arXiv research

Geodesics of contactomorphisms on a specific manifold are characterized by Hamiltonian functions.

H. Sato introduced a Schwarzian derivative of a contactomorphism of three-dimensional Euclidean space and with T. Ozawa described its basic properties. In this note their construction is extended to all odd dimensions and to non-flat contact projective structures. The contact projective Schwarzian derivative of a conta…

Contact group retracts to unitary subgroup.

New examples show transverse knots are determined by their branched covers.

In this paper we study the groups of contactomorphisms of a closed contact manifold from a topological viewpoint. First we construct examples of contact forms on spheres whose Reeb flow has a dense orbit. Then we show that the unitary group U(n+1) is homotopically essential in the group of contactomorphisms of the stan…

For a stratified group $G$, we construct a class of polarised Lie groups, which we call modifications of $G$, that are locally contactomorphic to it. Vice versa, we show that if a polarised group is locally contactomorphic to a stratified group $G$, whose Lie algebra has finite Tanaka prolongation, then it must be a mo…

Defines spectral selectors on lens spaces for contactomorphisms.

The group of volume preserving diffeomorphisms, the group of symplectomorphisms and the group of contactomorphisms constitute the classical groups of diffeomorphisms. The first homology groups of the compactly supported identity components of the first two groups have been computed by Thurston and Banyaga, respectively…

Study connects contact structures to cone geodesics and contactomorphisms.

We give examples of contactomorphisms in every dimension that are smoothly isotopic to the identity but that are not contact isotopic to the identity. In fact, we prove the stronger statement that they are not even symplectically pseudo-isotopic to the identity. We also give examples of pairs of contactomorphisms which…

In this paper, we prove that on any contact manifold, there exists an arbitrary C^{\infty}-small contactomorphism which does not admit a square root. In particular, there exists an arbitrary C^{\infty}-small contactomorphism which is not "autonomous". This result is the first step to study the topology of non-autonomou…

Defines Lorentzian distance on contactomorphisms, proving continuity and finite conditions.

A right-invariant metric $ρ_α$ on the compactly supported identity component $Cont_0(M,α)$ of the group of contactomorphisms of an arbitrary contact manifold $(M,α)$ is introduced in a similar way that the Hofer metric was defined on the group of Hamiltonian symplectomorphisms of a symplectic manifold. The restriction …

We calculate the weak homotopy type of the group of contactomorphisms of the three-sphere which coincide with the identity on (a neighborhood of) an overtwisted disk.

Study the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds.

In his 1992 article on generating functions Viterbo constructed a bi-invariant metric on the group of compactly supported Hamiltonian symplectomorphisms of R^2n. Using the set-up of arXiv:0901.3112 we extend the Viterbo metric to the group of compactly supported contactomorphisms of R^2n x S^1 isotopic to the identity.…

A point q in a contact manifold is called a translated point for a contactomorphism φ, with respect to some fixed contact form, if φ(q) and q belong to the same Reeb orbit and the contact form is preserved at q. In this article we discuss a version of the Arnold conjecture for translated points of contactomorphisms and…

We study the singularities of Legendrian subvarieties of contact manifolds in the complex-analytic category and prove two rigidity results. The first one is that Legendrian singularities with reduced tangent cones are contactomorphically biholomorphic to their tangent cones. This result is partly motivated by a problem…

In a recent preprint Yael Karshon showed that there exist non-conjugate tori in a group of symplectomorphisms of a Hirzebruch surface. She counted them in terms of the cohomology class of the symplectic structure. We show that a similar phenomenon exists in the contactomorphism groups of pre-quantum circle bundles over…

Simple groups identified for contactomorphisms with high regularity.

In this article we consider a variant of Rabinowitz Floer homology in order to define a homological count of discriminant points for paths of contactomorphisms. The growth rate of this count can be seen as an analogue of Givental's nonlinear Maslov index. As an application we prove a Bott-Samelson type obstruction theo…

We prove, for a class of contact manifolds, that the universal cover of the group of contact diffeomorphisms carries a natural partial order. It leads to a new viewpoint on geometry and dynamics of contactomorphisms. It gives rise to invariants of contactomorphisms which generalize the classical notion of the rotation …

New distances defined on Legendrian spaces without positive loops.

Clarifies properties of Ozsvath-Szabo contact invariant.

By a theorem of Banyaga the group of diffeomorphisms of a manifold $P$ preserving a regular contact form $α$ is a central $S^1$ extension of the commutator of the group of symplectomorphisms of the base $B = P/S^1$. We show that if $T$ is a Hamiltonian maximal torus in the group of symplectomorphism of $B$, then its pr…

I describe a general scheme which associates conjugacy classes of tori in the contactomorphism group to transverse almost complex structures on a compact contact manifold. Moreover, to tori of Reeb type whose Lie algebra contains a Reeb vector field one can associate a Sasaki cone. Thus, for contact structures of K-con…

New contactomorphisms found via Dehn twists on 3-manifold sums.

In this paper we examine the Riemannian geometry of the group of contactomorphisms of a compact contact manifold. We compute the sectional curvature of $\mathcal{D}_θ(M)$ in the sections containing the Reeb field and show that it is non-negative. We also solve explicitly the Jacobi equation along the geodesic correspon…

Introduces GFC for learning complex dynamical systems with geometric constraints.

Study confirms contact cosmetic surgery for most knots, with exceptions.

Study shows how to section map between holonomic and formal solutions.

Here we investigate some geometric properties of the contactomorphism group $\mathcal{D}_θ(M)$ of a compact contact manifold with the $L^2$ metric on the stream functions. Viewing this group as a generalization to the $\mathcal{D}(S^1)$, the diffeomorphism group of the circle, we show that its sectional curvature is al…

A point q in a contact manifold is called a translated point for a contactomorphism φ, with respect to some fixed contact form, if φ(q) and q belong to the same Reeb orbit and the contact form is preserved at q. The problem of existence of translated points is related to the chord conjecture and to the problem of leafw…

New contact Kirby moves complete the set for contact surgery diagrams.

The paper proves Morse estimates for translated points on unit tangent bundles.

We describe a contact analog of the symplectic cut construction. As an application we show that the group of contactomorphisms for a particular overtwisted contact structure on the three sphere contains countably many nonconjugate two tori.

We show that the contact structure on the link of a cusp singularity is contactomorphic to a Sol-manifold with the positive contact structure arising from the Anosov flow.

We construct (infinitely many) examples in all dimensions of contactomorphisms of closed overtwisted contact manifolds that are smoothly isotopic but not contact-isotopic to the identity.

In hyperbolic L-spaces, we find multiple pseudo-Anosov flows with unique properties.

We study in this paper the remnants of the contact partial order on the orbits of the adjoint action of contactomorphism groups on their Lie algebras. Our main interest is a class of non-compact contact manifolds, called convex at infinity.

The paper extends Thurston's method to new variants of Mather-Thurston theorem.

It has been shown by V. Colin that every tight contact 3-manifold can be written as a connected sum of prime manifolds. Here we prove that the summands in this decomposition are unique up to order and contactomorphism.

New Legendrian knots found with equivalent Stein traces.

In this note, we first classify all topological torus knots lying on the Heegaard torus in lens spaces, and then we study Legendrian representatives of these knots. We classify oriented positive Legendrian torus knots in the universally tight contact structures on the lens spaces up to contactomorphism.

Contact manifolds are odd-dimensional smooth manifolds endowed with a maximally non-integrable field of hyperplanes. They are intimately related to symplectic manifolds, i.e. even-dimensional smooth manifolds endowed with a closed non-degenerate 2-form. Although in symplectic topology a famous bi-invariant metric, the …

In this paper, we study contact structures on any open 3-manifold V which is the interior of a compact 3-manifold. To do this, we introduce proper contact isotopy invariants called the slope at infinity and the division number at infinity. We first prove several classification theorems for T^2 x [0, \infty), T^2 x R, a…

Given a manifold $M$ endowed with a contact 1-form $α$, a bi-invariant pseudo-metric $\varrho_α$ is introduced on $Cont(M,α)$, the compactly supported identity component of the group of all strict contactomorphisms of $(M,α)$. For $M$ open $\varrho_α$ is a metric.

In this note we show that $+1$-contact surgery on distinct Legendrian knots frequently produces contactomorphic manifolds. We also give examples where this happens for $-1$-contact surgery. As an amusing corollary we find overtwisted contact structures that contain a large number of distinct Legendrian knots with the s…