arXiv research

We extend the Heegaard Floer homological definition of spectral order for closed contact 3-manifolds due to Kutluhan, Matić, Van Horn-Morris, and Wand to contact 3-manifolds with convex boundary. We show that the order of a codimension zero contact submanifold bounds the order of the ambient manifold from above. As the…

We extract a nonnegative integer-valued invariant, which we call the "order of algebraic torsion", from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic tors…

Study non-squeezing phenomena in contact geometry using specific capacities.

We consider contact elements in the sutured Floer homology of solid tori with longitudinal sutures, as part of the (1+1)-dimensional topological quantum field theory defined by Honda--Kazez--Matić in \cite{HKM08}. The $\Z_2$ $SFH$ of these solid tori forms a "categorification of Pascal's triangle", and contact structur…

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 analyzes errors in mechanical systems with external forces.

It is well known that Cayley's ruled cubic surface carries a three-parameter family of twisted cubics sharing a common point, with the same tangent and the same osculating plane. We report on various results and open problems with respect to contact of higher order and dual contact of higher order for these curves.

The study proves properties of spectral selectors for contact manifolds and applies them to contact big fibers and geodesics.

Develops k-contact geometry theory for field theories.

We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result i…

Jet bundles as higher-order polarised $k$-contact manifolds

Using contact homology, we reobtain some recent results of Geiges and Gonzalo about the fundamental group of the space of contact structures on some 3-manifolds. We show that our techniques can be used to study higher dimensional contact manifolds and higher order homotopy groups.

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 …

Cayley's (ruled cubic) surface carries a three-parameter family of twisted cubics. We describe the contact of higher order and the dual contact of higher order for these curves and show that there are three exceptional cases.

We show that for a big class of contact manifolds the groups of order $\leq n$ invariants (with values in an arbitrary Abelian group) of Legendrian, of transverse and of framed knots are canonically isomorphic. On the other hand for an arbitrary cooriented contact structure on $S^1\times S^2$ with the nonzero Euler cla…

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…

In previous work we showed that the contact category algebra of a quadrangulated surface is isomorphic to the homology of a strand algebra from bordered sutured Floer theory. Being isomorphic to the homology of a differential graded algebra, this contact category algebra has an A-infinity structure, allowing us to comb…

Contact Lie systems analyze integral curves of Hamiltonian vector fields.

We address the problem of local geometry of third order ODEs modulo contact, point and fibre-preserving transformations of variables. Several new and already known geometries are described in a uniform manner by the Cartan method of equivalence. This includes conformal, Weyl and metric geometries in three and six dimen…

Local flatness theorem for paraquaternionic contact structures.

We collect our recent results ([5] and [8]) and we get the equivalence of the three notions of the title under some conditions. We then use this equivalence in order to prove some consequences about Sasakian manifolds, complex almost contact structures and complex k-contact structures.

Unified interpretation of sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds.

In this survey, we give an overview of Brieskorn manifolds and varieties, and their role in contact topology. We discuss open books, fillings and invariants such as contact and symplectic homology. We also present some new results involving exotic contact structures, invariants and orderability. The main tool for the r…

In the present paper we consider the problem of local equivalence of second order ODEs which are cubic in second derivative under the action of the pseudogroup of contact transformations. We show how it may be reduced to the equivalence problem of 2-webs in $\mathbb{R}^3$ under the action of finite-dimensional group, a…

Extends Moutard quadric concept to higher dimensions.

Contact path geometries are curved geometric structures on a contact manifold comprising smooth families of paths modeled on the family of all isotropic lines in the projectivization of a symplectic vector space. Locally such a structure is equivalent to the graphs in the space of independent and depedent variables of …

Study stochastic processes on surfaces in contact sub-Riemannian manifolds using Riemannian approximations.

Study shows infinite order in mapping class groups for certain 3D shapes.

We find a general solution to the unique 7th order ODE admitting ten dimensional group of contact symmetries. The integral curves of this ODE are rational contact curves in $\PP^3$ which give rise to rational plane curves of degree six. The moduli space of these curves is a real form of the homogeneous space $Sp(4)/SL(…

We classify codimension two analytic submanifolds X of projective space having the property that any line through a general point p having contact to order two with X at p automatically has contact to order three. We give applications to the study of the Debarre--de Jong conjecture, and of n-dimensional varieties whose…

Lagrangian contact supersymmetries (depending on derivatives of arbitrary order) are treated in very general setting. The cohomology of the variational bicomplex on an arbitrary graded manifold and the iterated cohomology of a generic nilpotent contact supersymmetry are computed. In particular, the first variational fo…

We develop a systematic approach to contact and Jacobi structures on graded supermanifolds. In this framework, contact structures are interpreted as symplectic principal GL(1,R)-bundles. Gradings compatible with the GL(1,R)-action lead to the concept of a graded contact manifold, in particular a linear (more generally,…

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

The study shows examples of contact 3-manifold binding sums that fail to preserve certain properties.

Develops a new calculus for contact structures on manifolds.

In the present paper we establish the necessary and sufficient conditions for two ordinary differential equations of the form $y"{}^2+A(x,y,y') y"+B(x,y,y')=0$ to be equivalent under the action of the pseudogroup of contact transformations. These conditions are formulated in terms of integrals of some one-dimensional d…

A PhD thesis written under supervision of Pawel Nurowski and defended at the Faculty of Physics of the University of Warsaw. We adress the problems of local equivalence and geometry of third order ODEs modulo contact, point and fibre-preserving transformations of variables. Several new and already known geometries are …

In her PhD thesis Milin developed an equivariant version of the contact homology groups constructed by Eliashberg, Kim and Polterovich and used it to prove an equivariant contact non-squeezing theorem. In this article we re-obtain the same result in the setting of generating functions, starting from the homology groups…

For each simple Lie algebra $\mathfrak{g}$ (excluding, for trivial reasons, type ${\sf C}$) we find the lowest possible degree of an invariant second-order PDE over the adjoint variety in $\mathbb{P}\mathfrak{g}$, a homogeneous contact manifold. Here a PDE $F(x^i,u,u_i,u_{ij})=0$ has degree $\le d$ if $F$ is a polynomi…

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.

We give a classification of $1^{st}$ order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so called metaplectic contact projective type. These bundles are associated via representations, which are derived from the so called higher symplectic, harmonic …

New Lie systems defined on $k$-contact manifolds, with applications.

In this paper, we compute contact homology of some quasi-regular contact structures, which admit Hamiltonian actions of Reeb type of Lie groups. We will discuss the toric contact case, (where the torus is of Reeb type), and the case of homogeneous contact manifolds. In both of these cases the quotients by the Reeb acti…

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

We introduce higher order mean curvatures of screen almost conformal (SAC) half-lightlike submanifolds of indefinite contact manifolds, admitting a semi-symmetric non-metric connection, and use them to generalize some known results of [6]. Also, we derive a new set of integration formulae via the divergence of some spe…

The aim of this research is the study of Gray curvature identities, introduced by Alfred Gray in \cite{kn:Gra76} for the class of almost hermitian manifolds. As known till now, there is no equivalent for the class of almost contact manifolds. For this purpose we use the Boohby-Wang fibration and the warped manifolds co…

By using cobordism theoretic arguments similar to those in the literature on positive scalar curvature metrics we prove the existence of contact structures on 5-dimensional spin manifolds whose fundamental group is a group of odd order (not divisible by 9) and finite cohomological period.

Let Y be a compact, oriented 3-manifold with a contact form a. For any Dirac operator D, we study the asymptotic behavior of the spectral flow between D and D+cl(-ira) as r very large. If a is the Thurston-Winkelnkemper contact form whose monodromy is the product of Dehn twists along disjoint circles, we prove that the…