Contact structures on product five-manifolds and fibre sums along circles

Two constructions of contact manifolds are presented: (i) products of S ¹ with manifolds admitting a suitable decomposition into two exact symplectic pieces and (ii) fibre connected sums along isotropic circles. Baykur has found a decomposition as required for (i) for all closed, oriented 4-manifolds. As a corollary, we can show that all closed, oriented 5-manifolds that are Cartesian products of lower-dimensional manifolds carry a contact structure. For symplectic 4-manifolds we exhibit an alternative construction of such a decomposition; this gives us control over the homotopy type of the corresponding contact structure. In particular, we prove that \mathbb CP²×S¹{{\mathbb {CP}}^2\times S^1} admits a contact structure in every homotopy class of almost contact structures. The existence of contact structures is also established for a large class of 5-manifolds with fundamental group \mathbbZ₂{{\mathbb{Z}}_2} .     

Brieskorn manifolds as contact branched covers of spheres

We show that Brieskorn manifolds with their standard contact structures are contact branched coverings of spheres. This covering maps a contact open book decomposition of the Brieskorn manifold onto a Milnor open book of the sphere.      

Surgery diagrams and open book decompositions of contact 3-manifolds

Summary We show an algorithmic way for finding a compatible open book decomposition on a contact 3-manifold given by contact (±1)-surgery.     

Foliations and Complemented Framed Structures on an Almost Contact Metric Manifold

On an odd dimensional manifold, we define a structure which generalizes several known structures on almost contact manifolds, namely Sasakian, trans-Sasakian, quasi-Sasakian, Kenmotsu and cosymplectic structures. This structure, hereinafter called a generalized quasi-Sasakian, shortly G.Q.S. structure, is defined on an almost contact metric manifold and satisfies an additional condition. Then we consider a distribution D₁{\mathcal{D}_{1}} wich allows a suitable decomposition of the tangent bundle of a G.Q.S. manifold. Necessary and sufficient conditions for the normality of the complemented framed structure on the distribution D₁{\mathcal{D}_{1}} defined on a G.Q.S manifold are studied. The existence of the foliation on G.Q.S. manifolds and of bundle-like metrics are also proven. It is shown that under certain circumstances a new foliation arises and its properties are investigated. Some examples illustrating these results are given in the final part of this paper.     

Ozsváth-Szabó invariants and fillability of contact structures

Recently Ozsváth and Szabó defined an invariant of contact structures with values in the Heegaard-Floer homology groups. They also proved that a version of the invariant with twisted coefficients is non trivial for weakly symplectically fillable contact structures. In this article we show that their non vanishing result does not hold in general for the contact invariant with untwisted coefficients. As a consequence of this fact Heegaard-Floer theory can distinguish between weakly and strongly symplectically fillable contact structures. The author is a member of EDGE, Research Training Network HPRN-CT-2000-00101, supported by The European Human Potential Programme.     

Contact handle decompositions

We review Giroux's contact handles and contact handle attachments in dimension three and show that a bypass attachment consists of a pair of contact 1 and 2-handles. As an application we describe explicit contact handle decompositions of infinitely many pairwise non-isotopic overtwisted 3-spheres. We also give an alternative proof of the fact that every compact contact 3-manifold (closed or with convex boundary) admits a contact handle decomposition, which is a result originally due to Giroux.     

Uzawa block relaxation domain decomposition method for a two-body frictionless contact problem

We propose a Uzawa block relaxation domain decomposition method for a two-body frictionless contact problem. We introduce auxiliary variables to separate subdomains representing linear elastic bodies. Applying a Uzawa block relaxation algorithm to the corresponding augmented Lagrangian functional yields a domain decomposition algorithm in which we have to solve two uncoupled linear elasticity subproblems in each iteration while the auxiliary variables are computed explicitly using Kuhn–Tucker optimality conditions.     

Convexity and Zariski decomposition structure

This is the first part of our work on Zariski decomposition structures, where we study Zariski decompositions using Legendre–Fenchel type transforms. In this way we define a Zariski decomposition for curve classes. This decomposition enables us to develop the theory of the volume function for curves defined by the second named author, yielding some fundamental positivity results for curve classes. For varieties with special structures, the Zariski decomposition for curve classes admits an interesting geometric interpretation.     

Harmonic almost contact structures via the intrinsic torsion

We proceed further in the study of harmonicity for almost contact metric structures already initiated by Vergara-Díaz and Wood. By using the intrinsic torsion, we characterise harmonic almost contact metric structures in several equivalent ways and show conditions relating harmonicity and classes of almost contact metric structures. Additionally, we study the harmonicity of such structures as a map into the quotient bundle of the oriented orthonormal frames by the action of the structural group U(n)×1. Finally, by using a Bochner type formula proved by Bör and Hernández Lamoneda, we display some examples which give the absolute minimum for the energy.     

Invariants of contact structures from open books

In this note we define three invariants of contact structures in terms of open books supporting the contact structures. These invariants are the support genus (which is the minimal genus of a page of a supporting open book for the contact structure), the binding number (which is the minimal number of binding components of a supporting open book for the contact structure with minimal genus pages) and the norm (which is minus the maximal Euler characteristic of a page of a supporting open book).

     

Existence and classification of overtwisted contact structures in all dimensions

We establish a parametric extension h-principle for overtwisted contact structures on manifolds of all dimensions, which is the direct generalization of the 3-dimensional result from [12]. It implies, in particular, that any closed manifold admits a contact structure in any given homotopy class of almost contact structures.     

A Note on Contact Shape Optimization with Semicoercive State Problems

This note deals with contact shape optimization for problems involving floating structures. The boundedness of solutions to state problems with respect to admissible domains, which is the basic step in the existence analysis, is a consequence of Korn's inequality in coercive cases. In semicoercive cases (meaning that floating bodies are admitted), the Korn inequality cannot be directly applied and one has to proceed in another way: to use a decomposition of kinematically admissible functions and a Korn type inequality on appropriate subspaces. In addition, one has to show that the constant appearing in this inequality is independent with respect to a family of admissible domains.     

An o-minimal structure without mild parameterization

We prove, by explicit construction, that not all sets definable in polynomially bounded o-minimal structures have mild parameterization. Our methods do not depend on the bounds particular to the definition of mildness and therefore our construction is also valid for a generalized form of parameterization, which we call G-mild. Moreover, we present a cell decomposition result for certain o-minimal structures which may be of independent interest. This allows us to show how our construction can produce polynomially bounded, model complete expansions of the real ordered field which, in addition to lacking G-mild parameterization, nonetheless still have analytic cell decomposition.     

Infinitely many universally tight torsion free contact structures with vanishing Ozsváth–Szabó contact invariants

Ozsváth–Szabó contact invariants are a powerful way to prove tightness of contact structures but they are known to vanish in the presence of Giroux torsion. In this paper we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsváth–Szabó invariant vanishes. We also discuss the relation between these invariants and an invariant on T³ and construct other examples of new phenomena in Heegaard–Floer theory. Along the way, we prove two conjectures of K. Honda, W. Kazez and G. Matić about their contact topological quantum field theory. Almost all the proofs in this paper rely on their gluing theorem for sutured contact invariants.     

A domain decomposition method for two-body contact problems with Tresca friction

The paper analyzes a continuous and discrete version of the Neumann-Neumann domain decomposition algorithm for two-body contact problems with Tresca friction. Each iterative step consists of a linear elasticity problem for one body with displacements prescribed on a contact part of the boundary and a contact problem with Tresca friction for the second body. To ensure continuity of contact stresses, two auxiliary Neumann problems in each domain are solved. Numerical experiments illustrate the performace of the proposed approach.     

Hypergraph decomposition and secret sharing

A secret sharing scheme is a protocol by which a dealer distributes a secret among a set of participants in such a way that only qualified sets of them can reconstruct the value of the secret whereas any non-qualified subset of participants obtain no information at all about the value of the secret. Secret sharing schemes have always played a very important role for cryptographic applications and in the construction of higher level cryptographic primitives and protocols.In this paper we investigate the construction of efficient secret sharing schemes by using a technique called hypergraph decomposition, extending in a non-trivial way the previously studied graph decomposition techniques. A major advantage of hypergraph decomposition is that it applies to any access structure, rather than only structures representable as graphs. As a consequence, the application of this technique allows us to obtain secret sharing schemes for several classes of access structures (such as hyperpaths, hypercycles, hyperstars and acyclic hypergraphs) with improved efficiency over previous results. Specifically, for these access structures, we present secret sharing schemes that achieve optimal information rate. Moreover, with respect to the average information rate, our schemes improve on previously known ones.In the course of the formulation of the hypergraph decomposition technique, we also obtain an elementary characterization of the ideal access structures among the hyperstars, which is of independent interest.     

An Ozsváth–Szabó Floer homology invariant of knots in a contact manifold

Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsváth–Szabó contact invariant we obtain an invariant of knots in a contact three-manifold. This invariant provides an upper bound for the Thurston–Bennequin plus rotation number of any Legendrian realization of the knot. We use it to demonstrate the first systematic construction of prime knots in contact manifolds other than S³ with negative maximal Thurston–Bennequin invariant. Perhaps more interesting, our invariant provides a criterion for an open book to induce a tight contact structure. A corollary is that if a manifold possesses contact structures with distinct non-vanishing Ozsváth–Szabó invariants, then any fibered knot can realize the classical Eliashberg–Bennequin bound in at most one of these contact structures.     

On a contact 3-structure

A contact 3-structure consists of three contact metric structures which satisfy the relation (2.1). On a product manifold of the real line and a manifold with a contact 3-structure, we can construct three almost Hermitian structures satisfying the quaternionic identities. From this view point we discuss a contact 3-structure. Owing to Hitchin's well known Lemma concerning to hyperk?hler structure (Lemma H), we show that a contact 3-structure is necessarily a Sasakian 3-structure. Received: 26 August 1999; in final form: 2 May 2000 / Published online: 4 May 2001     

Determination of a stochastic friction coefficient depending on a randomly rough surface

The goal of this contribution is to calculate the the friction coefficient for a scanned surface of a worn brake pad. The data shows that the asperities can be approximated by paraboloids which allows to calculate the contact force and area with the Hertz contact model if the deformation is elastic. The friction force is calculated with the Bowden-Tabor approach which suggests that the friction force is the force to shear apart contacting asperities. This is considered to be the dominant friction cause in dry contact. To generate many surfaces with similar peak statistics the spectral decomposition is used. The friction coefficient and it's stochastic properties is calculated for these surfaces. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)