Showing papers in "Journal of Geometry in 2006"
TL;DR: In this paper, the curvature tensor with respect to the affine connection of the total space satisfies the condition (2.12) for statistical manifold with almost contact sturctures.
Abstract: In this paper, we discuss statistical manifolds with almost contact sturctures. We define a Sasaki-like statistical manifold. Moreover, we consider Sasaki-like statistical submersions, and we study Sasaki-like statistical submersion with the property that the curvature tensor with respect to the affine connection of the total space satisfies the condition (2.12).
48 citations
TL;DR: In this paper, it was shown that any smooth (C1 resp.) statistical manifold can be embedded into the space of probability measures on a finite set, and that the smooth manifold can also be realized as a finite dimensional statistical model.
Abstract: In this note we prove that any smooth (C1 resp.) statistical manifold can be embedded into the space of probability measures on a finite set. As a result, we get positive answers to Lauritzen’s question and Amari’s question on a realization of smooth (C1 resp.) statistical manifolds as finite dimensional statistical models.
26 citations
TL;DR: In this paper, the authors obtained the Frenet equations of pseudo null and partially null curves, lying fully in the semi-Euclidean space and classifying all such curves with constant curvatures.
Abstract: In this paper, we obtain the Frenet equations of a pseudo null and a partially null curves, lying fully in the semi–Euclidean space
$$ R^{4}_{2} $$
, and classify all such curves with constant curvatures.
17 citations
TL;DR: In this paper, the authors show that the geometry of PG(3, q) over a non-singular quadric Q4,q of a polygonal quadric can be represented as a set of lines connecting the tangent cones of Q4 and the reguli of the hyperbolic quadrics hyperplane section of Q 4,q.
Abstract: We transfer the whole geometry of PG(3, q) over a non-singular quadric Q4,q of PG(4, q) mapping suitably PG(3, q) over Q4,q. More precisely the points of PG(3, q) are the lines of Q4,q; the lines of PG(3, q) are the tangent cones of Q4,q and the reguli of the hyperbolic quadrics hyperplane section of Q4,q. A plane of PG(3, q) is the set of lines of Q4,q meeting a fixed line of Q4,q. We remark that this representation is valid also for a projective space $${\mathbb{P}}_{{3,{\mathbf{k}}}} $$
over any field K and we apply the above representation to construct maximal partial spreads $${\mathfrak{F}}$$
in PG(3, q). For q even we get new cardinalities for $${\mathfrak{F}}.$$
For q odd the cardinalities are partially known.
13 citations
TL;DR: In this paper, a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M is given.
Abstract: We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when M is a Riemannian manifold and E is the tangent bundle of M endowed with the Levi-Civita connection The proof is based on an explicit geometric construction of the Thom class for 2-plane bundles
10 citations
TL;DR: In this paper, it was shown that the level set f−1(f (p)) is either a smooth curve passing through p, or a graph of two smooth curves tangent at p or a cusp shape, and that the sum of the indexes of the critical points of f equals χ (S), the Euler characteristic of S.
Abstract: In this paper we deal with analytic functions \( f:S \to \mathbb{R} \) defined on a compact two dimensional Riemannian surface S whose critical points are semi degenerated (critical points having a non identically vanishing Hessian). To any element p of the set of semi degenerated critical points Q we assign an unique index which can take the values −1, 0 or 1, and prove that Q is made up of finitely many (critical) points with non zero index and embedded circles. Further, we generalize the famous Morse result by showing that the sum of the indexes of the critical points of f equals χ (S), the Euler characteristic of S. As an intermediate result we locally describe the level set of f near a point p ∈Q. We show that the level set f−1(f (p)) is either a) the set {p}, or b) the graph of a smooth curve passing through p, or c) the graphs of two smooth curves tangent at p or d) the graphs of two smooth curves building at p a cusp shape.
9 citations
TL;DR: In this paper, an almost Hermitian manifold M admits a holomorphically planar conformal vector (HPCV) field (a generalization of a closed conformal field) V.
Abstract: We classify and characterize an almost Hermitian manifold M admitting a holomorphically planar conformal vector (HPCV) field (a generalization of a closed conformal vector field) V . We show that if V is nowhere vanishing and strictly non-geodesic, then it is homothetic and almost analytic. If, in addition,M satisfies Gray’s first condition, then M is Kaehler. For a semi-Kaehler manifold M admitting an HPCV field V we show that either V is closed, or M becomes almost Kaehler and V is homothetic and almost analytic.
8 citations
TL;DR: The translation planes of order 81 admitting SL(2, 5), generated by affine elations, are completely determined as mentioned in this paper, and there are seven mutually non-isomorphic translation planes, of which five are new.
Abstract: The translation planes of order 81 admitting SL(2, 5), generated by affine elations, are completely determined. There are seven mutually non-isomorphic translation planes, of which five are new. Each of these planes may be derived producing another set of seven mutually non-isomorphic translation planes admitting SL(2, 5), where the 3-elements are Baer. Of this latter set, five planes are new.
7 citations
TL;DR: In this article, the general formulae for reflection of a wavefront in a surface are derived and in three special cases explicit descriptions are provided: when the reflecting surface is a plane, when the incoming wave is a spherical, and in each case particular examples are computed exactly and the results plotted to illustrate the outgoing wavefront.
Abstract: Recent investigations of the space of oriented lines in
$$ \mathbb{R}^{3} $$
are applied to geometric optics. The general formulae for reflection of a wavefront in a surface are derived and in three special cases explicit descriptions are provided: when the reflecting surface is a plane, when the incoming wave is a plane and when the incoming wave is spherical. In each case particular examples are computed exactly and the results plotted to illustrate the outgoing wavefront.
6 citations
TL;DR: In this article, the configuration arising from intersection of two Hermitian surfaces in PG(3, q) is described, provided that the linear system they generate contains at least a degenerate variety.
Abstract: We provide a description of the configuration arising from intersection of two Hermitian surfaces in PG(3, q), provided that the linear system they generate contains at least a degenerate variety.
6 citations
TL;DR: In this article, the notion of extended parallelity is introduced in an arbitrary spine space, and rudimentary properties of the obtained geometry are presented, including the horizon and dilatation group relative to this parallelity.
Abstract: The notion of extended parallelity is introduced in an arbitrary spine space, and rudimentary properties of the obtained geometry are presented. The extended parallelity is used in the development of the theory of spine spaces. Also, the horizon and dilatation group relative to this parallelity are examined.
TL;DR: Tight bounds for polygons of miscellaneous shapes are presented: watched guards for monotone and spiral polygons, and vertex watchedguards for star-shaped polygons.
Abstract: The art gallery problem asks how many guards are sufficient to see every point of the interior of a polygon. A set of guards is called watched if each guard itself is seen by at least one of its colleagues. In 1994, Hernandez-Penalver wrote that \( \lfloor {\frac{{2n}} {5}} \rfloor \)watched guards always suffice to guard any polygon with n vertices. However in 2001, Michael and Pinciu, and independently Żylinski, presented a class of polygons that required more than \( \lfloor {\frac{{2n}} {5}} \rfloor \)watched guards – which disproved the Hernandez-Penalver’s result – and they established a new tight bound for watched guards: \( \lfloor {\frac{{3n - 1}} {7}} \rfloor \). Combinatorial bounds for watched guards in orthogonal polygons were independently given by Hernandez-Penalver , and by Michael and Pinciu, who proved the \( \lfloor {\frac{{n}} {3}} \rfloor \) -bound to be tight. In this paper, tight bounds for polygons of miscellaneous shapes are presented: \( \lfloor {\frac{{2n}} {5}} \rfloor \) watched guards for monotone and spiral polygons, and \( \lfloor {\frac{{3n - 1}} {7}} \rfloor \) vertex watched guards for star-shaped polygons.
TL;DR: In this article, it was shown that the total angular measure τ around a point in a two-dimensional Minkowski space need not be 2π, and that τ always lies between ρ 2π and 8.
Abstract: Recently, Dekster introduced a new angle measure for Minkowski spaces according to which the total angular measure τ around a point in a two-dimensional Minkowski space need not be 2π. In this paper, we shall show that while τ need not be 2π, τ always lies between \(\sqrt{2\pi}\) and 8.
TL;DR: In this article, the operation of convolution is defined, which associates with a partial Steiner triple system and an abelian group a new partial STEINER triple system.
Abstract: We define the operation of convolution, which associates with a partial Steiner triple system and an abelian group a new partial Steiner triple system, and we determine conditions under which some fundamental geometric properties remain invariant under the operation of convolution.
TL;DR: In this paper, it was shown that if X and Y are at least three-dimensional and f : X → Y is surjective and preserves equilateral triangles, then f is a similarity transformation.
Abstract: Let (X,||·||) and (Y,||·||) be normed linear spaces, dim X, dim Y ≥ 2. We say that f: X → Y preserves equilateral triangles if for all triples of points x, y, z ∈X with ||x − y|| = ||y − z|| = ||x − z|| we have
$$||f (x) -f (y)||=||f (y)-f (z)||=||f (x)- f (z)||.$$
We prove that if X and Y are at least three-dimensional and f : X → Y is surjective and preserves equilateral triangles, then f is a similarity transformation. We obtain also some new results in case where X = Y is an inner product space with dim X = 2.
TL;DR: For a pair of convex bodies K1 and K2 in Euclidean space, this paper showed that K1 is a translate of K2 if either of the following conditions holds.
Abstract: For a pair of convex bodies K1 and K2 in Euclidean space \( \mathbb{E}^n \), n ≥ 3, possibly unbounded, we show that K1 is a translate of K2 if either of the following conditions holds: (i) the orthogonal projections of K1 on 2-dimensional planes are translates of the respective orthogonal projections of K2, (ii) there are points p1 ∈K1 and p2 ∈K2 such that for every pair of parallel 2-dimensional planesL1and L2 through p1 and p2, respectively, the section K1 ∩ L1is a translate of K2 ∩ L2.
TL;DR: In this article, a functional equation motivated by the classical result of Gaspard Monge is presented and solved completely, which is a first step towards solving the problem of functional equation in general.
Abstract: We present and solve completely a functional equation motivated by a classical result of Gaspard Monge.
TL;DR: In this article, the maximum number of points in a cap of PG(N, q) that contains at least one free pair of points was shown to be sharp for all q when N ≤ 4.
Abstract: We say that two points x, y of a cap C form a free pair of points if any plane containing x and y intersects C in at most three points. For given N and q, we denote by m
2
+
(N, q) the maximum number of points in a cap of PG(N, q) that contains at least one free pair of points. It is straightforward to prove that m
2
+
(N, q) ≤ (qN-1 + 2q − 3)/(q − 1), and it is known that this bound is sharp for q = 2 and all N. We use geometric constructions to prove that this bound is sharp for all q when N ≤ 4. We briefly survey the motivation for constructions of caps with free pairs of points which comes from the area of statistical experimental design.
TL;DR: In this paper, the notion of structure torsion fields for helices is introduced, and the necessary and sufficient condition that they are generated by some Killing vector fields is established.
Abstract: In this paper, we study helices which are orbits of one parameter families of isometries on a symmetric space of rank one. We introduce the notion of structure torsion fields for helices, show the necessary and sufficient condition that they are generated by some Killing vector fields, and study their moduli space.
TL;DR: In this paper, the authors characterize conformal vector fields with respect to the Sasaki metric tensor field on the tangent bundle of a Riemannian manifold of dimension at least three.
Abstract: The purpose of this article is to characterize conformal vector fields with respect to the Sasaki metric tensor field on the tangent bundle of a Riemannian manifold of dimension at least three. In particular, if the manifold in question is compact, it is found that the only conformal vector fields are Killing vector fields.
TL;DR: In this article, it was shown that congruence by dissection of any circular disc C with any square S can be achieved if and only if C contains a contractive map.
Abstract: Let $$ \mathcal{G} $$
be a group of affine transformations of the Euclidean plane
$$ {\mathbb{R}}^2 $$
. Two topological discs D,
$$ {\rm E} \subseteq \mathbb{R}^{2} $$
are called congruent by dissection with respect to
$$ \mathcal{G} $$
if D can be dissected into a finite number of subdiscs that can be rearranged by maps from
$$ \mathcal{G} $$
to a dissection of E. Our main result says in particular that
$$ \mathcal{G} $$
admits congruence by dissection of any circular disc C with any square S if and only if
$$ \mathcal{G} $$
contains a contractive map and all orbits
$$ \mathcal{G}(x) $$
,
$$ x \in \mathbb{R}^{2} $$
, are dense in
$$ \mathbb{R}^{2} $$
. In this case any two discs D and E are congruent by dissection with respect to
$$ \mathcal{G} $$
and every disc D is congruent by dissection with n copies of D for every n ≥ 2. Moreover, we give estimates on minimal numbers of pieces that are needed to realize congruences by dissection.
TL;DR: In this article, it was shown that for any homogeneous polynomial f of degree 3 with 0 a regular value of the function can be computed on a closed minimal surface.
Abstract: In this paper we prove that if
$$ M \subset S^{3} $$
is a closed minimal surface, then,
$$ M
e S^{3} \cap f^{{ - 1}} (0) $$
, for any homogeneous polynomial f of degree 3 with 0 a regular value of the function
$$ f| {_{{S^{3} }} :S^{3} \to { \bf R}} . $$
.
TL;DR: In this article, every semi-definite plane with spread in PG(3,K), where K is a field admitting a quadratic extension K+, is shown to admit a transitive parabolic unital.
Abstract: Every semifield plane with spread in PG(3,K), where K is a field admitting a quadratic extension K+, is shown to admit a transitive parabolic unital.
TL;DR: In this article, the authors characterized submanifolds in the Euclidean space whose shape operator satisfies the equation (Aξ)2 ǫ = k||ξ ||2Id, where k  > 0 is constant.
Abstract: The object of this article is to characterize submanifolds $$M \subset \mathbb{R}^{n}$$
of the Euclidean space whose shape operator Aξ satisfies the equation (Aξ)2 = k||ξ ||2Id, where k > 0 is constant.
TL;DR: In this paper, the maximal cardinality of a hyperbolic antipodal set in Euclidean n-space has been studied, where the set is a set of points with the property that through any two of them there is a pair of parallel hyperplanes supporting the set.
Abstract: An antipodal set in Euclidean n-space is a set of points with the property that through any two of them there is a pair of parallel hyperplanes supporting the set. In this paper we discuss the various possible ways to translate this notion to hyperbolic space and find the maximal cardinality of a hyperbolic antipodal set (according to the different definitions).
TL;DR: In this paper, a simplicial convex 2m-polytope with n vertices is shown to be cyclic and satisfies Gale's Evenness Condition, and every subpolytope of P is cyclic.
Abstract: Let P denote a simplicial convex 2m-polytope with n vertices. Then the following are equivalent: (i) P is cyclic; (ii) P satisfies Gale’s Evenness Condition; (iii) Every subpolytope of P is cyclic; (iv) P has at least 2m+2 cyclic subpolytopes with n−1 vertices if n ≥ 2m+5; (v) P is neighbourly and has n universal edges.
TL;DR: In this article, a combinatorial characterization of the Corrado Segre variety of type {n,m} in terms of its incidence structure of points and lines is given.
Abstract: In this paper we give a combinatorial characterization of the Corrado Segre variety of type {n,m} in terms of its incidence structure of points and lines.
TL;DR: In this paper, it was shown that n ≤ p ǫ − 1, where n is the dimension of the geodesic foliation of dimension n and codimension p on a Riemannian manifold.
Abstract: Let $$\mathcal{F}$$
be a totally geodesic foliation of dimension n and codimension p on a Riemannian manifold (M, g). Suppose that g is a bundle-like metric for $$\mathcal{F}$$
and M has at least one point at which none of its mixed sectional curvatures vanishes. Under these conditions we prove that n ≤ p − 1. We show that this inequality is optimal, and none of the above conditions can be removed.
TL;DR: In this paper, a new model of isotropic plane called V-model is built, where V-points are points in the usual sense and V-straight lines are conics.
Abstract: A new model of isotropic plane called V-model is built, where V-points are points in the usual sense and V-straight lines are conics. A relationship with the affine model of isotropic plane is established. Furthermore, conics in the V-model are constructed.
TL;DR: This paper showed that Martin's theorem for the Euclidean 3-or higher-dimensional case does not depend on the Beckman-Quarles theorem, and can be rephrased as a result about axiomatizability and definability.
Abstract: We point out that the results in [12] are the model-theoretic counterpart of results established syntactically in [3] and [10], and that Martin’s theorem for the Euclidean 3- or higher-dimensional case, established in [5], does not depend on the Beckman-Quarles theorem, and can be rephrased as a result about axiomatizability and definability.