Showing papers in "Advances in Geometry in 2016"

Framed curves in the Euclidean space

Abstract: Abstract A framed curve in the Euclidean space is a curve with a moving frame. It is a generalization not only of regular curves with linear independent condition, but also of Legendre curves in the unit tangent bundle. We define smooth functions for a framed curve, called the curvature of the framed curve, similarly to the curvature of a regular curve and of a Legendre curve. Framed curves may have singularities. The curvature of the framed curve is quite useful to analyse the framed curves and their singularities. In fact, we give the existence and the uniqueness for the framed curves by using their curvature. As applications, we consider a contact between framed curves, and give a relationship between projections of framed space curves and Legendre curves.

Inequalities for casorati curvatures of submanifolds in real space forms

Abstract: By using T. Oprea's optimization methods on submanifolds, we give another proof of the inequalities relating the normalized $\delta-$Casorati curvature $\hat{\delta}_c(n-1)$ for submanifolds in real space forms. Also, inequalities relating the normalized $\delta-$Casorati curvature $\delta_C(n-1)$ for submanifolds in real space forms are obtained. Besides, we characterize a kind of Casorati ideal hypersurface of Euclidean 4-space. We also show that this kind of Casorati ideal hypersurface is rigid.

A counterexample to the containment I(3) ⊂ I2 over the reals

Abstract: The purpose of this note is to give counterexamples to the con- tainment I (3) I 2 over the real numbers.

Inequalities for hyperconvex sets

Abstract: Abstract An r-hyperconvex body is a set in the d-dimensional Euclidean space 𝔼d that is the intersection of a family of closed balls of radius r. We prove the analogue of the classical Blaschke–Santaló inequality for r-hyperconvex bodies, and we also establish a stability version of it. The other main result of the paper is an r-hyperconvex version of the reverse isoperimetric inequality in the plane.

Conformal Ricci solitons and related integrability conditions

Abstract: In this paper we introduce, in the Riemannian setting, the notion of conformal Ricci soliton, which includes as particular cases Einstein manifolds, conformal Einstein manifolds and (generic and gradient) Ricci solitons. We provide here some necessary integrability conditions for the existence of these structures that also recover, in the corresponding contexts, those already known in the literature for conformally Einstein manifolds and for gradient Ricci solitons. A crucial tool in our analysis is the construction of some appropriate and highly nontrivial (0, 3)-tensors related to the geometric structures, that in the special case of gradient Ricci solitons become the celebrated tensor D recently introduced by Cao and Chen. A significant part of our investigation, which has independent interest, is the derivation of a number of commutation rules for covariant derivatives (of functions and tensors) and of transformation laws of some geometric objects under a conformal change of the underlying metric.

Simple crystallizations of 4-manifolds

Abstract: Minimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We present a minimal crystallization of the standard PL K3 surface. In combination with known results this yields minimal crystallizations of all simply connected PL 4-manifolds of ``standard'' type, that is, all connected sums of CP2, S-2 x S-2, and the K3 surface. In particular, we obtain minimal crystallizations of a pair of homeomorphic but non-PL-homeomorphic 4-manifolds. In addition, we give an elementary proof that the minimal 8-vertex crystallization of CP2 is unique and its associated pseudotriangulation is related to the 9-vertex combinatorial triangulation of CP2 by the minimum of four edge contractions.

Edges versus circuits: A hierarchy of diameters in polyhedra

Abstract: Author(s): Borgwardt, S; De Loera, JA; Finhold, E | Abstract: The study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation initiated in [5] by introducing a vast hierarchy of generalizations to the notion of graph diameter. This hierarchy provides some interesting lower bounds for the usual graph diameter. After explaining the structure of the hierarchy and discussing these bounds, we focus on clearly explaining the differences and similarities among the many diameter notions of our hierarchy. Finally, we fully characterize the hierarchy in dimension two. It collapses into fewer categories, for which we exhibit the ranges of values that can be realized as diameters.

Neighborly inscribed polytopes and delaunay triangulations

Abstract: We construct a large family of neighborly polytopes that can be realized with all the vertices on the boundary of any smooth strictly convex body. In particular, we show that there are superexponentially many combinatorially distinct neighborly polytopes that admit realizations inscribed on the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, and provide the current best lower bound for the number of combinatorial types of inscribable polytopes (which coincides with the current best lower bound for the number of combinatorial types of polytopes). Via stereographic projections, this translates into a superexponential lower bound for the number of combinatorial types of (neighborly) Delaunay triangulations.

Generalized distance-squared mappings of the plane into the plane

Abstract: We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.

Homogeneous geodesics in pseudo-Riemannian nilmanifolds

Abstract: Fil: del Barco, Viviana Jorgelina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingenieria y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matematica; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Rosario; Argentina

Symmetric subspaces of SE(3)

Abstract: Abstract Being a Lie group, the group SE(3) of orientation preserving motions of the real Euclidean 3-space becomes a symmetric space (in the sense of O. Loos) when endowed with the multiplication µ(g, h) = gh−1g. In this note we classify all connected symmetric subspaces of SE(3) up to conjugation. Moreover, we indicate some of its important applications in robot kinematics.

Sharply 2-transitive groups

Abstract: We give an explicit construction of sharply 2-transitive groups with fixed point free involutions and without nontrivial abelian normal subgroup.

On spherical submanifolds with finite type spherical Gauss map

Abstract: Chen and Lue (2007) initiated the study of spherical submanifolds with nite type spherical Gauss map. In this paper, we rstly prove that a submanifold Mn of the unit sphere Sm−1 has non-mass-symmetric 1-type spherical Gauss map if and only ifMn is an open part of a small n-sphere of a totally geodesic (n + 1)sphere Sn+1 ⊂ Sm−1. Thenwe show that a non-totally umbilical hypersurfaceM of Sn+1 with nonzero constant mean curvature in Sn+1 has mass-symmetric 2-type spherical Gauss map if and only if the scalar curvature curvature ofM is constant. Finally, we classify constant mean curvature surfaces in S3 with mass-symmetric 2-type spherical Gauss map.

On the polyhedrality of global Okounkov bodies

Abstract: We prove that the existence of a finite Minkowski base for Okounkov bodies on a smooth projective variety with respect to an admissible flag implies rational polyhedrality of the global Okounkov body. As an application of this general result, we deduce that the global Okounkov body of a surface with finitely generated pseudo-effective cone with respect to a general flag is rational polyhedral. We give an alternative proof for this fact which recovers the generators more explicitly. We also prove the rational polyhedrality of global Okounkov bodies in the case of certain homogeneous 3-folds using inductive methods.

The second largest Erdős–Ko–Rado sets of generators of the hyperbolic quadrics Q+(4n + 1, q)

Abstract: An Erdos-Ko-Rado set of generators of a hyperbolic quadric is a set of generators which are pairwise not disjoint. In this article we classify the second largest maximal Erdos-Ko-Rado set of generators of the hyperbolic quadrics Q(+)(4 n + 1, q), q >= 3.

Cellular homology of real maximal isotropic Grassmannians

Abstract: Abstract In this paper we obtain a formula for the boundary map of the cellular ℤ-homology of the real maximal isotropic Grassmannians of type B, C and D which are obtained as minimal flag manifolds of split real forms of semi-simple Lie algebras of the respective type. We use the model of shifted Young diagrams presented in [9] to describe the Schubert varieties and the cellular decompositions of such flag manifolds. The results are applied to determine the orientability of the real maximal isotropic Grassmannians.

Comparison theorems on smooth metric measure spaces with boundary

Abstract: Abstract In this paper we study smooth metric measure spaces with boundary via the Bakry–Émery curvature and the weighted mean curvature of the boundary. We establish the weighted Laplacian comparison theorems and the upper bound estimates of the distance from any point of the manifold to its boundary. As applications, we derive lower bound estimates for the first Dirichlet eigenvalue.

On caustics by reflection of algebraic surfaces

Abstract: Given a point S (the light position) in P^3 and an algebraic surface Z (the mirror) of P^3, the caustic by reflection of Z from S is the Zariski closure of the envelope of the reflected lines got by reflection of the incident lines (Sm) on Z at m in Z. We use the ramification method to identify the caustic by reflection with the Zariski closure of the image, by a rational map, of an algebraic 2-covering space of Z. We also give a general formula for the degree (with multiplicity) of caustics (by reflection) of algebraic surfaces.

Balanced and absorbing subsets with empty interior

Abstract: Abstract Our first result says that every real or complex infinite-dimensional normed space has an unbounded absolutely convex and absorbing subset with empty interior. As a consequence, a real normed space is finite-dimensional if and only if every convex subset containing 0 whose linear span is the whole space has non-empty interior. In our second result we prove that every real or complex separable normed space with dimension greater than 1 contains a balanced and absorbing subset with empty interior which is dense in the unit ball. Explicit constructions of these subsets are given.

A fixed point theorem in Euclidean buildings

Abstract: We establish a xed point theorem for a certain type of nonexpanding maps in Euclidean buildings, which is inspired by a theorem of La aille in p-adic Hodge theory [10, Theorem 3.2].

Clifford's Theorem for graphs

Abstract: Let $\Gamma$ be a metric graph having a linear system $g^r_{2r}$ for some $2 \leq r \leq g-2$ then $\Gamma$ has a linear system $g^1_2$. This is similar to the well-known Clifford's Theorem from the theory of linear systems on smooth projective curves.

Positive polynomials on nondegenerate basic semi-algebraic sets

Abstract: Abstract A concept of nondegenerate basic closed semi-algebraic sets in ℝn is introduced. These are unbounded closed semi-algebraic sets for which we obtain some representations of polynomials with positive infima (the polynomials are further assumed to be bounded if n>2) and solutions of the moment problem. The key to obtain these results is an explicit description of the algebra of bounded polynomials on a nondegenerate basic semi-algebraic set via the combinatorial information of the Newton polyhedron corresponding to the generators of the semi-algebraic set.

Optimal cover of a disk with three smaller congruent disks

Abstract: Abstract In 2008 R. Connelly asked how one should place n small disks of radius r to cover the largest possible area of a disk of radius R > r. More specifically, is there always an optimal configuration with n-fold rotational symmetry for small values of n? The answer is known to be positive for n = 2, negative for n = 5, and it has been conjectured to be positive for n = 3 and 4. In this paper, we present a systematic way to list all possible combinatorial structures of optimal configurations, and we prove that for n = 3 there is always an optimal configuration with rotational symmetry of order three.

Totally isotropic subspaces of small height in quadratic spaces

Abstract: Let K be a global field or Q, F a nonzero quadratic form on K N , N � 2, and V a subspace of K N. We prove the existence of an infinite collection of finite families of small-height maximal totally isotropic subspaces of (V,F) such that each such family spans V as a K-vector space. This result generalizes and extends a well known theorem of J. Vaaler (16) and further contributes to the effective study of quadratic forms via height in the general spirit of Cassels' theorem on small zeros of quadratic forms. All bounds on height are explicit. 1. Introduction and statement of results In his celebrated 1955 paper (3) J. W. S. Cassels proved that an integral quadratic form which is isotropic over Q has a non-trivial integral zero of bounded height (we detail the necessary notation of heights and quadratic forms in Section 2 below). At the heart of Cassels' argument lies a beautiful geometric idea, based on Minkowski's Linear Forms Theorem and an appropriately chosen orthogonal reflection of the quadratic space in question. The investigation of small-height zeros of quadratic forms has since been taken up by a number of authors (see (8) for an overview) with geometric ideas continuing to play a key role. In particular, geometry of numbers techniques along with orthogonal projections were used by Schlickewei (13) to prove the existence of a small-height maximal totally isotropic subspace in an isotropic rational quadratic space, and the method was then extended by Schlickewei and W. M. Schmidt (14) to prove the existence of a family of such small-height subspaces which generate the entire quadratic space. The Schlickewei-Schmidt results were then extended over number fields by Vaaler (15, 16) by the use of adelic geometry of numbers and local projection operators at all places. Vaaler's method (15) has also been extended over global function fields of odd characteristic in (4) to prove the existence of a small-height maximal totally isotropic subspace in an isotropic quadratic space. An important underlying idea used in all such arguments over global fields is Nortcott's principle, guaranteeing finiteness of sets of projective points of bounded height and degree. Northcott's principle no longer holds over Q, where an analogue of Vaaler's first result (15) was established in (6) by an application of another geometric principle, namely a version of arithmetic Bezout's theorem due to Bost, Gillet, and Soule (2). Estimates on small-height zeros were further applied to produce effective versions of the Witt decomposition theorem over global fields and Q in (5), (6), and (4).

On the density function on moduli spaces of toric 4-manifolds

Abstract: The optimal density function assigns to each symplectic toric manifold $M$ a number $0 < d \leq 1$ obtained by considering the ratio between the maximum volume of $M$ which can be filled by symplectically embedded disjoint balls and the total symplectic volume of $M$. In the toric version of this problem, $M$ is toric and the balls need to be embedded respecting the toric action on $M$. The goal of this note is first to give a brief survey of the notion of toric symplectic manifold and the recent constructions of moduli space structure on them, and recall how to define a natural density function on this moduli space. Then we review previous works which explain how the study of the density function can be reduced to a problem in convex geometry, and use this correspondence to to give a simple description of the regions of continuity of the maximal density function when the dimension is $4$.

Hilbert schemes of some threefold scrolls over F_e

Abstract: Hilbert schemes of suitable smooth, projective manifolds of low degree which are 3-fold scrolls over the Hirzebruch surface F1 are studied. An irreducible component of the Hilbert scheme parametriz- ing such varieties is shown to be generically smooth of the expected dimension and the general point of such a component is described.