Showing papers in "Geometriae Dedicata in 1999"
TL;DR: In this paper, the authors define and study both bi-slant and semisupermanifolds of an almost contact metric manifold and, in particular, of a Sasakian manifold.
Abstract: We define and study both bi-slant and semi-slant submanifolds of an almost contact metric manifold and, in particular, of a Sasakian manifold We prove a characterization theorem for semi-slant submanifolds and we obtain integrability conditions for the distributions which are involved in the definition of such submanifolds We also study an interesting particular class of semi-slant submanifolds
189 citations
TL;DR: A characterization of continuous isometry covariant valuations on convex sets is presented and the main result generalizes previous results of Hadwiger and Hadwinger and Schneider.
Abstract: We present a characterization of continuous isometry covariant valuations on convex sets. The main result generalizes previous results of Hadwiger and Hadwiger and Schneider.
116 citations
Abstract: Ruled real hypersurfaces of complex space forms are investigated by using the fact that such hypersurfaces can be constructed by moving a 1-codimensional complex totally geodesic submanifold of the ambient space along a curve Among other results, a classification of minimal ruled real hypersurfaces and an example of a homogeneous ruled real hypersurface are given
86 citations
TL;DR: In this paper, the authors classify homogeneous surfaces in real and complex affine three-space by choosing affine coordinates so that the surface is defined by a function whose Taylor series is in a preferred normal form.
Abstract: We classify homogeneous surfaces in real and complex affine three-space. This is achieved by choosing affine coordinates so that the surface is defined by a function whose Taylor series is in a preferred normal form.
52 citations
TL;DR: In this paper, the densest packing of 19 congruent circles in a circle with the help of a technique developed by Bateman and Erdos was shown. But the density of the packing was not shown.
Abstract: Dense packings of n congruent circles in a circle were given by Kravitz in 1967 for n = 2,..., 16. In 1969 Pirl found the optimal packings for n ≤ 10, he also conjectured the dense configurations for 11 ≤ n ≤ 19. In 1994, Melissen provided a proof for n = 11. In this paper we exhibit the densest packing of 19 congruent circles in a circle with the help of a technique developed by Bateman and Erdos.
51 citations
TL;DR: In this paper, the authors studied the problem of describing the cone of the effective divisors in the second symmetric product of a curve with general moduli using a degeneration to a rational g-nodal curve.
Abstract: We study the problem of describing the cone of the effective divisors in the second symmetric product of a curve with general moduli using a degeneration to a rational g-nodal curve.
40 citations
TL;DR: In this paper, the concepts of binormal and asymptotic directions for submanifolds embedded with codimension 2 into Euclidean spaces were defined and necessary conditions for the convexity and sphericity of these sub-mansifolds were obtained.
Abstract: We define the concepts of binormal and asymptotic directions for submanifolds embedded with codimension 2 into Euclidean spaces and obtain necessary conditions, in terms of the existence of such directions, for the convexity and the sphericity of these submanifolds.
40 citations
TL;DR: In this article, it was shown that for any two convex open bounded bodies K and T there exists a diffeomorphism f : K → T preserving volume ratio (i.e. with constant determinant of the Jacobian) and such that the Minkowski sum K + T { x + f (x) | x ∈ K }.
Abstract: We prove that for any two convex open bounded bodies K and T there exists a diffeomorphism f : K → T preserving volume ratio (i.e. with constant determinant of the Jacobian) and such that the Minkowski sum K + T { x + f (x) | x ∈ K }. As an application of this method, we prove some of the Alexandov–Fenchel inequalities.
40 citations
TL;DR: In this paper, it was shown that every Akivis algebra is isomorphic to a subalgebra of Ak (B) for a certain algebra B by considering the usual commutator [x, y] = xy − yx and associator A(x,y,z) = (xy)z − x(yz).
Abstract: An Akivis algebra is a vector space V endowed with a skew-symmetric bilinear product [x,y] and a trilinear product A(x,y,z) that satisfy the identity
$$\begin{gathered} [[x,y],z] + [[y,z],x] + [[z,x],y] \hfill \\ = {\mathcal{A}}(x,y,z) + {\mathcal{A}}(y,z,x) + {\mathcal{A}}(z,x,y) - {\mathcal{A}}(y,x,z) - {\mathcal{A}}(x,z,y) - {\mathcal{A}}(z,y,x). \hfill \\ \end{gathered}$$
These algebras were introduced in 1976 by M.A. Akivis as local algebras of three-webs. For any (nonassociative) algebra B one may obtain an Akivis algebra Ak (B) by considering in B the usual commutator [x,y] = xy − yx and associator A(x,y,z) = (xy)z − x(yz). Akivis posed the problem whether every Akivis algebra is isomorphic to a subalgebra of Ak (B) for a certain B. We prove that this problem has a positive answer.
34 citations
TL;DR: In this article, a revision of Albert's results using a mixture of group-theoretical and algebraic methods was developed using an autotopism group on the non-vertex points of one or more sides of a generalized twisted field plane.
Abstract: We investigate autotopisms and isotopisms of generalized twisted field planes. A revision of Albert′s results is developed using a mixture of group-theoretical and algebraic methods. In particular, transitivity conditions of the autotopism group on the nonvertex points of one or more sides of the autotopism triangle are determined. Also, correlations and polarities in generalized twisted field planes are examined in detail. These results allow us to obtain several improvements of recent results in semifield planes.
34 citations
TL;DR: In this article, the authors studied three and four-dimensional Riemannian manifolds with Ricci-curvature homogeneous eigenvalues, that is, having constant Ricci eigen values.
Abstract: The aim of this paper is to study three- and four-dimensional Einstein-like Riemannian manifolds which are Ricci-curvature homogeneous, that is, have constant Ricci eigenvalues. In the three-dimensional case, we present the complete classification of these spaces while, in the four-dimensional case, this classification is obtained in the special case where the manifold is locally homogeneous. We also present explicit examples of four-dimensional locally homogeneous Riemannian manifolds whose Ricci tensor is cyclic-parallel (that is, are of type A) and has distinct eigenvalues. These examples are invalidating an expectation stated by F. Podesta and A. Spiro, and illustrating a striking contrast with the three-dimensional case (where this situation cannot occur). Finally, we also investigate the relation between three- and four-dimensional Einstein-like manifolds of type A and D'Atri spaces, that is, Riemannian manifolds whose geodesic symmetries are volume-preserving (up to sign).
TL;DR: In this paper, the smallest number of homothetic "reduced copies" of a convex body by which it is possible to cover the whole body was shown to be 16.
Abstract: Let A be a convex body in Euclidean space E3. We denote by H(A) the smallest number of homothetic 'reduced copies′ of A by which it is possible to cover the whole of A. The conjecture of Hadwiger is H(A) ≤ 8. We prove that H(A) ≤ 16.
TL;DR: In this paper, a polynomial expansion for measure-valued functionals which are translation covariant on the set of convex bodies is presented, where the coefficients are measures on product spaces.
Abstract: We prove a polynomial expansion for measure-valued functionals which are translation covariant on the set of convex bodies. The coefficients are measures on product spaces. We then apply this construction to the curvature measures of convex bodies and obtain mixed curvature measures for bodies in general relative position. These are used to generalize an integral geometric formula for nonintersecting convex bodies. Finally, we introduce support measures relative to a quite general structuring body B and describe connections between the different types of measures.
TL;DR: In this article, Paoluzzi and Zimmermann constructed a family of compact orientable hyperbolic 3-manifolds with totally geodesic boundary, and classified them up to homeomorphism.
Abstract: L. Paoluzzi and B. Zimmermann constructed a family of compact orientable hyperbolic 3-manifolds with totally geodesic boundary, and classified them up to homeomorphism. Our main purpose is to determine the canonical decompositions of these manifolds. Using the result, we can obtain an alternative proof of the classification theorem of these manifolds and determine their isometry groups. We also determine their unknotting tunnels. Some of these manifolds are related to certain spatial graphs, so-called Suzuki′s Brunnian graphs. The properties of these manifolds enable us to obtain those of the graphs. Moreover, we give an affirmative answer to Kinoshita′s problem concerning these graphs. In the Appendix, we calculate the volume of these manifolds.
TL;DR: In this article, it was shown that any polyhedron in two dimensions admits a type of potential theoretic skeleton called mother body, and that the mother bodies of any polyhedral structure in any number of dimensions are in one-to-one correspondence with certain kinds of decompositions of the polyhedra into convex subpolyhedra.
Abstract: We prove that any polyhedron in two dimensions admits a type of potential theoretic skeleton called mother body. We also show that the mother bodies of any polyhedron in any number of dimensions are in one-to-one correspondence with certain kinds of decompositions of the polyhedron into convex subpolyhedra. A consequence of this is that there can exist at most finitely many mother bodies of any given polyhedron. The main ingredient in the proof of the first mentioned result consists of showing that any polyhedron in two dimensions contains a convex subpolyhedron which sticks to it in the sense that every face of the subpolyhedron has some part in common with a face of the original polyhedron.
TL;DR: In this article, a necessary condition for the existence of such a tiling for Zn when n ≥ 2 was given. And this condition is sufficient when n=2. But it is not sufficient when Zn ≥ 3.
Abstract: We investigate tilings of the integer lattice in the Euclidean n-dimensional space. The tiles considered here are the union of spheres defined by the Manhattan metric. We give a necessary condition for the existence of such a tiling for Zn when n ≥ 2. We prove that this condition is sufficient when n=2. Finally, we give some tilings of Zn when n ≥ 3.
TL;DR: In this paper, it was shown that a spacelike surface in L 3 with non-zero constant mean curvature and foliated by pieces of circles in spacelite planes is a surface of revolution.
Abstract: We prove that a spacelike surface in L 3 with nonzero constant mean curvature and foliated by pieces of circles in spacelike planes is a surface of revolution. When the planes containing the circles are timelike or null, examples of nonrotational constant mean curvature surfaces constructed by circles are presented. Finally, we prove that a nonzero constant mean curvature spacelike surface foliated by pieces of circles in parallel planes is a surface of revolution.
TL;DR: A left-symmetric algebra is said to be filiform if both natural descending series induced by the product tend to zero, but as slow as possible as discussed by the authors, which is the analogue of the more common notion for Lie algebras.
Abstract: A left-symmetric algebra is said to be filiform if both natural descending series induced by the product tend to zero, but as slow as possible. This is the analogue of the more common notion for Lie algebras. In this paper, we develop some theoretical aspects of this class of left-symmetric algebras and we also present a classification of them up till dimension 5.
TL;DR: In this article, a weighted version of the generalized theorem of Napoleon was extended to higher dimensions, leading to a natural configuration of (n−1)-spheres corresponding with T by an almost arbitrarily chosen point.
Abstract: The famous theorem of Napoleon was recently extended to higher dimensions. With the help of weighted vertices of an n-simplex T in \(\mathbb{E}^n\), n ≥ 2, we present a weighted version of this generalized theorem, leading to a natural configuration of (n−1)-spheres corresponding with T by an almost arbitrarily chosen point. Besides, the Euclidean point of view, also affine aspects of the theorem become clear, and in addition a critical discussion on the role of the Fermat–Torricelli point in this framework is given.
TL;DR: In this paper, the authors consider sets of planes of a projective space such that any two planes of the planes intersect in exactly one point and show that these sets can be classified in most cases.
Abstract: Let ℘ be a projective space. In this paper we consider sets ℰ of planes of ℘ such that any two planes of ℰ intersect in exactly one point. Our investigation will lead to a classification of these sets in most cases. There are the following two main results:
TL;DR: In this paper, the notion of flex curve F(ℙ)(f; P) at a nonsingular point P of a plane curve Ca was defined. And the moduli space of projective curves of degree 12 with 27 cusps has at least three irreducible components.
Abstract: In this paper, we define the notion of the flex curve F(ℙ)(f; P) at a nonsingular point P of a plane curve Ca. We construct interesting plane curves using a cyclic covering transform, branched along F(ℙ)(f; P). As an application, we show the moduli space of projective curves of degree 12 with 27 cusps has at least three irreducible components. Simultaneously, we give an example of Alexander-equivalent Zariski pair of irreducible curves.
TL;DR: In this paper, the necessary and sufficient condition for G-invariant metrics g on ℝn\{0} to be extendend to the origin was given.
Abstract: Let G be one of the connected subgroups of the orthogonal group of ℝn which acts transitively on the unit sphere Sn−1. We get the necessary and sufficient condition for G-invariant metrics g on ℝn\{0} to be extendend to the origin. For n=2 this is a classical result of Berard–Bergery. The curvature tensor and the sectional curvature of any such Riemannian G-manifold (ℝn, g) are described in terms of the length of the Killing vector fields, as well as the second fundamental form of the regular orbits G(P)=Sn−1. As an application we describe all G-invariant metrics which are Kahler, hyperKahler or have constant principal curvatures. Some of these results are generalized to the case of any cohomogeneity one G-manifold which, in a neighbourhood of a singular orbit, can be identified with a twisted product.
TL;DR: In this paper, a general integrability result for matched pairs of Lie algebroids was proved for the case of Poisson Lie groups, and the method used is an extension of a method introduced by Lu and Weinstein in the case by using double groupoids which satisfy an etale form of the vacancy condition.
Abstract: We prove a general integrability result for matched pairs of Lie algebroids. (Matched pairs of Lie algebras are also known as double Lie algebras or twilled extensions of Lie algebras.) The method used is an extension of a method introduced by Lu and Weinstein in the case of Poisson Lie groups, and yields double groupoids which satisfy an etale form of the vacancy condition.
TL;DR: In this paper, a graph theoretic analogue of Cheng's eigenvalue comparison theorems for the Laplacian of complete Riemannian manifolds is given.
Abstract: We give a graph theoretic analogue of Cheng's eigenvalue comparison theorems for the Laplacian of complete Riemannian manifolds. As its applications, we determine the infimum of the (essential) spectrum of the discrete Laplacian for infinite graphs.
TL;DR: In this paper, a class of compact Kahlerian solvmanifolds is defined, namely, a finite quotient of a complex torus that is a holomorphic fiber bundle over the torus with fiber a complex manifold, and it is shown that under some restriction, a compact solvmannifold is kahlerian if and only if it belongs to this class.
Abstract: We give some explicit examples of compact Kahlerian solvmanifolds and, by extending them naturally, we find a class of compact Kahlerian solvmanifolds; namely, a finite quotient of a complex torus that is a holomorphic fiber bundle over a complex torus with fiber a complex torus. Then we see that under some restriction a compact solvmanifold is Kahlerian if and only if it belongs to this class. We are thus led to a conjecture that this result would hold without any restriction.
TL;DR: The limit Area/Length for a sequence of convex sets expanding over the whole hyperbolic plane is less than or equal 1, and exactly 1 when the sets considered are convex with respect to horocycles as mentioned in this paper.
Abstract: It is known that the limit Area/Length for a sequence of convex sets expanding over the whole hyperbolic plane is less than or equal 1, and exactly 1 when the sets considered are convex with respect to horocycles. We consider geodesics and horocycles as particular cases of curves of constant geodesic curvature λ with 0 ≥ λ ≤ 1 and we study the above limit Area/Length as a function of the parameter λ.
TL;DR: In this article, a conjecture that might strengthen the Brunn-Minkowski inequality was formulated and discussed, and the conjecture was shown to strengthen the Minkowski-Brunn inequality.
Abstract: We formulate and discuss a conjecture that might strengthen the Brunn–Minkowski inequality.
TL;DR: In this paper, Liouville-type theorems for p-harmonic maps (p ≥' 2) between Riemannian manifolds with convex functions are investigated.
Abstract: p-harmonic maps (p ≥' 2) between Riemannian manifolds are investigated. Some theorems of Liouville type are given for such maps when target manifolds have convex functions.
TL;DR: In this paper, the authors studied the manifold of complete flags of a compact simple Lie group G and showed that the projective spaces of planar normal sections on a natural embedding of a flag manifold M are invariant to the torus action.
Abstract: We continue the study of the variety X[M] of planar normal sections on a natural embedding of a flag manifold M. Here we consider those subvarieties of X[M] that are projective spaces. When M=G/T is the manifold of complete flags of a compact simple Lie group G, we obtain our main results. The first one characterizes those subspaces of the tangent space T[T] (M), invariant by the torus action and which give rise to real projective spaces in X[M]. The other one is the following. Let \(\mathfrak{p}\) be the tangent space of the inner symmetric space G/K at [K] . Then RP (\(\mathfrak{p}\)) is maximal in X[M] if and only if π2(G/K) does not vanish.
TL;DR: In this article, a new optimal inequality for affine spheres for which the corresponding equality is assumed has been found, which complements the classification of the elliptic case of the affine case.
Abstract: In analogy to an inequality of Chen [2], Scharlach and co-workers [7] have found a new, optimal inequality for (equi-) affine spheres. We classify those three-dimensional hyperbolic affine spheres for which the corresponding equality is assumed. This complements the classification of the elliptic case [3].