Showing papers in "Journal of Geometry in 2012"
TL;DR: In this paper, the first positive eigenvalue of the sub-Laplacian takes the smallest possible value, up to a homothety of the pseudohermitian structure, and the manifold is the standard Sasakian unit sphere.
Abstract: We prove a CR version of the Obata’s result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian manifold which satisfies a Lichnerowicz type condition and has a divergence free pseudohermitian torsion. We show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value then, up to a homothety of the pseudohermitian structure, the manifold is the standard Sasakian unit sphere. We also give a version of this theorem using the existence of a function with traceless horizontal Hessian on a complete, with respect to Webster’s metric, pseudohermitian manifold.
26 citations
TL;DR: In this paper, the authors studied almost contact curves of type AW(k) in normal almost contact metric 3-manifolds and gave natural equations of planar biminimal curves.
Abstract: We study almost contact curves in normal almost contact metric 3-manifolds satisfying \({\triangle{H} = \lambda{H}}\) or \({\triangle^\bot {H} = \lambda{H}}\) . Moreover we study almost contact curve of type AW(k) in normal almost contact metric 3-manifolds. We give natural equations of planar biminimal curves.
16 citations
TL;DR: In this article, the authors studied the translation surfaces in the 3D Euclidean and Lorentz-Minkowski space under the condition that the surface of the translation surface satisfying the preceding relation is a surface of Scherk.
Abstract: In this paper we study the translation surfaces in the 3-dimensional Euclidean and Lorentz-Minkowski space under the condition $${\Delta ^{III}r_{i} = \mu _{i}r_{i},\mu _{i} \in \mathbb{R}}$$
, where Δ
III
denotes the Laplacian of the surface with respect to the third fundamental form III. We show that in both spaces a translation surface satisfying the preceding relation is a surface of Scherk.
16 citations
TL;DR: In this article, the singularities and geometric properties of pseudo-spherical evolutes of curves on a spacelike surface in three dimensional Lorentz-Minkowski space were investigated.
Abstract: In this paper we introduce the notion of pseudo-spherical evolutes of curves on a spacelike surface in three dimensional Lorentz–Minkowski space which is analogous to the notion of evolutes of curves on the hyperbolic plane. We investigate the singularities and geometric properties of pseudo-spherical evolutes of curves on a spacelike surface.
16 citations
TL;DR: In this article, the authors classify the factorable surfaces in the three-dimensional Euclidean space under the condition Δri = λiri, where Δ denotes the Laplace operator.
Abstract: In this paper we classify the factorable surfaces in the three-dimensional Euclidean space \({\mathbb{E}^{3}}\) and Lorentzian \({\mathbb{E}_{1}^{3}}\) under the condition Δri = λiri, where \({\lambda_{i}\in\mathbb{R}}\) and Δ denotes the Laplace operator and we obtain the complete classification for those ones.
14 citations
TL;DR: In this paper, the concept of Clifford parallelism was introduced, which consists of all regular spreads of the real projective 3-space whose focal lines form a regulus contained in an imaginary quadric (D1 = Klein's definition).
Abstract: Parallelity in the real elliptic 3-space was defined by W. K. Clifford in 1873 and by F. Klein in 1890; we compare the two concepts. A Clifford parallelism consists of all regular spreads of the real projective 3-space $${{\rm PG}(3,\mathbb{R})}$$
whose (complex) focal lines (=directrices) form a regulus contained in an imaginary quadric (D1 = Klein’s definition). Our new access to the topic ‘Clifford parallelism’ is free of complexification and involves Klein’s correspondence λ of line geometry together with a bijective map γ from all regular spreads of $${{\rm PG}(3,\mathbb{R})}$$
onto those lines of $${{\rm PG}(5,\mathbb{R})}$$
having no common point with the Klein quadric; a regular parallelism P of $${{\rm PG}(3,\mathbb{R})}$$
is Clifford, if the spreads of P are mapped by γ onto a plane of lines (D2 = planarity definition). We prove the equivalence of (D1) and (D2). Associated with γ is a simple dimension concept for regular parallelisms which allows us to say instead of (D2): the 2-dimensional regular parallelisms of $${{\rm PG}(3,\mathbb{R})}$$
are Clifford (D3 = dimensionality definition). Submission of (D2) to λ−1 yields a complexification free definition of a Clifford parallelism which uses only elements of $${{\rm PG}(3,\mathbb{R})}$$
: A regular parallelism P is Clifford, if the union of any two distinct spreads of P is contained in a general linear complex of lines (D4 = line geometric definition). In order to see (D1) and (D2) simultaneously at work we discuss the following two examples using, at the one hand, complexification and (D1) and, at the other hand, (D2) under avoidance of complexification. Example 1. In the projectively extended real Euclidean 3-space a rotational regular spread with center o is submitted to the group of all rotations about o; we prove, that a Clifford parallelism is generated. Example 2. We determine the group $${Aut_e({\bf P}_{\bf C})}$$
of all automorphic collineations and dualities of the Clifford parallelism P
C
and show $${Aut_e({\bf P}_{\bf C})\hspace{1.5mm} \cong ({\rm SO}_3\mathbb{R} \times {\rm SO}_3\mathbb{R})\rtimes \mathbb{Z}_2}$$
.
14 citations
TL;DR: In this article, the authors discuss the asymptotic parametrizations, their Lelieuvre vector fields, and especially the case of constant negative Gaussian curvature.
Abstract: In the category of semidiscrete surfaces with one discrete and one smooth parameter we discuss the asymptotic parametrizations, their Lelieuvre vector fields, and especially the case of constant negative Gaussian curvature. In many aspects these considerations are analogous to the well known purely smooth and purely discrete cases, while in other aspects the semidiscrete case exhibits a different behaviour. One particular example is the derived T-surface, the possibility to define Gaussian curvature via the Lelieuvre normal vector field, and the use of the T-surface’s regression curves in the proof that constant Gaussian curvature is characterized by the Chebyshev property. We further identify an integral of curvatures which satisfies a semidiscrete Hirota equation.
14 citations
TL;DR: The first Betti number of compact nearly cosymplectic manifolds was shown to be zero or even in this article, which is the first known result for a general class of manifolds.
Abstract: We shall show that the first Betti number of some class of compact nearly cosymplectic manifolds is zero or even.
11 citations
TL;DR: In this article, the generalized Bochner curvature tensor is expressed as a linear combination of B1, B2, and B3 such that (6.4) holds.
Abstract: We consider an almost Hermitian manifold and apply the conformal change of metric to its holomorphic curvature tensor. In such a way we find that the generalized Bochner curvature tensor can be expressed as a linear combination of B1, B2, and B3 such that (6.4) holds. Each of the tensors B1, B2, B3 is conformally invariant and satisfies the condition (1.2) of Kahler type.
10 citations
TL;DR: In this article, a generalization of K-contact and (k,μ)-contact manifolds is studied, and it is shown that if such manifolds of dimensions ≥ 5 are conformally flat, then they have constant curvature + 1.
Abstract: We study a generalization of K-contact and (k, μ)-contact manifolds, and show that if such manifolds of dimensions ≥ 5 are conformally flat, then they have constant curvature +1. We also show under certain conditions that such manifolds admitting a non-homothetic closed conformal vector field are isometric to a unit sphere. Finally, we show that such manifolds with parallel Ricci tensor are either Einstein, or of zero $${\xi}$$
-sectional curvature.
10 citations
TL;DR: In this paper, the authors give intrinsic and extrinsic conditions for a linear Weingarten submanifold to be totally umbilical, where R and H are the normalized scalar curvatures and the length of the mean curvature vector respectively.
Abstract: Let M
n
be a spacelike linear Weingarten submanifold in a de Sitter space $${S^{n+p}_{p}(1)}$$
with R = aH + b, where R and H are the normalized scalar curvature and the length of the mean curvature vector respectively. In this paper, we give intrinsic and extrinsic conditions for M
n
to be totally umbilical, respectively.
TL;DR: In this paper, the authors define and classify splints of root systems of complex semisimple Lie algebras and show that splints play a role in determining branching rules of a module over a complex semi-simple Lie algebra when restricted to a subalgebra.
Abstract: We define and classify splints of root systems of complex semisimple Lie algebras. In a few instances, splints play a role in determining branching rules of a module over a complex semisimple Lie algebra when restricted to a subalgebra. In these particular cases, the set of submodules with respect to the subalgebra themselves may be regarded as the character of an auxiliary Lie algebra which may or may not be another Lie subalgebra.
TL;DR: In this paper, it was shown that if an angle in a triangle is increased without changing the lengths of its arms, then the length of the opposite side increases, and that this has a very satisfactory analogue for orthocentric tetrahedra.
Abstract: Propositions 24 and 25 of Book I of Euclid’s Elements state the fairly obvious fact that if an angle in a triangle is increased (without changing the lengths of its arms), then the length of the opposite side increases. In less technical terms, the wider you open your mouth, the farther apart your lips are. In this paper, we see that this has a very satisfactory analogue for orthocentric (but not for general) tetrahedra.
TL;DR: In this paper, it was shown that the minimal weighted t-fold (n − k)-blocking set contained in a PG(n, q) block set always contains a unique blocking set.
Abstract: A weighted t-fold (n − k)-blocking set B of PG(n, q) always contains a minimal weighted t-fold (n − k)-blocking set. We prove that, if \({|B| < (t+1)q^{n-k} + \theta_{n-k-1}}\) , then the minimal weighted t-fold (n − k)-blocking set contained in B is unique.
TL;DR: The Figueroa polar unitals do not satisfy a necessary condition, introduced by Wilbrink in 1983, for a unitary block design to be classical, and hence they are not classical.
Abstract: The finite Figueroa planes are non-Desarguesian projective planes of order q
3 for all prime powers q > 2. These planes were constructed algebraically in 1982 by Figueroa, and Hering and Schaeffer, and synthetically in 1986 by Grundhofer. All Figueroa planes of finite square order are shown to possess a unitary polarity by de Resmini and Hamilton in 1998, and hence admit unitals. Using the result of O’Nan in 1971 on the non-existence of his configuration in a classical unital, and the intrinsic characterization by Taylor in 1974 of the notion of perpendicularity induced by a unitary polarity in the classical plane (introduced by Dembowski and Hughes in 1965), we show that these Figueroa polar unitals do not satisfy a necessary condition, introduced by Wilbrink in 1983, for a unitary block design to be classical, and hence they are not classical.
TL;DR: In this paper, the authors studied submanifolds of an almost complex manifold with Norden metric which are non-degenerate with respect to one Norden measure and lightlike with regard to the other on the manifold.
Abstract: In this paper we study submanifolds of an almost complex manifold with Norden metric which are non-degenerate with respect to the one Norden metric and lightlike with respect to the other Norden metric on the manifold. Relations between the induced geometric objects of some of these submanifolds are given. Examples of the considered submanifolds are constructed.
TL;DR: In this paper, it is shown that the calculus of reflections can be used to introduce a relation of order in hyperbolic geometry, which can be seen as an extension of the axiom system of Menger.
Abstract: It is well known that the calculus of reflections (developed by Hjelmslev, Bachmann et al.) enables the derivation of a large part of Euclidean and non-Euclidean geometry without using assumptions about order and continuity. We show in this article that the calculus of reflections can conversely be used to introduce a relation of order in hyperbolic geometry. Our investigations are based on the famous ‘Endenrechnung’ of Hilbert which was formulated purely in terms of the calculus of reflections by F. Bachmann. We then discuss some implications of these results and show that the calculus of reflections enables (1) the introduction of an order relation in a Pappian projective line and (2) to define an axiom system for hyperbolic planes which seems to be as simple as the famous axiom system of Menger who only used the notion of point-line incidence to axiomatize plane hyperbolic geometry.
TL;DR: In this article, the set of lines which either belong to or are tangent to a non-singular Hermitian surface in the projective space of dimension 3 and order q 2 is characterized.
Abstract: We give a characterization of the set of the lines which either belong to or are tangent to a non-singular Hermitian surface in the projective space of dimension 3 and order q
2.
TL;DR: In this article, the uniqueness of a tangent plane that minimizes the area and then the minimality of the spherical discs among closed subsets with the same spherical area were studied.
Abstract: We consider area minimizing problems for the image of a closed subset in the unit sphere under a projection from the center of the sphere to a tangent plane, the central projection. We show, for any closed subset in the sphere, the uniqueness of a tangent plane that minimizes the area, and then the minimality of the spherical discs among closed subsets with the same spherical area.
TL;DR: In this paper, the authors consider convex improper affine maps of the 3D affine space and classify their singularities using a generating family with properties that closely resemble the area function for non-convex improper maps.
Abstract: In this paper we consider convex improper affine maps of the three-dimensional affine space and classify their singularities. The main tool developed is a generating family with properties that closely resembles the area function for non-convex improper affine maps.
TL;DR: The strongest possible inequalities of the form q(R, r) ≤ s2 ≤ q(r, r), r ≤ QB(R, r) that hold for all triangles becoming equalities for the equilaterals where q, Q real quadratic forms, occur for the Gerretsen forms qB(R r) = 16Rr − 5r2 and QB(r r, r). as discussed by the authors showed that all these minimal forms are strongest in Blundon's sense.
Abstract: Blundon has proved that if R, r and s are respectively the circumradius, the inradius and the semiperimeter of a triangle, then the strongest possible inequalities of the form q(R, r) ≤ s2 ≤ Q(R, r) that hold for all triangles becoming equalities for the equilaterals where q, Q real quadratic forms, occur for the Gerretsen forms qB(R, r) = 16Rr − 5r2 and QB(R, r) = 4R2 + 4Rr + 3r2; strongest in the sense that if Q is a quadratic form and s2 ≤ Q(R, r) ≤ QB(R, r) for all triangles then Q(R, r) = QB(R, r), and similarly for qB(R, r). In this paper we prove that QB (resp. qB) is just one of infinitely many forms that appear as minimal (resp. maximal) elements in the partial order induced by the comparability relation in a certain set of forms, and we conclude that all these minimal forms are strongest in Blundon’s sense. We actually find all possible such strongest forms. Moreover we find all possible quadratic forms q, Q for which q(R, r) ≤ s2 ≤ Q(R, r) for all triangles and which hold as equalities for the equilaterals.
TL;DR: In this paper, the authors define Ceva's triangle as the triangle formed by three cevians each joining a vertex of T to the point which divides the opposite side in the ratio ρ: (1 − ρ).
Abstract: For a given triangle T and a real number ρ we define Ceva’s triangle \({\mathcal{C}_{\rho}(T)}\) to be the triangle formed by three cevians each joining a vertex of T to the point which divides the opposite side in the ratio ρ: (1 – ρ). We identify the smallest interval \({\mathbb{M}_T \subset \mathbb{R}}\) such that the family \({\mathcal{C}_{\rho}(T), \rho \in \mathbb{M}_T}\), contains all Ceva’s triangles up to similarity. We prove that the composition of operators \({\mathcal{C}_\rho, \rho \in \mathbb{R}}\), acting on triangles is governed by a certain group structure on \({\mathbb{R}}\). We use this structure to prove that two triangles have the same Brocard angle if and only if a congruent copy of one of them can be recovered by sufficiently many iterations of two operators \({\mathcal{C}_\rho}\) and \({\mathcal{C}_\xi}\) acting on the other triangle.
TL;DR: In this paper, it was shown that the n-dimensional equizonal ovaloids are analytic when n is even and are of exactly Cn-1 smoothness if n is odd.
Abstract: We show that the n-dimensional equizonal ovaloids are analytic when n is even and are of exactly Cn-1 smoothness when n is odd This substantially improves the previously published result on the smoothness of the even-dimensional equizonal ovaloids and slightly corrects the previously published statement regarding the smoothness of the odd-dimensional equizonal ovaloids Our methods should be generally useful in determining the degree of smoothness of surfaces and hypersurfaces of revolution generated by piecewise-defined profile curves In particular, they include a novel and elegant application of Bernstein’s theory of absolutely monotonic functions
TL;DR: In this article, it was shown that types 12 and 14 can only occur in finite miquelian Minkowski planes of order 3 or 5, and that types 13 and 18 in finite Minkowowski planes can only exist in miquedelian planes.
Abstract: Monica Klein classified Minkowski planes with respect to linearly transitive subgroups of Minkowski homotheties. She obtained 23 possible types. In this paper we investigate Minkowski planes with respect to groups of automorphism of certain Klein types 12 and higher. We show that types 12 and 14 can only occur in finite miquelian Minkowski planes of order 3 or 5, and we provide examples for such groups. Furthermore, we prove that types 13 and 18 in finite Minkowski planes can only occur in miquelian planes.
TL;DR: In this paper, it was shown that convex bodies can be distinguished by their shadow pictures given on any two straight lines, which can be used to distinguish them from any convex body.
Abstract: We prove that convex bodies can be distinguished by their shadow pictures given on any two straight lines.
TL;DR: In this article, the authors apply the Omori-Yau generalized maximum principle to complete spacelike hypersurfaces immersed in a generalized Robertson-Walker spacetime, which is supposed to obey a standard convergence condition.
Abstract: We apply the well know Omori–Yau generalized maximum principle (Omori in J Math Soc Jpn 19:205–214, 1967; Yau in Commun Pure Appl Math 28:201–228, 1975), as well as a suitable extension of it that was established in a joint work with Caminha (Caminha and de Lima in Gen Relat Grav 41:173–189, 2009), in order to investigate Bernstein-type properties of complete spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime, which is supposed to obey a standard convergence condition.
TL;DR: In this article, the intrinsic geometry of Lie hypersurfaces, such as Ricci curvatures, scalar curvatures and sectional curvatures was studied, and the authors showed that the set of all Lie hypergraphs in the complex hyperbolic space is bijective to a closed interval, giving a deformation of homogeneous hypersurface from the ruled minimal one to the horosphere.
Abstract: A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation of homogeneous hypersurfaces from the ruled minimal one to the horosphere. In this paper, we study intrinsic geometry of Lie hypersurfaces, such as Ricci curvatures, scalar curvatures, and sectional curvatures.
TL;DR: In this paper it was proved that the quaternionic Kahler manifolds with the considered metric structure are Einstein for dimension at least 8, and the class of the non-hyper-Kahler quaternion quaternions of the considered type is determined.
Abstract: Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kahler manifolds are considered. Some necessary and sufficient conditions for the investigated manifolds to be isotropic hyper-Kahlerian and flat are found. It is proved that the quaternionic Kahler manifolds with the considered metric structure are Einstein for dimension at least 8. The class of the non-hyper-Kahler quaternionic Kahler manifolds of the considered type is determined.
TL;DR: The Pasch axiom is shown to be equivalent to the conjunction of its outer form with the statement that K ≥ 5 (or K ≥ 3,3) is not planar as mentioned in this paper.
Abstract: The Pasch axiom is shown to be equivalent, given the linear order axioms, to the conjunction of its outer form with the statement that K
5 (or K
3,3) is not planar.
TL;DR: In this paper, the authors consider the various measures on trihedral angles and show that no two of these measures are monotone with respect to each other, and they show that for any measures f, g, there exist trihedral angle α, β, γ, θ such that f(α) > f(β), g(α), f(γ) f(θ), g (γ) > g(γ))
Abstract: We consider the various measures on trihedral angles that have appeared in the literature and we show that no two of these measures are monotone with respect to each other. In other words, for any measures f, g, there exist trihedral angles α, β, γ, θ such that f(α) > f(β), g(α) f(θ), g(γ) > g(θ). This is done through an elementary and systematic method based on multivariable calculus.