the automorphism group of at least one chiral polytope of rank 4 for every \(q\ge 5\). Finally, we determine for which values of q the group \( PSL \left( 2,q\right) \) is the automorphism group of a chiral polytope of rank 4, except when \(q=p^d\equiv 3\pmod {4}\) where \(d>1\) is not a prime power, in which case the problem remains unsolved.