Showing papers in "Journal of Geometry in 1997"
TL;DR: In this article, it is shown that for a Lie incidence geometry of type Bn,n and Cn,1 over a field of characteristic not two, An,k,Dn, 1, Dn, n, E6,1 or E7,1, the concepts of e-indepedence of a point set and a basis of a basis are defined.
Abstract: For a rank two incidence system\((\mathcal{P}, \mathcal{L})\) and a projective embedding\(e:(\mathcal{P}, \mathcal{L}) \to \mathbb{P}\mathbb{G}(V)\) the concepts ofe- indepedence of a point set and a basis of\((\mathcal{P}, \mathcal{L})\) are defined. It is then demonstrated that for a Lie incidence geometry\((\mathcal{P}, \mathcal{L})\) of type Bn,n and Cn,1 over a field of characteristic not two, An,k,Dn,1,Dn,n,E6,1 or E7,1,\(Aut(\mathcal{P}, \mathcal{L})\) is transitive on certain classes of subsets called frames. As a consequence we obtain a characterization of the apartments of these geometries and demonstrate that the frames are bases.
46 citations
TL;DR: In this paper, the authors established some Bonnesen-style isoperimetric inequalities for plane polygons via an analytic isoper-imetric inequality and an isoprocedoric inequality in pseudo-perimeters of polygons.
Abstract: In this paper, we establish some Bonnesen-style isoperimetric inequalities for plane polygons via an analytic isoperimetric inequality and an isoperimetric inequality in pseudo-perimeters of polygons.
22 citations
TL;DR: In this paper, the curvature tensor of indefinite almost contact manifolds is investigated by means of the Jacobi operator along spacelike, timelike and null geodesies, spaces of constant curvature are characterized as well as spaces of pointwise constant ϕ-sectional curvature.
Abstract: The curvature tensor of indefinite almost contact manifolds is investigated. By means of the study of the Jacobi operator along spacelike, timelike and null geodesies, spaces of constant curvature are characterized as well as spaces of pointwise constant ϕ-sectional curvature. As an extension of these conditions we introduce the socalled ϕ-isotropic spaces and show a local classification of such manifolds.
18 citations
TL;DR: In this paper, the authors extend their earlier work on overlarge sets of Fano planes, obtaining three results of particular interest: they find seven new partial geometries pg(8,7,4) and nine new strongly regular graphs, by means of switching cliques of points with spreads of lines.
Abstract: We extend our earlier work on overlarge sets of Fano planes, obtaining three results of particular interest. We find seven new partial geometries pg(8,7,4) and nine new strongly regular graphs, by means of switching cliques of points with spreads of lines. One of these new strongly regular graphs supports four different partial geometries. Then we give a new construction of the recently discovered eightfold cover of the complete graph K16.
18 citations
TL;DR: In this article, the Buekenhout unitals in the Hall plane are studied by deriving PG(2,q2) and observing the effect on the unitals of H(q2).
Abstract: The unitals in the Hall plane are studied by deriving PG(2,q2)and observing the effect on the unitals of PG(2,q2).The number of Buekenhout and Buekenhout-Metz unitals in the Hall plane is determined. As a corollary we show that the classical unital is not embeddble in the Hall plane as a Buekenhout unital and that the Buekenhout unitals of H(q2)are not embeddable as Buekenhout unitals in the Desarguesian plane. Finally, we generalize this technique to other translation planes.
12 citations
TL;DR: In this article, the generalized geodesic coordinates in ann-dimensional Weyl space Wn and the Bianchi identities for Wn were obtained using the prolonged derivative and prolonged covariant derivative of satellites.
Abstract: In this paper, using the prolonged derivative and prolonged covariant derivative of satellites we define the so-called generalized geodesic coordinates in ann-dimensional Weyl space Wn and obtain the Bianchi identities for Wn. We then give the definition of a Recurrent-Weyl space (WnR) and study some properties of hypersurfaces of (WnR). As is well-known, in some papers [1] dealing with physical problems, one usually chooses a fixed metric and develop a theory under this restriction. We hope that the method used in this work will enable one to remove this restriction.
11 citations
TL;DR: In this paper, the authors proved that a constant mean curvature compact embedded surface with planar boundary is a graph near the boundary, over the compact planar domain determined by the boundary.
Abstract: In this note, we prove that a constant mean curvature compact embedded surface with planar boundary, which is a graph near the boundary, over the compact planar domain determined by the boundary, is indeed a graph globally.
9 citations
8 citations
TL;DR: In this paper, the Suzuki-Tits ovoid is defined over a perfect fieidK (or equivalently living inside a self polar symplectic quadrangle) and the Suzuki group has exactly two orbits in the set of lines of PPG(3,K).
Abstract: We characterize in a geometrical way those Suzuki-Tits ovoids which are defined over a perfect fieidK (or equivalently living inside a self polar symplectic quadrangle). We simplify our axioms in the particular cases that (1) the associated Suzuki group has exactly two orbits in the set of lines ofPG(3,K), and (2) the ovoid is finite.
7 citations
TL;DR: Semi-convex spaces as discussed by the authors are a generalization of convexity spaces that are more appropriate for investigating issues of visibility and kernels, and they have been shown to be useful for metrics in the plane.
Abstract: We introduce the notion of a semi-convex space as a unifying framework for the treatment of various notions of convexity in the plane Semi-convex spaces are a generalization of convexity spaces that are more appropriate for investigating issues of visibility We define the notion of visibility within the general framework of semi-convex spaces, and investigate the relationship between visibility, kernels, and skulls We prove the Kernel Theorem and the Cover Kernel Theorem, both of which relate kernels and skulls Based on these results for semi-convex spaces we prove a theorem about metrics in the plane and demonstrate the utility of our theory with two examples of semi-convex spaces based on geodesic convexity and staircase convexity
7 citations
TL;DR: In this article, it was shown that for most of the standard Lie incidence geometries, all geometric hyperplanes arise from a necessarily universal embedding, by adding E7,1 to the list.
Abstract: This paper continues a program to show that for most of the standard Lie incidence geometries, all geometric hyperplanes arise from a necessarily absolutely universal embedding, by addingE7,1 to the list. It follows from [5, 12] that any projective embedding of this point line geometry is a homomorphic image of the one afforded by the 56-dimensional module for the groupE7(K).
TL;DR: In this paper, the authors discuss elliptic Plucker transformations of three-dimensional elliptic spaces, which are permutations on the set of lines such that any two related (orthogonally intersecting or identical) lines go over to related lines in both directions.
Abstract: We discuss elliptic Plucker transformations of three-dimensional elliptic spaces. These are permutations on the set of lines such that any two related (orthogonally intersecting or identical) lines go over to related lines in both directions. It will be shown that for “classical” elliptic 3-spaces a bijection of its lines is already a Plucker transformation, if related lines go over to related lines. Moreover, if the ground field admits only surjective monomorphisms, then “bijection” can be replaced by “injection”.
TL;DR: In this article, it was shown that every five-dimensional simply connected naturally reductive space is weakly symmetric in the sense of A.Selberg [11] and that the converse holds for all dimensions not greater than five.
Abstract: We prove that every five-dimensional simply connected naturally reductive space is weakly symmetric in the sense of A.Selberg [11]. (For dimension less than five it was shown in [1] by a different method). We also show that the converse holds for all dimensions not greater than five.
TL;DR: In contrast to what happens for the Grassmann varieties, in the Segre varieties there are primes which are not hyperplane sections as mentioned in this paper, which destroy the order structure of the product spaces.
Abstract: Aprime of a product spaceS (in particular: a Segre variety) is a proper subsetH of the point set ofS, such that any line either is contained inH or meetsH in exactly one point. In contrast to what happens for the Grassmann varieties [1], in the Segre varieties there are primes which are not hyperplane sections. Some of such primes “destroy” the order structure. This is a consequence of the description of the primes that we give in this paper. On the other hand, it is not possible to give an intrinsic characterization of all hyperplane sections. So, we consider some special ones among them, that we call “singular primes”, which allow a satisfactory study of the order structure of the product spaces.
TL;DR: In this paper, the authors completely classified proper slant surfaces with constant Gaussian curvature and nonzero constant mean curvature in C2 into four classes: 1) proper slants with constant curvatures, 2) proper slopes, 3) proper slope slopes and 4) proper curves.
Abstract: In this paper, we completely classify proper slant surfaces with constant Gaussian curvature and nonzero constant mean curvature in C2.
TL;DR: In this paper, a modified theorem of Mobius is considered and its equivalence to the commutativity of the basic field of the projective space is proved, which is the case for the case of this paper.
Abstract: A modified theorem of Mobius (see [1]) is considered. Its equivalence to the commutativity of the basic field of the projective space is proved.
TL;DR: In this paper, the authors construct topological spreads of PG(3,ℝ) which admit one elliptic and no further regulization, and apply a method due to Thas and Walker to determine the group of automorphic collineations.
Abstract: Throughout this paper, the underlying projective space is 3-dimensional and Pappian. A spreadL admits aregulization ∑, if ∑ is a collection of reguli contained inL and if each element ofL, except at most two lines, is contained either in exactly one regulus of ∑ or in all reguli of ∑. Replacement of each regulus of ∑ by its complementary regulus (exceptional lines remain unchanged) produces thecomplementry congruence
L
∑
of
L
with respect to ∑. IfL
∑
is an elliptic linear congruence of lines, then we call ∑ anelliptic regulization. Applying a method due to Thas and Walker we construct topological spreads of PG(3,ℝ) which admit one elliptic and no further regulization. For each of these spreads we determine the group of automorphic collineations. Among others we obtain also spreads which are the complete intersection of a general linear complex of lines and of a cubic complex of lines.
TL;DR: In this paper, upper bounds on the number of points of large complete caps in the plane PG(n, q), (q odd,n≥3) were derived, and the previously known bounds of Hirschfeld and Segre were improved using the cardinality of the second largest complete cap.
Abstract: In this paper upper bounds on the number of points of large complete caps in PG(n, q), (q odd,n≥3) are derived. The previously known bounds of Hirschfeld and Segre are improved using recent bounds on the cardinality of the second largest complete cap in the plane PG(2,q).
TL;DR: In this paper, the authors study submanifolds in the Euclidean space whose mean curvature vector field is a Jacobi field and obtain non-trivial (non-minimal) examples.
Abstract: We study submanifolds in the Euclidean space whose mean curvature vector field is a Jacobi field. First, we characterize them and produce non-trivial (non-minimal) examples and then, we look for additional conditions which imply minimality.
TL;DR: In this article, it was shown that for any p > 2, there is a unique example with ¦B/L¦ =p + 1, where p ≤ 7, and all examples are classical ones.
Abstract: In [1] it is shown that ifB is a minimum size blocking set in PG(2,p),p prime, such that there is a lineL meetingB in at least s|B|−p−p+45/20 points, thenB/L consists ofp, p + 1 orp + 2 points. It has been known for a long time that for anyp > 2 there is a unique example with ¦B/L¦= p [2]. In [3] the authors prove that ¦B/L¦ =p + 1 can only occur whenp ≤7. Here we show that if¦B/L¦= p+2, thenp = 3, 5 or 7, and all examples are classical ones. Besides combinatorial arguments we use polynomials over finite fields and a formula that generalizes the Newton formulae relating power sums and elementary symmetric polynomials.
TL;DR: In this paper, it was shown that any plane convex figure with twice continuously differentiable boundary, different from a circular disk, can be immobilized with three points and that the figure can be fixed with the sufficient secondorder curvature criterion for immobilization holds.
Abstract: In “Immobilization of Smooth Convex Figures” J. Bracho et al [2] prove a conjecture by Kuperberg, namely that any plane convex figure with twice continuously differentiable boundary, different from a circular disk, can be immobilized with three points. Here we extend the theorem to say that the figure can be immobilized firmly (meaning that the sufficient secondorder curvature criterion for immobilization holds), give a proof of their main lemma, show that fixing points are subsets of level sets of certain functions, and prove that for triples firmly fixing a figure, at least one point can be shifted uniquely to adjust for small shifts in the other two and maintain the figure firmly fixed.
TL;DR: In this paper, small complete k-arcs with k−1 points in common with a normal rational curve have been constructed for increasing values of the dimension n, and it is even possible to construct smaller complete arcs.
Abstract: InPG(2,q), small completek-arcs, which have a large number of points in common with a conic have been constructed. This article generalizes these results. InPG(n,q), n ≥ 3, small completek-arcs, which havek−1 points in common with a normal rational curve, are constructed. For increasing values of the dimensionn, it is even possible to construct smaller complete arcs.
TL;DR: In this article, the authors study incidence structures with a sharply point transitve group of automorphisms and state conditions for the possibility to reconstruct the lines from information about the action of the group.
Abstract: We study incidence structures with a sharply point transitve group of automorphisms (i.e., incidence groups), and state conditions for the possibility to reconstruct the lines from information about the action of the group. This treatment generalizes the familiar description of translation planes. It also includes H. Groh's characterization [Abh. Math. Sem. Hamburg48 (1979) 171–202, Theorem 5.2] of the real affine plane and the real half plane. In a second part, we generalize H. Groh's topological construction ofarc planes [loc. cit.] to the discrete case (and arbitrary groups with partitions), and prove that the regular action that was used in the construction yields also an action on the new set of lines (that is, an action by collineations). Finally, we consider possible applications to stable planes.
TL;DR: In this paper, the supremun of the sectional curvature of a submanifold N of a Riemannian or Kaehlerian manifold M such that N is contained in a tube about a sub manifold P of M is given.
Abstract: We give some estimates for the supremun of the sectional curvature of a submanifold N of a Riemannian or Kaehlerian manifoldM such that N is contained in a tube about a submanifoldP ofM. These estimates depend on the sectional curvature ofM, the Weingarten map ofP and the radius of the tube. Then we apply them to get theorems of non immersibility.
TL;DR: In this article, a sufficient condition on W⊂Sn was given such that the map ϕWKn:Kn →P(T Sn), defined by ϕwKn (M)=∪{CwM¦ weW}, is injective.
Abstract: We give a generalization of results obtained in [15]. LetKn denote the set of embedded hypersurfaces in ℝn+1; for all xeSn and MeKn we denote by CxM the apparent contour ofM in the directionx. Then we give a sufficient condition on W⊂Sn such that the map ϕWKn:Kn →P(T Sn) , defined by ϕWKn (M)=∪{CwM¦ weW}, is injective.
TL;DR: In this article, the authors classify affine projectively flat surfaces with non-nondiagonalizable shape operator and show that the shape operator is non-over-the-field invariant.
Abstract: The aim of this paper is to classify affine projectively flat surfaces with nondiagonalizable (over the field ℂ) shape operator.
TL;DR: In this paper, a special class of models of R3-spaces in the sense of Betten is studied, and the continuity of the geometric operations which involve planes and the topology of the space of planes and of line pencils is investigated.
Abstract: We study a special class of models of R3-spaces in the sense of Betten. We single out some of the properties of these models, and use these properties as additional axioms for general R3-spaces. Then we investigate the consequences of these new axioms in general R3-spaces. We prove the continuity of the geometric operations which involve planes, and we characterize the planes in incidence geometric terms. Using these results, we study the topology of the space of planes and of line pencils, and we prove the continuity of collineations. The obtained results are applied to our concrete examples.
TL;DR: In this article, a ternary ring associated with a projective plane with the incidence relation I and coordinatized by a quadrangle (X,Y,Q,E) is defined.
Abstract: In this paper, we use the definitions and notions of [5]: (R,F) denotes a ternary ring associated with a projective plane rc [1,8] with the incidence relation I and coordinatized by a quadrangle (X,Y,Q,E); with X = (0), Q = (0,0) and E = (1,1) (0,1 being two distiguished elements in R) such that, y = F(x,m,k) (x,y)I[m,k],Vx,m,k ER. We use also the definitions of three binary operations defined out of (R,F) as follows [5]: a+b = F(a,l,b), a.b = F(a,b,0) and a*b = F(1,a,b). The line joining the two points P and Q will be denoted by [P Q].
TL;DR: In this article, the concept of n−1-isomorphism between Steiner systems was introduced, which coincides with the notion of isomorphism whenever n = 1. But it is not known whether any two Steiner system S1 and S2 are isomorphic.
Abstract: In this work we introduce the concept of n−1-isomorphism between Steiner systems (this coincides with the concept of isomorphism whenever n=1).Precisely two Steiner systems S1and S2are said to be n−1-isomorphic if there exist n partial systems Si(1),...,Si(n)contained in Si, i∃.{1,2},such that S1(k)and S2(k)are isomorphic for each ke{1,..., n}.The n−1-isomorphisms are also used to study nets replacements, see Ostrom [8], and to study the transformation methods of designs and other incidence structures introduced in [9] and generalized in [1] and [10].