arXiv research

Standard contact structure proven for line fibration in 3D space.

Local flatness theorem for paraquaternionic contact structures.

We construct an open book decomposition compatible with a contact structure given by a rational contact surgery on a Legendrian link in the standard contact $S^3$. As an application we show that some rational contact surgeries on certain Legendrian knots induce overtwisted contact structures.

A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry an…

We prove various results on contact structures obtained by contact surgery on a single Legendrian knot in the standard contact three--sphere. Our main tool are the contact Ozsvath--Szabo invariants.

Contact round surgery of contact 3-manifolds is introduced in this paper. By using this method, an alternative proof of the existence of a contact structure on any closed orientable 3-manifold is given. It is also proved that any contact structure on any closed orientable 3-manifold is constructed from the standard con…

Study proves all left-invariant contact structures on 3D Lie groups are tight.

In this paper we construct complex contact structures on $\mathbb{C}^{2n+1}$ for any $n\ge 1$ with the property that every holomorphic Legendrian map $\mathbb{C}\to \mathbb{C}^{2n+1}$ is constant. In particular, these contact structures are not globally contactomorphic to the standard complex contact structure on $\mat…

Study contact structures on lens spaces, classifying rational knots.

The abstract discusses embedding manifolds in open books and contact structures.

We make some elementary observations concerning subcritically Stein fillable contact structures on 5-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case the fundamental group is finite cyclic, and we show that on the 5-sphere the standard contact structure is the unique s…

Characterizes invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.

The paper defines and studies contact surgery numbers for contact 3-manifolds.

Solutions to a differential equation link to contact structures.

These notes provide an introduction to Giroux's theory of convex surfaces in contact 3-manifolds and its simplest applications. They put a special emphasis on pictures and discussions of explicit examples. The first goal is to explain why all the information about a contact structure in a neighborhood of a generic surf…

We show that an oriented elliptic 3-manifold admits a universally tight positive contact structure iff the corresponding group of deck transformations on $S^3$ preserves a standard contact structure pointwise. We also relate univerally tight contact structures on 3-manifolds covered by $S^3$ to the exceptional isomorph…

We show that there exists a transverse link in the standard contact structures on the 3-sphere such that all contact 3-manifolds are contact branched covers over this transverse link.

As shown by H. Gluck in 1962, the diffeotopy group of S^1 \times S^2 is isomorphic to Z_2 + Z_2 + Z_2. Here an alternative proof of this result is given, relying on contact topology. We then discuss two applications to contact topology: (i) it is shown that the fundamental group of the space of contact structures on S^…

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.

Contractibility of tight contact structures on $S^3$ proven.

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,ξ) admits a …

Study contact structures on specific manifolds, proving infinite non-isotopic structures and tight/overtwisted classifications.

A contact metric manifold is said to be $H$-contact, if the characteristic vector field is harmonic. We prove that the unit tangent bundle of a Riemannian manifold $M$ equipped with the standard contact metric structure is $H$-contact if and only if $M$ is $2$-stein.

A partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere is given. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Biquard connection is zero and this occurs precisely on 3-Sasakian manifolods. All con…

Study contact structures on projective spaces, proving infinite non-isotopic structures.

Contact group retracts to unitary subgroup.

A correspondence is studied by H. Matsuda between front projections of Legendrian links in the standard contact structure for 3-space and rectangular diagrams. In this paper, we introduce braided rectangular diagrams, and study a relationship with Legendrian links in the standard contact structure for 3-space. We show …

We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of …

Study on transverse knots and their neighborhoods, proving unique standard neighborhoods and destabilization results.

This paper completely answers the question of when contact (r)-surgery on a Legendrian knot in the standard contact structure on the 3-sphere yields a symplectically fillable contact manifold for r in (0,1]. We also give obstructions for other positive r and investigate Lagrangian fillings of Legendrian knots.

The study explores embedding closed contact manifolds in higher dimensions.

Classifies tight contact structures with special symmetries.

In this article, we prove a generalization of a theorem of Lisca-Matic to Stein cobordisms and develop a method for distinguishing certain Stein cobordisms using rotation numbers. Using these results along with standard techniques from convex surface theory and classifications of tight contact structures on certain 3-m…

The paper constructs contact forms on a sphere with constant Webster curvature and applies them to CR Yamabe problems.

We prove that every closed, connected contact 3-manifold can be obtained from the 3-sphere with its standard contact structure by contact surgery of coefficient plus or minus 1 along a Legendrian link. As a corollary, we derive a result of Etnyre and Honda about symplectic cobordisms (in slightly stronger form).

New contact structures on spheres and tori with weakly compatible metrics.

Study symplectic fillings of lens spaces, focusing on virtually overtwisted contact structures.

In two previous papers, the two first-named authors introduced a notion of contact r-surgery along Legendrian knots in contact 3-manifolds. They also showed how (at least in principle) to convert any contact r-surgery into a sequence of contact plus or minus 1 surgeries, and used this to prove that any (closed) contact…

Authors prove a contact structure result using branched covers and overtwisted disks.

Let p and n be positive integers with p>1, and let E(p,n) be the oriented 3-manifold obtained by performing pn(p-1)-1 surgery on a positive torus knot of type (p, pn+1). We prove that E(2,n) does not carry tight contact structures for any n, while E(p,n) carries tight contact structures for any n and any odd p. In part…

Every real 3-manifold can be turned into a real contact structure.

The study proves CR structures on specific three-manifolds are equivalent to standard structures.

The Newman-Penrose-Perjes formalism is applied to smooth contact structures on riemannian 3-manifolds. In particular it is shown that a contact 3-manifold admits an adapted riemannian metric if and only if it admits a metric with a divergence-free, constantly twisting, geodesic congruence. The shear of this congruence …

Stabilization operation for high-dimensional contact manifolds, proving many links are non-simple.

Classification of fillings for a specific type of contact surgery on the Hopf link.

According to Giroux, contact manifolds can be described as open books whose pages are Stein manifolds. For 5-dimensional contact manifolds the pages are Stein surfaces, which permit a description via Kirby diagrams. We introduce handle moves on such diagrams that do not change the corresponding contact manifold. As an …

It is shown that Legendrian (resp. transverse) cable links in the 3-sphere with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston-Bennequin invariant and rotation number in the Legendrian…

New invariant stops certain types of geometric transformations.