Showing papers in "Journal of Geometry in 2018"

A study of Wintgen like inequality for submanifolds in statistical warped product manifolds

Abstract: In this paper, we study statistical manifolds and their submanifolds. We first construct two new examples of statistical warped product manifolds and give a method how to construct Kenmotsu-like statistical manifold and cosymplectic-like statistical manifold based on the existence of Kaehler-like statistical manifold. Then we obtain the general Wintgen inequality for statistical submanifolds of statistical warped product manifolds.

The decomposition of almost paracontact metric manifolds in eleven classes revisited

Abstract: This paper is a continuation of our previous work, where eleven basic classes of almost paracontact metric manifolds with respect to the covariant derivative of the structure tensor field were obtained. First we decompose one of the eleven classes into two classes and the basic classes of the considered manifolds become twelve. Also, we determine the classes of $$\alpha $$ -para-Sasakian, $$\alpha $$ -para-Kenmotsu, normal, paracontact metric, para-Sasakian, K-paracontact and quasi-para-Sasakian manifolds. Moreover, we study 3-dimensional almost paracontact metric manifolds and show that they belong to four basic classes from the considered classification. We define an almost paracontact metric structure on any 3-dimensional Lie group and give concrete examples of Lie groups belonging to each of the four basic classes, characterized by commutators on the corresponding Lie algebras.

Generalized normalized $$\varvec{\delta }$$ δ -Casorati curvature for statistical submanifolds in quaternion Kaehler-like statistical space forms

Abstract: In 2017, C. W. Lee et al. derived optimal Casorati inequalities with normalized scalar curvature for statistical submanifolds of statistical manifolds of constant curvature. In this paper, we generalizes those inequalities. In fact, we obtain the bounds for the generalized normalized $$\delta $$ -Casorati curvatures for statistical submanifolds in quaternion Kaehler-like statistical space forms.

Translating solitons of the mean curvature flow in the space $\mathbb{H}^2\times\mathbb{R}$

Abstract: In this paper we study the theory of translating solitons of the mean curvature flow of immersed surfaces in the product space $${\mathbb {H}}^2\times {\mathbb {R}}$$ . We relate this theory to the one of manifolds with density, and exploit this relation by regarding these translating solitons as minimal surfaces in a conformal metric space. Explicit examples of these surfaces are constructed, and we study the asymptotic behavior of the existing rotationally symmetric examples. Finally, we prove some uniqueness and non-existence theorems.

Curvature properties of Robinson–Trautman metric

Abstract: The curvature properties of Robinson–Trautman metric have been investigated. It is shown that Robinson–Trautman metric is a Roter type metric, and in a consequence, admits several kinds of pseudosymmetric type structures such as Weyl pseudosymmetric, Ricci pseudosymmetric, pseudosymmetric Weyl conformal curvature tensor etc. Moreover, it is proved that this metric is a 2-quasi-Einstein, the Ricci tensor is Riemann compatible and its Weyl conformal curvature 2-forms are recurrent. It is also shown that the energy momentum tensor of the metric is pseudosymmetric and the conditions under which such tensor is of Codazzi type and cyclic parallel have been investigated. Finally, we have made a comparison between the curvature properties of Robinson–Trautman metric and Som–Raychaudhuri metric.

The axiomatization of affine oriented matroids reassessed

Abstract: In an unpublished manuscript of 1992, Johan Karlander has given an axiomatization of affine oriented matroids, which can be thought of as oriented matroids with a hyperplane at infinity. A closer examination of the text revealed an invalid construction and an incorrect argument in the proof of his main theorem. This paper provides an alternative argument to fix and slightly simplify the proof of the main theorem.

Geometry of warped product lightlike submanifolds of indefinite nearly Kaehler manifolds

Abstract: The aim of present paper is to study geometry of warped product SCR-lightlike submanifolds of indefinite nearly Kaehler manifolds. We prove the non-existence of warped product SCR-lightlike submanifolds of the type $$N_{\bot } \times _{f} N_{T}$$ in an indefinite nearly Kaehler manifold. We find a necessary and sufficient condition for a SCR-lightlike submanifold of an indefinite nearly Kaehler manifold to be a SCR-lightlike warped product submanifold of the type $$N_{T} \times _{f} N_{\bot }$$ . We give some characterizations in terms of the canonical structures T and $$\omega $$ on a SCR-lightlike submanifold of an indefinite nearly Kaehler manifold under which it reduces to a SCR-lightlike warped product submanifold. We also prove that for a proper SCR-lightlike warped product submanifold of an indefinite nearly Kaehler manifold, the induced connection $$ abla $$ can never be a metric connection. Finally, we obtain a sharp inequality for the squared norm of the second fundamental form in terms of the warping function for a SCR-lightlike warped product submanifold of an indefinite nearly Kaehler manifold.

AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves

Abstract: We investigate several types of linear codes constructed from two families of maximal curves over finite fields recently constructed by Skabelund as cyclic covers of the Suzuki and Ree curves. Plane models for such curves are provided, and the Weierstrass semigroup at an $$\mathbb {F}_{q}$$ -rational point is shown to be symmetric.

Gradient Ricci Solitons with a Conformal Vector Field

Abstract: We show that a connected gradient Ricci soliton ( $$M,g,f,\lambda $$ ) with constant scalar curvature and admitting a non-homothetic conformal vector field V leaving the potential vector field invariant, is Einstein and the potential function f is constant. For locally conformally flat case and non-homothetic V we show without constant scalar curvature assumption, that f is constant and g has constant curvature.

Monge surfaces and planar geodesic foliations

Abstract: A Monge surface is a surface obtained by sweeping a generating plane curve along a trajectory that is orthogonal to the moving plane containing the curve. Locally, they are characterized as being foliated by a family of planar geodesic lines of curvature. We call surfaces with the latter property PGF surfaces, and investigate the global properties of these two naturally defined objects. The only compact orientable PGF surfaces are tori; these are globally Monge surfaces, and they have a simple characterization in terms of the directrix. We show how to produce many examples of Monge tori and Klein bottles, as well as tori that do not have a closed directrix.

Some geometric properties of translating solitons in Euclidean space

Abstract: A translating soliton is a surface in Euclidean space $$\mathbb {R}^3$$ that is minimal for a log-linear density $$\phi (x,y,z)=\alpha x+\beta y+\gamma y$$ , where $$\alpha ,\beta ,\gamma $$ are real numbers not all zero. We characterize all translating solitons that are foliated by circles and the ones that are graphs of type $$z=f(x)+g(y)$$ .

On standard two-intersection sets in PG ( r , q )

Abstract: In this paper, we extend and analyze in a finite projective space of any dimension the notion of standard two-intersection sets previously introduced in the projective plane by Penttila and Royle (Des Codes Cryptogr 6:229–245, 1995), see also Blokhuis and Lavrauw (J Combin Theory Ser A 99:377–382, 2002). Moreover, given a pair of suitable distinct standard two-intersection sets in a finite projective space it is possible to get further standard two-intersection sets by applying elementary set-theoretical operations to the elements of the pair.

On the p -harmonic and p -biharmonic maps

Abstract: In this paper, we study the existence of p-harmonic maps into Riemannian manifolds admitting a conformal vector field. We also prove a Liouville type theorem for p-biharmonic maps.

On metric connections with torsion on the cotangent bundle with modified Riemannian extension

Abstract: Let M be an n-dimensional differentiable manifold equipped with a torsion-free linear connection $$ abla $$ and $$T^{*}M$$ its cotangent bundle. The present paper aims to study a metric connection $$\widetilde{ abla }$$ with nonvanishing torsion on $$T^{*}M$$ with modified Riemannian extension $${}\overline{g}_{ abla ,c}$$ . First, we give a characterization of fibre-preserving projective vector fields on $$(T^{*}M,{}\overline{g} _{ abla ,c})$$ with respect to the metric connection $$\widetilde{ abla }$$ . Secondly, we study conditions for $$(T^{*}M,{}\overline{g}_{ abla ,c})$$ to be semi-symmetric, Ricci semi-symmetric, $$\widetilde{Z}$$ semi-symmetric or locally conharmonically flat with respect to the metric connection $$ \widetilde{ abla }$$ . Finally, we present some results concerning the Schouten–Van Kampen connection associated to the Levi-Civita connection $$ \overline{ abla }$$ of the modified Riemannian extension $$\overline{g} _{ abla ,c}$$ .

Burmester theory in Cayley–Klein planes with affine base

Abstract: In this paper, we study the Burmester theory in Euclidean, Galilean and pseudo-Euclidean planes and extend the classical Burmester theory to the Cayley–Klein planes with affine base by a unified method. For this purpose, we use the generalized complex numbers and define a generalized form of Bottema’s instantaneous invariants. By this way, we expose the instantaneous geometric properties of motion of rigid bodies in the Cayley–Klein planes with affine base.

A universal linear algebraic model for conformal geometries

Abstract: This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown analytically, through a framework comparing the measurement of distances and angles in Cayley–Klein geometries, including Lorentzian geometries, as done by F. Bachmann and later R. Struve. On the other hand, such a relationship may also be expressed in a purely linear algebraic manner, as explained by D. Hestens, H. Li and A. Rockwood. The model described in this article unifies these approaches via a generalization of Lie sphere geometry. Like the work of N. Wildberger, it is a purely algebraic construction, and as such it works over any field of odd characteristic. It is shown that measurement of distances and angles is an inherent property of the model that is easy to identify, and the possible models are classified over the real, complex and finite fields, and partially in characteristic 2, revealing a striking analogy between the real and finite geometries. This is an abbreviated version of a previous manuscript, with certain expository parts removed for the sake brevity. The original manuscript is available on the website arXiv, with more motivation, examples, properties. Several definitions and theorems can be extended to include the characteristic 2 case, which are also omitted here.

On the constructibility of the axes of an ellipsoid: the construction of Chasles in practice

Abstract: In this paper we discuss Chasles’s construction on ellipsoid to draw the semi-axes from a complete system of conjugate diameters. We prove that there is such situation when the construction is not planar (the needed points cannot be constructed with compasses and ruler) and give some others in which the construction is planar.

Invariant and anti-invariant submanifolds of special quasi-sasakian manifolds

Abstract: The present paper deals with the study of Chaki-pseudo parallel and Deszcz-pseudo parallel invariant submanifolds of SQ-Sasakian manifolds with respect to Levi–Civita connection as well as semisymmetric metric connection. It is obtained that these two classes are equivalent with a certain condition. In addition, invariant and anti-invariant submanifolds of SQ-Sasakian manifolds with respect to Levi–Civita connection as well as semisymmetric metric connection whose metrics are Ricci solitons are studied.

The scalar curvature of a projectively invariant metric on a convex domain

Abstract: Considering a projectively invariant metric on a strongly convex bounded domain, we study the asymptotic expansion of the scalar curvature with respect to the distance function, and use the Fubini–Pick invariant to describe the second term in the expansion. In particular for the two-dimensional convex domain, we also show that the third term in the expansion is zero.

Duals of timelike Sabban curves in de Sitter n -space

Abstract: In this paper, we define generalized Sabban frames of non-lightlike curves on $$S_{1}^{n}$$ and investigate the singularities of de Sitter dual hypersurfaces of timelike Sabban curves.

The homogeneous ruled real hypersurface in a complex hyperbolic space

Abstract: We characterize the homogeneous ruled real hyperurface of a complex hyperbolic space in the class of ruled real hypersurfaces having constant mean curvature.

An inequality for contact CR-submanifolds in Sasakian space forms

Abstract: The theory of $$\delta $$ -invariants was initiated by Chen (Arch Math 60:568–578, 1993) in order to find new necessary conditions for a Riemannian manifold to admit a minimal isometric immersion in a Euclidean space. Chen (Int J Math 23(3):1250045, 2012) defined a CR $$\delta $$ -invariant $$\delta (D)$$ for anti-holomorphic submanifolds in complex space forms. Afterwards, Al-Solamy et al. (Taiwan J Math 18:199–217, 2014) established an optimal inequality for this invariant for anti-holomorphic submanifolds in complex space forms. In this article, we prove an optimal inequality for the contact CR $$\delta $$ -invariant on contact CR-submanifolds in Sasakian space forms. An example for the equality case is given.

Putting convex d -polytopes inside frames

Abstract: We introduce a construction of large neighborly and nearly-neighborly families of d-polytopes in $$E^{d}$$ , obtained by putting smaller such families in rods, to form suitable frames for a larger collection. We use it to exhibit, among general results, a nearly-neighborly family, consisting of $$9\times 4^{d-2}$$ combinatorially d-boxes in $$E^{d}$$ , for every $$d \ge 3$$ (like 36 3-boxes in $$E^{3})$$ and a neighborly family of $$3\times 2^{d-1}$$ of prisms over d-simplices (like 12 triangular prisms in $$E^{3})$$ .

On strongly convex projectively flat and dually flat complex Finsler metrics

Abstract: In this paper, we prove that a strongly convex complex Finsler metric F on a domain $$D\subset \mathbb {C}^n$$ is projectively flat (resp. dually flat) if and only if F comes from a strongly convex complex Minkowski metric.

On automorphism groups of toroidal circle planes

Abstract: Schenkel proved that the automorphism group of a flat Minkowski plane is a Lie group of dimension at most 6 and described planes whose automorphism group has dimension at least 4 or one of whose kernels has dimension 3. We extend these results to the case of toroidal circle planes.

Construction of locally compact nearfields

Abstract: The aim of this work is the construction of a wide class of disconnected locally compact nearfields. They are all Dickson nearfields and derived from local fields by means of couplings described explicitly.

Paradoxical sets and sets with two removable points

Abstract: A set $$E \subset \mathbb {R}^n$$ is called uniformly discrete if there exists an $$\varepsilon >0 $$ such that no two points of E are closer than $$\varepsilon $$ . Applying a theorem of T. Tao on the absence of paradoxical decompositions of uniformly discrete sets we will prove, under an additional assumption, that such a set $$E \subset \mathbb {R}^n$$ has at most one point p such that $$E {\setminus } \{p\}$$ and E are congruent. We prove also that if $$E \subseteq \mathbb {R}^n$$ is a discrete set and G is a discrete subgroup of the group of isometries of $$\mathbb {R}^n$$ then there is at most one point $$p \in E$$ such that there exists a $$\varphi \in G$$ with $$\varphi (E) = E {\setminus } \{p\}$$ . Related unsolved problems will be pointed out.

Spheres and circles in a tangent space of a 4-dimensional Riemannian manifold with respect to an indefinite metric

Abstract: Our study is in the tangent space at an arbitrary point on a 4-dimensional Riemannian manifold. The manifold is equipped with an additional tensor structure of type (1, 1), whose fourth power is the identity and the second power is an almost product structure. The metric and the additional structure are compatible, such that an isometry is induced in every tangent space. They determine an associated metric, which is necessarily indefinite. We study spheres and circles, which are given with respect to the associated metric, in some special subspaces of a single tangent space of the manifold.

Non-existence of sets of type $$\mathbf (0, 1, 2 , {\varvec{n}}_{{\varvec{d}}})_{{\varvec{d}}}$$ in PG($${\varvec{r,q}}$$r,q) with $$\mathbf 3 \le {\varvec{d}}\le {\varvec{r}}-\mathbf 1 $$3≤d≤r-1 and $${\varvec{r}}\ge \mathbf 4 $$r≥4

Abstract: This paper deals with sets of type $$(0,1,2,n_{d})_{d}$$ in PG(r, q), $$1\le d\le r-1$$ . The non-existence of sets of type $$(0,1,2,n_{d})_{d}$$ , $$3\le d\le r-1$$ in PG(r, q) with $$r\ge 4$$ is proved.

A compactness theorem in Riemannian manifolds

Abstract: In this paper, we use the m-Bakry–Emery Ricci tensor on a complete n-dimensional Riemannian manifold to obtain a compactness theorem including a diameter estimate. The proof is based on the Riccati comparison theorem.