arXiv research

Develops k-contact geometry theory for field theories.

New HyperKahler structure found for 3-contact distributions on Sasakian manifolds.

Study new $k$-contact distributions and Lie systems.

The article proves the existence of horizontal immersions into fat distributions and contact structures.

Local normal forms for symmetrical contact structures on 3-manifolds.

Smooth contact mappings in a flat (2,3,5)-distribution are shown to be smoother.

We generalise the notion of contact manifold by allowing the contact distribution to have codimension two. There are special features in dimension six. In particular, we show that the complex structure on a three-dimensional complex contact manifold is determined solely by the underlying contact distribution.

Study cone structures on contact manifolds to understand their geometric properties.

We study the sectional curvature of plane distributions on 3-manifolds. We show that if the distribution is a contact structure it is easy to manipulate this curvature. As a corollary we obtain that for every transversally oriented contact structure on a closed 3-dimensional manifold $M$ there is a metric, such that th…

In this paper, the HyperKahler contact distribution of a 3-Sasakian manifold is studied. To analyze the curvature properties of this distribution, the special metric connection $\bar{\nabla}$ is defined. This metric connection is completely determined by HyperKahler contact distribution. We prove that HyperKahler conta…

The study finds many tight contact structures on hyperbolic 3-spheres.

New example disproves complex contact theory for fat distributions with Reeb directions.

The study defines and analyzes semi-invariant submanifolds in complex contact metric manifolds.

The study generalizes a specific geometric correspondence to higher dimensions.

Solutions to a differential equation link to contact structures.

In this paper, the notion of an almost contact Kählerian structure is introduced. The interior geometry of almost contact Kählerian spaces is investigated. On the zero-curvature distribution of an almost contact metric structure, as on the total space of a vector bundle, an almost contact Kählerian structure is obtaine…

We prove Gray--Moser stability theorems for complementary pairs of forms of constant class defining symplectic pairs, contact-symplectic pairs and contact pairs. We also consider the case of contact-symplectic and contact-contact structures, in which the constant class condition on a one-form is replaced by the conditi…

There are two well-known parabolic split $G_2$-geometries in dimension five, $(2,3,5)$-distributions and $G_2$-contact structures. Here we link these two geometries with yet another $G_2$-related contact structure, which lives on a seven-manifold. We present a natural geometric construction of a Lie contact structure o…

The paper studies a new structure on contact manifolds and its foliation properties.

The paper classifies 3D contact partially hyperbolic diffeomorphisms.

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

Study weak quasi contact metric manifolds to generalize K-contact and Sasakian manifolds criteria.

Positive paths connect diffeomorphisms on contact manifolds.

Survey of recent results in weak almost contact structures.

The paper introduces a statistical version of contact CR-product for Sasakian statistical manifolds.

It is known that for every smooth great circle fibration of the 3-sphere, the distribution of tangent 2-planes orthogonal to the fibres is a contact structure, in fact a tight one, but we show here that, beginning with the 5-sphere, there exist smooth great circle fibrations of all odd-dimensional spheres for which the…

New contact structures defined on differentiable stacks.

A contact manifold is a manifold equipped with a distribution of codimension one that satisfies a `maximal non-integrability' condition. A standard example of a contact structure is a strictly pseudoconvex CR manifold, and operators of analytic interest are the tangential Cauchy-Riemann operator and the Szego projector…

I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on ma…

We show that the CR structure on the twistor space of a quaternionic contact structure described by Biquard is normal if and only if the Ricci curvature of the Biquard connection commutes with the endomorphisms in the quaternionic structure of the contact distribution.

Study on null submanifolds in indefinite complex contact geometry.

Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any Riemannian manifold, which is universal with respect to isometric embeddings. I…

In this paper we generalize the main notions from the geometry of (almost) contact manifolds in the category of Lie algebroids. Also, using the framework of generalized geometry, we obtain an (almost) contact Riemannian Lie algebroid structure on a vertical Liouville distribution over the big-tangent manifold of a Riem…

We regard a contact metric manifold whose Reeb vector field belongs to the $(κ,μ)$-nullity distribution as a bi-Legendrian manifold and we study its canonical bi-Legendrian structure. Then we characterize contact metric $(κ,μ)$-spaces in terms of a canonical connection which can be naturally defined on them.

There are introduced and studied a pair of associated Schouten-van Kampen affine connections adapted to the contact distribution and an almost contact B-metric structure generated by the pair of associated B-metrics and their Levi-Civita connections. By means of the constructed non-symmetric connections, the basic clas…

Study investigates induced geometry on surfaces in 3D contact manifolds.

On a manifold with an almost contact metric structure $(\varphi,\vecξ,η,g,X,D)$ the notions of the interior and the $N$-prolonged connections are introduced. Using the $N$-prolonged connection, a new almost contact metric structure is defined on the distribution $D$. The properties of this structure are studied.

We study the sub-Riemannian exponential for contact distributions on manifolds of dimension greater or equal to 5. We compute an approximation of the sub-Riemannian Hamiltonian flow and show that the conjugate time can have multiplicity 2 in this case. We obtain an approximation of the first conjugate locus for small r…

We explore the consequences of curvature and torsion on the topology of quaternionic contact manifolds with integrable vertical distribution. We prove a general Myers theorem and establish a Cartan-Hadamard result for almost qc-Einstein manifolds.

We study the behavior of the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the eta-invariant and the determinant of the L…

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…

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

Study of contact-complex Riemannian submersions in various manifold classes.

Study proves h-principles for curves in bracket-generating distributions.

The contact graph of a CAT(0) cubical complex has unbounded structure and a Gaussian CLT for random walks.

Functor connects symplectic and contact structures via cutting and blowups.

The paper characterizes Sasakian manifolds using weak nearly Sasakian structures.

We obtain some results on symmetries of sub-Riemannian surfaces. In case of contact sub-Riemannian surface we base on invariants found by Hughen \cite{Hughen}. Using these invariants, we find conditions under which a sub-Riemannian surface does not admit symmetries. If a surface admits symmetries, we show how invariant…