Showing papers in "Journal of Geometry in 2002"
TL;DR: In this paper, the authors studied generic properties of cylindrical helices and Bertrand curves as applications of singularity theory for plane curves and spherical curves and showed that these properties can be applied to Bertrand and plane curves.
Abstract: We study generic properties of cylindrical helices and Bertrand curves as applications of singularity theory for plane curves and spherical curves.
108 citations
TL;DR: In this paper, it was shown that any linear k-blocking set is either a canonical subgeometry or a projection of some canonical sub-geometry, i.e.
Abstract: In this paper we prove that any linear k-blocking set is either a canonical subgeometry or a projection of some canonical subgeometry.
29 citations
TL;DR: In this paper, the authors classify space-like submanifolds in pseudo-Euclidean space which are either T-constant (respectively, N -constant) if the tangential component of their position function has constant length.
Abstract: It is well-known that the position function is the simplest and the most natural geometric object associated with a submanifold in a Euclidean space or, more generally, in a pseudo-Euclidean space. We call a submanifold T-constant (respectively, N-constant) if the tangential component (respectively, the normal component) of its position function has constant length. The main purpose of this paper is to classify space-like submanifolds in pseudo-Euclidean space which are either T-constant or N-constant.
25 citations
TL;DR: This paper shows a procedure to realize a statistical manifold, which is $\alpha$-conformally equivalent to a manifold with an $\ alpha$-transitively flat connection, as a statistical submanifold.
Abstract: In this paper, we show a procedure to realize a statistical manifold, which is \(\alpha\)-conformally equivalent to a manifold with an \(\alpha\)-transitively flat connection, as a statistical submanifold.
22 citations
TL;DR: In this paper, it was shown that every collineation of the Segre product of strongly connected partial line spaces is (up to permutation of indices) the product of collineations of its components.
Abstract: In this paper we prove that every collineation of the Segre product of strongly connected partial line spaces is (up to permutation of indices) the product of collineations of its components (Thm. 1.10). Spaces of pencils are strongly connected, so the claim holds for Segre products of them (Thm. 1.14). In the second part we study the extendability of collineations of Segre products of spaces of pencils under some natural embeddings.
22 citations
TL;DR: In this paper, a generalization of the Fundamental Theorem of Projective Geometry for Grassmannian automorphisms has been proposed, which is closely related to the geometrical interpretation of automomorphisms of classical groups.
Abstract: We study transformations preserving certain linear structure in Grassmannians and give a generalization of the Fundamental Theorem of Projective Geometry. This result is closely related to the geometrical interpretation of automorphisms of classical groups.
19 citations
TL;DR: In this paper, an improved upper bound for the size of caps in finite projective spaces was presented. But this upper bound was only for the case where the maximum size of a cap in the projective space is bounded.
Abstract: This article presents improved upper bounds for the sizes of caps in finite projective spaces. In [13], Nagy and Szonyi obtained an upper bound for the size \( m_2(4,q) \) of a cap in \( PG(4,q) \), q odd. We show that this upper bound can be improved slightly. A similar result is true for the upper bound on the size of caps in \( PG(N,q) \), q even, \( N \geq 5 \). As supplementary results, we prove the uniqueness of the extendability of large caps to complete caps, and improve the formula of Hill which links the maximal size of a cap in \( PG(N-1,q) \) to the maximal size of a cap in \( PG(N,q) \).
19 citations
TL;DR: In this paper, the separation properties of sets with axiomatically defined convexity structures were investigated and a general separation theorem for pairs of convexities was proposed. But this theorem is not applicable to lattices, real vector spaces and modules.
Abstract: The purpose of this paper is to investigate some separation properties of sets with axiomatically defined convexity structures. We state a general separation theorem for pairs of convexities, improving some known results. As an application, we discuss separation properties of lattices, real vector spaces and modules.
11 citations
TL;DR: In this paper, it is proved that a non-degenerate conic in a Baer subplane of PG(2,q2) is a normal rational curve of order 2, 3, or 4 in the Andre/Bruck and Bose representation.
Abstract: The Andre/Bruck and Bose representation ([1], [5,6]) of PG(2,q
2) in PG(4,q) is a tool used by many authors in the proof of recent results. In this paper the Andre/Bruck and Bose representation of conics in Baer subplanes of PG(2,q
2) is determined. It is proved that a non-degenerate conic in a Baer subplane of PG(2,q
2) is a normal rational curve of order 2, 3, or 4 in the Andre/Bruck and Bose representation. Moreover the three possibilities (classes) are examined and we classify the conics in each class.
10 citations
TL;DR: In this article, the authors considered real hypersurfaces of a Kaehler manifold with a time-like normal unit regarding the B-metric and obtained four basic classes of such hypersurface as almost contact B -metric manifolds.
Abstract: There are considered real hypersurfaces of a Kaehler manifold with a time-like normal unit regarding the B-metric and there are obtained four basic classes of such hypersurfaces as almost contact B-metric manifolds. The generated sixteen classes of the considered hypersurfaces are described with respect to the second fundamental form. There is constructed an example of a 3-dimensional manifold of the 11th basic class as a hypersurface of
8 citations
TL;DR: In this article, a Minkowski plane with a centrally symmetric, strictly convex C-1-curve is defined and the existence of left-or right-orthocentric triangles is discussed.
Abstract: Let (A
2
,
$ \mathcal{C} $
) be a Minkowski plane with a centrally symmetric, strictly convex C
1-curve $ \mathcal{C} $
as the unit circle. Then $ \mathcal{C} $
induces in (A
2
,
$ \mathcal{C} $
) a left-orthogonality structure '
$ \dashv $
' by setting tangents of $ \mathcal{C} $
(and their parallels) left-orthogonal to the corresponding radii (and their paralles). If a line g is left-orthogonal to another one h, then h is right-orthogonal to $ g, (h \vdash g) $
. Based on those concepts of orthogonality in (A
2
,
$ \mathcal{C} $
) left- and right-altitudes of a triangle are defined and one can discuss the existence of left- or right-orthocentric triangles. In general Minkowski planes these concepts of orthocenters are independent of a third type of a triangle-orthocenter, which is based on a circle-geometric definition due to Asplund and Grunbaum, c.f. [1].¶¶Further results are the following: In every plane A
2
,
$ \mathcal{C} $
there exist triplets of directions $ \overline{g}_i $
such that the triangles $ \mathcal {T} $
having sides g
i
parallel to $ \overline{g}_i $
are left-orthocentric. A plane A
2
,
$ \mathcal{C} $
is euclidean, iff each triangle $ \mathcal {T} $
is left-orthocentric. Constructing the altitudes of an altitude-triangle of a non (left- or right-)-orthocentric triangle $ \mathcal {T} $
starts iteration processes with attractors (resp. repulsors) which can be called ‘limit orthocenters’ to the given triangle $ \mathcal {T} $
.
TL;DR: In this article, it was shown that for commutative Dickson planes, the sharply transitive normal subgroup K of a collineation group G of a translation plane is transitive.
Abstract: A parabolic unital \( \cal U \) of a translation plane is called transitive, if the collineation group G fixing \( \cal U \) fixes the point at infinity of \( \cal U \) and acts transitively on the affine points of \( \cal U \). It has been conjectured that if a transitive parabolic unital \( \cal U \) consists of the absolute points of a unitary polarity in a commutative semi-field plane, then the sharply transitive normal subgroupK of G is not commutative. So far, this has been proved for commutative twisted field planes of odd square order, see [1],[5]. Here we prove this conjecture for commutative Dickson planes.
TL;DR: The Gergonne center of a triangle is defined as the intersection of the cevians through the points where the incircle touches the sides of the triangle as mentioned in this paper, which does not admit a direct generalization to tetrahedra.
Abstract: The Gergonne center of a triangle is the intersection of the cevians through the points where the incircle touches the sides. This does not admit a direct generalization to tetrahedra since the cevians of a tetrahedron through the points where the insphere touches the faces are not necessarily concurrent. This article introduces an alternative definition of the Gergonne center that coincides with the previous definition for the triangle and that admits a generalization to tetrahedra. The same is done for the Nagel center.
TL;DR: It is shown that the general degree of Veronese surface forms a singular subvariety of certain algebraic varieties of certaingebraic varieties.
Abstract: Recently, the classical Veronese surface reemerged in the context of the application oriented field of Computer Aided Geometric Design due to its interesting relation to rational triangular Bezier surfaces. This motivated the investigation of Veronese varieties presented in this paper. It is shown that the general degree
$ n $
Veronese surface forms a singular subvariety of certain algebraic varieties. This general result is then further examined and extended in the low-degree cases
$ n=2,3,4 $
.
TL;DR: In this article, the existence of real light-like hypersurfaces of an indefinite quaternion Kaehler manifold has been shown to be non-existence under some conditions.
Abstract: In this paper, we introduce real lightlike hypersurfaces of indefinite quaternion Kaehler manifold Fundamental properties of real lightlike hypersurfaces of an indefinite quaternion Kaehler manifold are investigated We prove the non existence of real lightlike hypersurfaces in indefinite qaternionic space form under some conditions
TL;DR: In this paper, the natural opposition relation on the set of flags of a generalized polygon was characterized and a new definition of generalized polygons was given. And the relation on any rank 2 geometry of finite diameter is equivalent to the opposition relation in a general polygon.
Abstract: In this paper, we characterize the natural opposition relation on the set of flags of a generalized polygon. We also investigate when a certain relation on any rank 2 geometry of finite diameter is equivalent to the opposition relation in a generalized polygon. As a consequence we obtain a new definition of generalized polygons. Finally, we also characterize the opposition relation in twin trees, which are the analogues of polygons with infinite diameter.
TL;DR: In this paper, the authors examined the bifurcation set in the space of linear transformations for the number of vertices on the image curve, showing that generally there is a codimension-one set of linear transformation making an arbitrary point into a vertex, and obtaining conditions that the point be capable of being transformed into a higher vertex.
Abstract: The number of vertices of a smooth Jordan curve with nowhere vanishing curvature can change under the action of a nonsingular real linear transformation. We examine the bifurcation set in the space of linear transformations for the number of vertices on the image curve, showing that generally there is a codimension-one set of linear transformations making an arbitrary point into a vertex, and obtaining conditions that the point be capable of being transformed into a higher vertex. We demonstrate that there is always an open set of linear transformations such that the image curves have at least six vertices.
TL;DR: In this article, the authors investigated the one-parametric set of projective subspaces that is generated by a set of rational curves in projective relation and connected the algebraic degree of the set with the dimension of all rational curves that can be used to generate them.
Abstract: We investigate the one-parametric set \( \mathbb{G} \) of projective subspaces that is generated by a set of rational curves in projective relation. The main theorem connects the algebraic degree \( \delta \) of \( \mathbb{G} \), the number of degenerate subspaces in \( \mathbb{G} \) and the dimension of the variety of all rational curves that can be used to generate \( \mathbb{G} \). It generalizes classical results and is related to recent investigations on projective motions with trajectories in proper subspaces of the fixed space.
TL;DR: In this article, it was shown that four Griffiths points are collinear if the vertices of a given quadrangle are on a circle, based on Desargues theorem.
Abstract: J. Tabov has proved [1] that four Griffiths points are collinear if the vertices of a given quadrangle are on a circle. In this article we prove some generalization of this result in a very simple geometrical way (based on Desargues theorem).
TL;DR: In this article, it was shown that a 2-net can be embedded in a half-ordered chain structure in the sense that the set of chains is not empty and there are two halforders of the set K which are related, and vice versa, if there are given two related half orders of K then there exists a splitting of K by K.
Abstract: Let \( (\mathcal{P},\frak{G}_{1}, \frak{G}_{2}) \) be a 2-net where the set \(\) of chains is not empty and let Ks be a splitting of \(\mathcal{P}\) by a chain \( K\in\frak{C} \). Then there correspond two halforders \(\xi_{l},\xi_{r}\) of the set K which are related, and vice versa, if there are given two related halforders of K then there exists a splitting of \( \mathcal{P} \) by K. The questions "when is \( \xi_{l}=\xi_{r} \)?", "when is \( \xi_{l} \)convex or an order?" will be studied. Moreover it will be shown that \( (\mathcal{P},\frak{G}_{1}, \frak{G}_{2},\{K_{s}\}) \) can be embedded in a halfordered chain structure in the sense of [1].
TL;DR: In this article, α-proportional and proportional nets in n-dimensional Weyl space Wn and Riemannian space Vn were introduced, and the fundamental forms of the spaces WN and Vn in the parameters of the αproportions were found.
Abstract: α-proportional and proportional nets in n-dimensional Weyl space Wn and Riemannian space Vn are introduced. The fundamental forms of the spaces Wn and Vn in the parameters of the α-proportional and proportional nets are found. The theorem - Weyl space \( W_n, n>2 \) which contains a proportional net is Riemannian Vn - is proved.
TL;DR: In this paper, the authors studied the eigenvectors of Legendre submanifolds and contact CR-hypersurfaces in a CR-surface with a mean curvature vector.
Abstract: Legendre submanifolds and contact CR-hypersurfaces in \( \mathbf{R}^{2n+1}(-3) \), whose \(\phi\)-mean curvature vectors are eigenvectors, are studied.
TL;DR: In this paper, the integrability conditions for a natural distribution determined by principal distributions contained in the distribution were investigated and a curvature-adapted real hypersurface M with constant principal curvatures in a quaternionic hyperbolic space was characterized.
Abstract: In this paper we study the quaternionic distribution \({\cal D}\) of a curvature-adapted real hypersurface M with constant principal curvatures in a quaternionic hyperbolic space \( \mathbb{H} H^{n} \). We investigate integrability conditions for some natural distribution determined by means of principal distributions contained in the distribution \({\cal D}\) and give a characterization of these real hypersurfaces in \( \mathbb{H} H^{n} \).
TL;DR: A short proof for the main result of [1] is given in this article, showing that a unital in PG(2,q) is classical if and only if it is preserved by a cyclic linear collineation group of order q - \sqrt{q} + 1 \.
Abstract: A short proof is given for the main result of [1]:¶¶THEOREM 1. A unital in PG(2,q) is classical if and only if it is preserved by a cyclic linear collineation group of order \( q - \sqrt{q} +1 \).
TL;DR: In this paper, the authors define the class of affine partial line spaces and characterize the structure of strong subspaces of an arbitrary affine part line space in terms of polygonal paths.
Abstract: In the paper we define the class of affine partial line spaces. The puncted Segre products of affine spaces, defined in the paper, are primary examples of affine partial line spaces which are not simply affine spaces (cf. 2.1 and 2.4). In Section 3 we study the structure of strong subspaces of an arbitrary affine partial line space. Finally, we characterize the class of all puncted Segre products of affine spaces 4.8, 4.9) in terms of polygonal paths.
[...]
TL;DR: The equivalence classes of triangles and tetrahedra with respect to the group of the space dilatations and translations can be expressed by quaternions and ordered pairs of quaternion, respectively as discussed by the authors.
Abstract: The equivalence classes of triangles and tetrahedra with respect to the group of the space dilatations and translations can be expressed by quaternions and ordered pairs of quaternions, respectively These quaternions and ordered pairs of quaternions are called space shapes of triangles and shapes of tetrahedra Using shapes, we discuss the similarity of two tetrahedra and obtain a common way for description of affine invariants of three collinear points and four coplanar points in the Euclidean space We also examine a two-parameter set of tetrahedra with the same centroid
TL;DR: In this article, it was shown that a translation plane of order ≥ 3 admits symmetric index 3 homology groups, with a finite number of exceptions, and a list of possibly exceptional orders was determined.
Abstract: It is shown that a translation plane of order
$ p^t $
which admits two homology groups of order
$ (p^t-1)/3 $
must in fact admit symmetric homology groups of this order. It is further shown that a plane admitting such symmetric index 3 homology groups is, with a finite number of exceptions, a generalized Andre plane. A list of the possibly exceptional orders is determined.
TL;DR: In this article, the authors studied 2n-dimensional conformally flat almost Hermitian manifolds with J-(anti)-invariant Ricci tensor and showed that Ricci's tensor is a Ricci-like tensor.
Abstract: We study some of 2n-dimensional conformally flat almost Hermitian manifolds with J-(anti)-invariant Ricci tensor.
TL;DR: In this article, it was shown that all except one of these degenerations characterize miquelian Laguerre planes, and the remaining degeneration characterizes elation Laguergre planes.
Abstract: For Mobius planes Schaeffer [9] has proved that all seven point degenerations of Miquel's Theorem characterize miquelian Mobius planes. For Laguerre planes we have several degenerations of Miquel's Theorem with eight and seven points. We prove that all except one of these degenerations characterize miquelian Laguerre planes. The remaining degeneration characterizes elation Laguerre planes.
TL;DR: In this paper, the authors consider immersions and construct a subspace of the normal holonomy group, which corresponds to a set of embedded manifolds which are either parallel to f, tubes around f, or in general, partial tubes around the manifold.
Abstract: We consider immersions: $ f:M^{m} \rightarrow {\hbox\mathbb{R}}^n $
and construct a subspace $ \Omega (f) $
of $ {\hbox{\mathbb{R}}}^{n-m} $
which corresponds to a set of embedded manifolds which are either parallel to f, tubes around f or, in general, partial tubes around f. This space is invariant under the action of the normal holonomy group, $ \mathcal{H} ol (f) $
We investigate the case where $ \mathcal{H} ol (f) $
is non-trivial and obtain some results on the number of connected components of $ \Omega (f) $
.