Showing papers in "Journal of Geometry in 1995"
TL;DR: In this article, the second order parallel tensors and vector fields keeping curvature tensor invariant on Ricci-symmetric contact metric manifolds are characterized on a class of contact manifolds.
Abstract: Some results on Ricci-symmetric contact metric manifolds are obtained. Second order parallel tensors and vector fields keeping curvature tensor invariant are characterized on a class of contact manifolds. Conformally flat contact manifolds are studied assuming certain curvature conditions. Finally some results onk-nullity distribution of contact manifolds are obtained.
38 citations
TL;DR: In this article, a characterisation of the Buekenhout-Metz unitals in PG(2,q2) was given for q even and q = 3.
Abstract: In [7], for the casesq even andq=3, a characterisation of the Buekenhout-Metz unitals inPG(2,q2) was given. We complete this characterisation by proving the result forq>3.
31 citations
TL;DR: A class of cubic curves which are invariant under the operation of isogonal conjugacy has attracted much attention, and a class of these is investigated by a variety of methods as mentioned in this paper.
Abstract: There are many interesting cubic curves which arise from the geometry of the triangle. In particular, those which are invariant under the operation of isogonal conjugacy have attracted much attention, and a class of these is here investigated by a variety of methods.
25 citations
TL;DR: In this paper, the dual-projective curvature tensor for a connection semi-compatible with the metrich is determined as an invariant of the dual projective group.
Abstract: Given a pseudo Riemannian metrich and a torsion-free affine connection ∇ on a smoothn-manifold M,a dual geodesic curve of ∇ is defined as a curve whose tangent 1-form is parallel along the curve. The corresponding dual-projective group is defined as a group of transformations of connections preserving dual-geodesic curves. The class of connections semi-compatible with the metrich and pairs of semi-conjugate connections are defined using the relations between their geodesics and dual-geodesics. The dual-projective curvature tensor for a connection semi-compatible withh is determined as an invariant of the dual projective group. Dual-projectively flat connections semi-compatible withh are characterized as connections with vanishing dual-projective curvature tensor. As an application we recover the fundamental theorem for non-degenerate hypersurface immersions.
23 citations
TL;DR: In this paper, the results of three computer searches for hyperovals in small projective planes were collated, each of which resulted in new hypervals, including GF(2) o-polynomials.
Abstract: In this paper we collate the results of three computer searches for hyperovals in small projective planes, each of which resulted in new hyperovals. The three searches involve finding all hyperovals with non-trivial automorphisms in PG(2,64), all hyperovals with GF(2) o-polynomials in PG(2, 128) and PG(2, 256) and hyperovals stabilised by a particular group in PG(2, 256).
22 citations
TL;DR: A formula for the Euler number of a generic singular surface in a 3-manifold is given in this article, and an estimate of the number of connected components of the complement of such a surface is given.
Abstract: A formula for the Euler number of a generic singular surface in a 3-manifold is given. We also give an estimate of the number of connected components of the complement of such a surface.
20 citations
TL;DR: In this paper, the problem of determining the surfaces in the Euclidean space which can admit at least one nontrival isometry that preserves the principal curvatures (Bonnet surfaces) was considered locally and examined in the general case.
Abstract: In this paper we deal with the Bonnet problem of determining the surfaces in the Euclidean three dimensional space which can admit at least one nontrival isometry that preserves the principal curvatures(Bonnet surfaces). The problem is considered locally and examined in the general case. The main results are: (a) Necessary and sufficient condition for a surface to be a Bonnet surface is that it admits a special isothermal parameter system. (b) Complete solution of the problem in the class of the isothermic surfaces. Moreover: These results and the methods used provide a new efficient and elegant manner of proving the, already known, fact that all helicoidal surfaces are Bonnet surfaces and determine the already known developable Bonnet surfaces.
19 citations
TL;DR: In this article, the authors introduced the notion of reflection groups with midpoints and associated to each of them an incidence structure, which is an incidence space which dimension can assume any value.
Abstract: The notion “reflection group” (Γ,\(\mathcal{D}\)) was introduced in order to give group theoretical characterizations of absolute planes. Here we consider “reflection groups with midpoints” and associate to each of them an incidence structure\(\mathfrak{G}\). Then\((\mathcal{D},\mathfrak{G})\) is an incidence space which dimension can assume any value. The motion group Γ together with the set\(\mathcal{D}\) of all reflections in points of a Euclidean or hyperbolic geometry are examples of reflection groups with midpoints. Furthermore the set\(\mathcal{D}\) can be turned into a K-loop. The precise results are summarized in the theorems at the end of the paper.
17 citations
TL;DR: In this article, the authors show a construction of the Grassmannian from the Plucker mapping of the projective space, which is called projected Grassmannians, and show that for any field F the projection is impossible exactly forn = 3,4.
Abstract: In this paper we show a construction, inductive onn, of the Grassmannian
$$\mathcal{G}_{n,1,F}$$
representing the lines of the projective spaceP
n,F
:
$$\mathcal{G}_{n,1,F}$$
is the union of those planes, any of which is obtained by joining a line of a certain (n−1)-flat S
n−1 with the corresponding point on
$$\mathcal{G}_{n - 1,1,F}$$
via the Plucker mapping; it is assumed that the
$$\left( {\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right) - 1} \right)$$
-flat spanned by this
$$\mathcal{G}_{n - 1,1,F}$$
is skew with S
n−1. Every embedded Grassmann space can be obtained from a Grassmannian by projecting it on a subspaceS from another one, sayS′ (the vertex of the projection). This is a consequence of the description of all projective embeddings of the Grassmann spaces given by HAVLICEK [4] and WELLS [9]. When the vertexS′ is not empty, we say that the embedded Grassmann space is a projected Grassmannian. Since examples of projected Grassmannians do exist, the embedded
$$\mathcal{G}_{n,h,F}$$
's cannot, in general, be projectively characterized by using only their intrinsic incidence properties. We show that for any fieldF, the projection is impossible exactly forn=3,4. In every other case the dimension of the vertex can depend onF.
17 citations
16 citations
TL;DR: In this paper, it was shown that given any general values for three different bisectors (internal or external) of a triangle, we can not draw the triangle using a ruler and a pair of compasses alone.
Abstract: Leta, b, andc be the three sides of a triangleABC, a i ,b i ,c i anda e ,b e , ce be the lengths of the three internal and external bisectors of the three anglesA, B, andC respectively. It is easy to express the bisectors as formulae of the sides. In this paper, we solve a problem proposed by H. Zassenhaus: for any three different bisectors in {ai, bi, ci, ae, be, ce}, finding the relations between each side of the triangle and the three chosen bisectors. We also prove that given any general values for three different bisectors (internal or external) of a triangle, we can not draw the triangle using a ruler and a pair of compasses alone. The formulae mentioned above are derived automatically using a general method of mechanical formula derivation.
TL;DR: In this paper, the authors prove uncountably many new analytic and geometric isoperimetric inequalities associated with the solutions of second order ODEs and ODE solutions of ordinary differential equations.
Abstract: We prove uncountably many new analytic and geometric isoperimetric inequalities associated with the solutions of second order ordinary differential equations.
TL;DR: In this paper, the authors show that the affine unital associated to the unital of Gruning's unitals is also isomorphic to the hyperbolic hermitian curve, and they give necessary and sufficient conditions for the constructed unitals to be projectively equivalent.
Abstract: A new transformation method for incidence structures was introduced in [8],an open problem is to characterize classical incidence structures obtained by transformation of others. In this work we give some, sufficient conditions to transform, with the procedure of [8],a unital embedded in a projective plane into another one. As application of this result we construct unitals in the Hall planes by transformation of the hermitian curves and we give necessary and sufficient conditions for the constructed unitals to be projectively equivalent. This allows to find different classes of not projectively equivalent Buekenhout's unitals, [2],and to find the class of unitals descovered by Gruning, [4],easily proving its embeddability in the dual of a Hall plane. Finally we prove that the affine unital associated to the unital of [4]is isomorphic to the affine hyperbolic hermitian curve.
TL;DR: In this article, a family of maximal arcs in the Hall planes inherited from the Desarguesian planes were constructed. But not in the even-order Hall planes, as was shown in this paper.
Abstract: In 1974 J.A. Thas constructed a class of maximal arcs in certain translation planes of square order, including the Desarguesian ones, but not the Hall planes. We construct a family of maximal arcs in the Hall planes inherited from the Thas maximal arcs in the Desarguesian planes. In particular, maximal arcs are shown to exist in all Hall planes of even order.
TL;DR: In this paper, a generalization of the 4 vertex theorem for C3 closed simple convex space curves including singular and zero curvature points is presented, which is called the 4-vertex theorem generalization.
Abstract: We prove a generalization of the 4 vertex theorem forC3 closed simple convex space curves including singular and zero curvature points.
TL;DR: In this paper, it was shown that Γ is either isomorphic to the compact exceptional Lie group G2, or the covering dimension of the ternary fixed field is at most 11.
Abstract: Let τ be an eight-dimensional, connected, locally compact ternary field and let Γ denote a connected closed subgroup of its automorphism group which is taken with the compact-open topology. It is proved that Γ is either isomorphic to the compact exceptional Lie group G2, or the (covering) dimension of Γ is at most 11. This bound can be decreased to 10, if the ternary fixed fieldF
Γ of Γ is connected.
TL;DR: In this article, a classification of surfaces in the 3-dimensional non-flat Lorentzian space forms satisfying the equation ∆H = λH was given, which consists of minimal, totally umbilical and B-scroll surfaces.
Abstract: In a series of early papers, with the aim of knowing of the shape of a pseudo-Riemannian hypersurface satisfying a certain differential equation in the induced Laplacian, we found a remarkable family of hypersurfaces in the Lorentz-Minkowski space whose mean curvature vector is an eigenvector of the Laplacian. Actually, the last two authors showed in [8] that the equation ∆H = λH , for a real constant λ, characterizes the family of surfaces in L3 made up of the quite interesting B-scrolls and the so-called standard examples, as well as minimal surfaces. Looking at those results obtained for surfaces in L3, the following geometric question was stated in [9] for hypersurfaces in Ln+1 (n > 2): Does the equation ∆H = λH mean that both the mean and the scalar curvatures of the hypersurface are constant? We were able to give a partial solution to that problem, since we had needed to do an additional hypothesis on the degree of the minimal polynomial of the shape operator. It is worth pointing out that the additional assumption was mainly made to control the position vector field of the hypersurface into Ln+1. Now when the ambient space is a non-flat pseudoRiemannian space form, Sn+1 ν (r) or Hn+1 ν (r), then the hypersurface is of codimension two in Rn+2 ν or R ν+1 , respectively, but Sn+1 ν (r) and Hn+1 ν (r) being both totally umbilical hypersurfaces in the corresponding pseudo-Euclidean space, it seems reasonable to hope for a richer classification of hypersurfaces into those spaces by means of the equation ∆H = λH . Or even, one looks for getting a complete answer to the stated problem in non-flat ambient spaces. In this paper we give a classification of surfaces in the 3-dimensional non-flat Lorentzian space forms satisfying the equation ∆H = λH . We show that the family of such surfaces consists of minimal, totally umbilical and B-scroll surfaces. As for hypersurfaces we suppose that their shape operators have no complex eigenvalues. This condition does not seem as restrictive as one could think, in view of examples and results given in section 5. Actually, we find that family is set up by minimal, totally umbilical and so-called generalized umbilical hypersurfaces, which are nothing but a natural generalization of B-scrolls.
TL;DR: The use of Laguerre transformations to discover theorems in the Euclidean and Minkowski planes has been studied in this paper, where they have been applied to the discovery of theorements in the following problems:
Abstract: Examples of the use of Laguerre transformations to discover theorems in the Euclidean and Minkowski planes.
TL;DR: In this article, the smallest integer g(n,d) such that any setS of at leastg n,d points, in general position in Ed, contains n points which are the vertices of an empty convexd-polytopeP, that is, S∩intP = 0.
Abstract: Letn andd be integers,n>d ≥ 2. We examine the smallest integerg(n,d) such that any setS of at leastg(n,d) points, in general position in Ed, containsn points which are the vertices of an empty convexd-polytopeP, that is, S∩intP = 0. In particular we show thatg(d+k, d) = d+2k−1 for 1 ≤k ≤ iLd/2rL+1.
TL;DR: In this paper, it was shown that for a nonempty proper subset S of a real topological linear space L that kerS = ∩{convAz: ze bdryS} and for a close connected nonconvex subset S, that kS =∩(convAz : ze slncS), where slncs denotes the sets of boundary points and boundary points of S respectively, and Az = {s e S: z is clearly visible from s via S}.
Abstract: It is proved for a nonempty proper subset S of a real topological linear space L that kerS =∩{convAz: ze bdryS} and for a close connected nonconvex subset S of L that kerS =∩{convAz: ze slncS}, where bdryS and slncS denote the sets of boundary points and strong local nonconvexity points of S, respectively, and Az = {s e S: z is clearly visible from s via S}.This extends previous results and, combined with standard techniques, yields among others two Krasnosel'skii-type characterizations for the dimension of kerS in Rd in case of a nonempty proper set S with bdryS bounded and a closed connected nonconvex set S with lncS bounded.The assumption of boundedness of S turns out to be irrelevant in these criteria.
TL;DR: Hadamard designs which can be associated with a hyperoval of a projective plane of even order have been investigated in this article, where they are shown to contain restrictions that are isomorphic to the 2-design of points and hyperplanes of projective geometry over GF(2).
Abstract: Hadamard designs which can be associated with a hyperoval of a projective plane of even order are investigated. In particular, when Ω is a translation hyperoval, these designs are shown to contain restrictions that are isomorphic to the 2-design of points and hyperplanes of a projective geometry overGF(2).
TL;DR: In this article, a general method allowing the construction of geometries whose diagram is an extension of the diagram of a given geometry is presented, and some applications of this construction process are described.
Abstract: We present a general method allowing the construction geometries whose diagram is an extension of the diagram of a given geometry. Some applications of this construction process are described.
[...]
TL;DR: In this paper, the first step towards the classification of finite flag-transitive geometries of rank 3 with affine planes and dual affine point-residues was taken.
Abstract: This paper is intended to be a first step towards the classification of finite flag-transitive geometries of rank 3 with affine planes and dual affine point-residues. We describe those of diameter 1. In the case of diameter > 1, we describe minimal quotients, assuming that the number of lines through two points is large enough.
TL;DR: In this paper, it was shown that a contact form with riemannian characteristic flow is k-contact, which implies the existence of a K-contact form with a prescribed unit Killing vector field as characteristic vector field.
Abstract: We prove that a contact form with riemannian characteristic flow is K-contact. We also present a purely riemannian hypothesis which implies the existence of a K-contact form with a prescribed unit Killing vector field as characteristic vector field. Our hypothesis is weaker than that previously presented by Hatakeyama, Ogawa and Tanno.
TL;DR: In the Euclidean 3-space, complete and connected smooth surfaces having two proper helical geodesics through each point are circular cylinders as discussed by the authors, and they can be represented by a circle.
Abstract: In the Euclidean 3-space, complete and connected smooth surfaces having two proper helical geodesics through each point are circular cylinders.
TL;DR: In this paper, partial linear spaces all of whose lines contain at least three points and in which every pair of intersecting lines generates a subspace isomorphic to a projective or dual affine plane are considered.
Abstract: We consider partial linear spaces all of whose lines contain at least three points and in which every pair of intersecting lines generates a subspace isomorphic to a projective or dual affine plane. In particular, we classify in this paper those spaces that contain a projective plane.
TL;DR: In this paper, the authors studied semidefinite planes of order qn(q=pr,p prime) with a collineation whose order is ap-primitive divisor of qn−1.
Abstract: In this work we study semifield planes of orderqn(q=pr,p prime) with a collineation whose order is ap-primitive divisor ofqn−1.
TL;DR: In this paper, an application of constructions from so-called Bass-Serre theory to group spaces in noncommutative geometry is discussed motivated by the question how to make "visible" the interplay of different structures coming together in the notion of a group space.
Abstract: An application of constructions from so-called Bass-Serre theory to group spaces in noncommutative geometry is discussed motivated by the question how to make “visible” the interplay of different structures coming together in the notion of a group space. A compact summary of basic notions and constructions from Bass-Serre theory up to the structure theorem is developed; it should serve as a brief guideline for further applications. Concluding prospects indicate a way for generalizing the methods from a categorical point of view.