Narrow Results

We give a construction of stationary discs and the indicatrix for manifolds of higher codimension which is a partial analog of L. Lempert's theory of stationary discs for strictly convex hypersurfaces. This leads to new invariants of the CR structure in higher codimension linked with the contact structure of the conormal bundle.

We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, on some Seifert fibered 3-manifolds we describe open books which are horizontal with respect to their plumbing description. As another application we describe horizontal open books isomorphic to Milnor open books for some complex surface singularities. Moreover we give examples of tight contact 3-manifolds supported by planar open books. As a consequence, the Weinstein conjecture holds for these tight contact structures [1].

A new surgery method for contact manifolds is introduced in this paper. The surgeries are defined by using round handles with symplectic structures which are also introduced in this paper. These symplectic round handles have the global Liouville vector fields. Then the symplectic structures are exact. In the 3-dimensional case, the surgery is described only in terms of 3-dimensional contact topology.

An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step, we construct an explicit Stein cobordism from any contact 3-manifold to the canonical contact structure of a proper almost rational singularity introduced by Némethi. We also show that the construction cannot always work in the reverse direction: in fact, the U-filtration depth of contact Ozsváth–Szabó invariant obstructs the existence of a Stein cobordism from a proper almost rational singularity to a rational one. Along the way, we detect the contact Ozsváth–Szabó invariants of those contact structures fillable by an AR plumbing graph, generalizing an earlier work of the first author.

We give a generalized Thurston–Bennequin-type inequality for links in $S^3$ using a Bauer–Furuta-type invariant for four-manifolds with contact boundary introduced by the first aut hor. As a special case, we also give an adjunction inequality for smoothly embedded orientable surfaces with negative intersection in a closed oriented smooth four-manifold whose non-equivariant Bauer–Furuta invariant is nonzero.

Suppose that $\Sigma$ is a closed and oriented surface equipped with a Riemannian metric. In the literature, there are three seemingly distinct constructions of open books on the unit (co)tangent bundle of $\Sigma$, having complex, contact and dynamical flavors, respectively. Each one of these constructions is based on either an admissible divide or an ordered Morse function on $\Sigma$. We show that the resulting open books are pairwise isotopic provided that the ordered Morse function is adapted to the admissible divide on $\Sigma$. Moreover, we observe that if $\Sigma$ has positive genus, then none of these open books are planar and furthermore, we determine the only cases when they have genus one pages.

We prove a simple necessary and sufficient condition for a two-bridge knot $K(p,q)$ to be quasipositive, based on the continued fraction expansion of $p/q$. As an application, coupled with some classification results in contact and symplectic topology, we give a new proof of the fact that smoothly slice two-bridge knots are non-quasipositive. Another proof of this fact using methods within the scope of knot theory is presented in Appendix A, by Stepan Orevkov.

We discuss three mathematical structures which arise in topologically massive Abelian gauge theory. First, the Euclidean topologically massive Abelian gauge theory defines a contact structure on a manifold. We briefly discuss three solutions and the related contact structures on the flat 3-torus, the AdS space, the 3-sphere which respectively correspond to Bianchi type I, VIII, IX spaces. We also present solutions on Bianchi type II, VI and VII spaces. Secondly, we discuss a family of complex (anti-)self-dual solutions of the Euclidean theory in Cartesian coordinates on which are given by (anti)holomorpic functions. The orthogonality relation of contact structures which are determined by the real parts of these complex solutions separates them into two classes: the self-dual and the anti-self-dual solutions. Thirdly, we apply the curl transformation to this theory. An arbitrary solution is given by a vector tangent to a sphere whose radius is determined by the topological mass in transform space. Meanwhile a gauge transformation corresponds to a vector normal to this sphere. We discuss the quantization of topological mass in an example.

We show that a null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K with respect to S satisfies sl(K, S) = -χ(S). In particular, every null-homologous topological knot type in an overtwisted contact manifold can be represented by the boundary of a Legendrian ribbon. Finally, we show that a contact structure is tight if and only if every Legendrian ribbon minimizes genus in its relative homology class.

In this paper, the support genera of all Legendrian right-handed trefoil knots and some other Legendrian knots are computed. We give examples of Legendrian knots in the three-sphere with the standard contact structure which have positive support genera with arbitrarily negative Thurston–Bennequin invariant. This answers a question in [S. Onaran, Invariants of Legendrian knots from open book decompositions, Int. Math. Res. Not.10 (2010) 1831–1859].

For n = 1, 2 and 3, we construct a pair of transverse knots T₁ and , in the standard contact 3-sphere, satisfying the following properties:

(1) the topological knot type of T₁ is the same as that of ,

(2) the self-linking number of T₁ is equal to that of ,

(3) is obtained from a transverse knot T₂ by n stabilizations, and

(4) T₁ is not transversely isotopic to .

The Chekanov–Eliashberg differential graded algebra of a Legendrian knot L is a rich source of Legendrian knot invariants, as is the theory of generating families. The set P(L) of homology groups of augmentations of the Chekanov–Eliashberg algebra is an invariant, as is a count of objects from the theory of generating families called graded normal rulings. This paper gives two results demonstrating the usefulness of computing the homology group of an augmentation using a combinatorial interpretation of a generating family called a Morse complex sequence (MCS). First, we show that if the projection of L to the xz-plane has exactly 4 cusps, then |P(L)| ≤ 1. Second, we show that two augmentations associated to the same graded normal ruling by the many-to-one map between augmentations and graded normal rulings defined by Ng and Sabloff [The correspondence between augmentations and rulings for Legendrian knots, Pacific J. Math.224(1) (2006) 141–150] need not have isomorphic homology groups.

We study the extrinsic geometry of Legendrian knots in the standard tight contact structure on ${ℝ}^3.$ In particular, we show that the total curvature of a Legendrian knot $L$ in ${ℝ}^3$ is bounded below by $\pi$ times, the total number of cusps in the front projection of $L$. We also show that a Legendrian $(p,q)$-torus knot has the total curvature bounded below by $2\pi q$ while that of the Legendrian knots $E(l,k)$ is bounded below by $2\pi (k+l+1)$. Furthermore, we find an explicit relation between the Thurston–Bennequin number of a Legendrian knot $L$ and the geometric self-linking number, the curvature and the torsion of the knot $L$.

In this paper we develop a method for studying tight contact structures on lens spaces. We then derive uniqueness and non-existence statements for tight contact structures with certain (half) Euler classes on lens spaces.

We propose in this paper a method for studying contact structures in 3-manifolds by means of branched surfaces. We explain what it means for a contact structure to be carried by a branched surface embedded in a 3-manifold. To make the transition from contact structures to branched surfaces, we first define auxiliary objects called σ-confoliations and pure contaminations, both generalizing contact structures. We study various deformations of these objects and show that the σ-confoliations and pure contaminations obtained by suitably modifying a contact structure remember the contact structure up to isotopy.

After defining tightness for all pure contaminations in a natural way, generalizing the definition of tightness for contact structures, we obtain some conditions on (the embedding of) a branched surface in a 3-manifold sufficient to guarantee that any pure contamination carried by the branched surface is tight. We also find conditions sufficient to prove that a branched surface carries only overtwisted (non-tight) contact structures.

Our long-term goal in developing these methods is twofold: Not only do we want to study tight contact structures and pure contaminations, but we also wish to use them as tools for studying 3-manifold topology.

In this work we analyze and compare the model of the material (elastic) element and the entropy form developed by Coleman and Owen with that one obtained by localizing the balance equations of the continuum thermodynamics. This comparison allows one to determine the relation between the entropy function S of Coleman–Owen and that one imported from the continuum thermodynamics. We introduce the Extended Thermodynamical Phase Space (ETPS) and realize the energy and entropy balance expressions as 1-forms in this space. This allows us to realizes I and II laws of thermodynamics as conditions on these forms. We study the integrability (closure) conditions of the entropy form for the model of thermoelastic element and for the deformable ferroelectric crystal element. In both cases closure conditions are used to rewrite the dynamical system of the model in term of the entropy form potential and to determine the constitutive relations among the dynamical variables of the model.

In a related study (to be published) these results will be used for the formulation of the dynamical model of a material element in the contact thermodynamical phase space of Caratheodory and Hermann similar to that of homogeneous thermodynamics.

In this paper, we introduce a Riemannian metric and a family of framed f-structures on the slit tangent bundle of a Finsler manifold Fⁿ = (M, F). Then we prove that there exists an almost contact structure on the tangent bundle, when this structure is restricted to the Finslerian indicatrix. We show that this structure is Sasakian if and only if Fⁿ is of positive constant curvature 1. Finally, we prove that (i) Fⁿ is a locally flat Riemannian manifold if and only if , (ii) the Jacobi operator is zero or commuting if and only if (M, F) have the zero flag curvature.

In this work we investigate a material point model (MP-model) and exploit the geometrical meaning of the "entropy form" introduced by Coleman and Owen. We show that a modification of the thermodynamical phase space (studied and exploited in numerous works) is an appropriate setting for the development of the MP-model in different physical situations. This approach allows to formulate the MP-model and the corresponding entropy form in terms similar to those of homogeneous thermodynamics. Closeness condition of the entropy form is reformulated as the requirement that admissible processes curves belong to the (extended) constitutive surfaces foliating the extended thermodynamical phase space of the model over the space X of basic variables. Extended constitutive surfaces Σ_S,κ are described as the Legendre submanifolds Σ_S of the space shifted by the flow of Reeb vector field. This shift is controlled by the entropy production function κ. We determine which contact Hamiltonian dynamical systems ξ_K are tangent to the surfaces Σ_S,κ, introduce conformally Hamiltonian systems μξ_K where conformal factor μ characterizes the increase of entropy along the trajectories. These considerations are then illustrated by applying them to the Coleman–Owen model of thermoelastic point.

The classical phase space of general relativistic classical test particle (here called, for short, "phase space") is defined as the first jet space of motions regarded as timelike one-dimensional submanifolds of spacetime. By the projectability assumption, we define the subsheaf of special phase functions with a special Lie bracket and we compare the Lie algebra of special phase functions with the structures obtained on the phase space by the standard Hamiltonian approach.

The existence of compact simply-connected K-contact, but not Sasakian, manifolds has been unknown only for dimension 5. The aim of this paper is to show that the Kollár's simply-connected example which is a Seifert bundle over the complex projective space ℂℙ² and does not carry any Sasakian structure is actually a K-contact manifold. As a consequence, we affirmatively answer the above existence problem in dimension 5, establishing that there are infinitely many compact simply-connected K-contact manifolds of dimension 5 which do not carry a Sasakian structure.