arXiv research

This paper constructs a family of coordinate systems about a point on a quaternionic contact manifold, called quaternionic contact pseudohermitian normal coordinates. Once defined, conformal variations of the quaternionic contact structure induce changes on the coordinates which are studied in an effort to simplify the…

Normalizes pseudo-Einstein contact forms for easier analysis.

Let $\CV$ be a vector field distribution on manifold $M$. We give an efficient algorithm for the construction of local coordinates on $M$ such that $\CV$ may be locally expressed as some partial prolongation of the contact distribution $\Cal C^{(1)}_q$, on the first order jet bundle of maps from $\Bbb R$ to $\Bbb R^q$,…

We present a compared analysis of some properties of 3-Sasakian and 3-cosymplectic manifolds. We construct a canonical connection on an almost 3-contact metric manifold which generalises the Tanaka-Webster connection of a contact metric manifold and we use this connection to show that a 3-Sasakian manifold does not adm…

Study finds lowest-degree invariant PDEs over rational contact manifolds.

Geometric study of thermodynamics using cotangent bundles.

Contact Riemannian manifolds, whose complex structures are not necessarily integrable, are generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection plays the role of the Tanaka-Webster connection of a pseudohermitian manifold. Conformal transformations and the Yamabe problem are a…

The study explores autonomous systems and their connections to contact geometry and Frobenius manifolds.

New geometric framework for non-conservative field theories with time-dependent terms.

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

Torus knots $(4,5)$ and $(5,6)$ provide counterexamples to a conjecture about knot contact homology.

Study knot contact homology and character varieties to prove a conjecture about knots.

Develop contact Tulczyjew formalism for dissipative dynamics on skew algebroids.

The paper introduces Carnot coordinates for Carnot manifolds, simplifying nilpotent approximation.

A new metriplectic system on contact manifolds is introduced for thermodynamic consistency.

Study of contact whirl curves in Sasakian Lorentzian 3-manifolds.

Let $M$ be a three-dimensional contact manifold and $ψ:D\setminus\{0\}\to M\times{\Bbb R}$ a finite-energy pseudoholomorphic map from a punctured disc in ${\Bbb C}$, that is asymptotic to a periodic orbit of the Reeb vector field. This article examines conditions under which smooth coordinates may be defined in a tubul…

Cyclidic nets are introduced as discrete analogs of curvature line parametrized surfaces and orthogonal coordinate systems. A 2-dimensional cyclidic net is a piecewise smooth $C^1$-surface built from surface patches of Dupin cyclides, each patch being bounded by curvature lines of the supporting cyclide. An explicit de…

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…

Equivalence of conformal maps proved in sub-Riemannian manifolds.

Locally symplectic structure found on Kerr space-time.

We provide necessary and sufficient conditions on the derived type of a vector field distribution $\Cal V$ in order that it be locally equivalent to a partial prolongation of the contact distribution $\Cal C^{(1)}_q$, on the first order jet bundle of maps from $\Bbb R$ to $\Bbb R^q$, $q\geq 1$. This result fully genera…

In the same way that a contact manifold determines and is determined by a symplectic cone, a Sasaki manifold determines and is determined by a suitable Kahler cone. Kahler-Sasaki geometry is the geometry of these cones. This paper presents a symplectic action-angle coordinates approach to toric Kahler geometry and how …

This research extends Sasakian geometry to non-necessarily coorientable contact structures.

Survey article analyzes pseudoholomorphic curves on symplectization via contact instantons.

Study null geodesics on specific spacetimes, finding contact manifolds and Engel geometry applications.

We investigate the number of geodesics between two points $p$ and $q$ on a contact sub-Riemannian manifold M. We show that the count of geodesics on $M$ is controlled by the count on its nilpotent approximation at $p$ (a contact Carnot group). For contact Carnot groups we make the count explicit in exponential coordina…

For curved projective manifolds we introduce a notion of a normal tractor frame field, based around any point. This leads to canonical systems of (redundant) coordinates that generalise the usual homogeneous coordinates on projective space. These give preferred local maps to the model projective space that encode geome…

Elie Cartan's general equivalence problem is recast in the language of Lie algebroids. The resulting formalism, being coordinate and model-free, allows for a full geometric interpretation of Cartan's method of equivalence via reduction and prolongation. We show how to construct certain normal forms (Cartan algebroids) …

Paper tackles dynamic behavior of variable topology mechanisms, presenting new transition conditions.

Unified geometric framework for integrability of conservative and dissipative systems.

A new geometric approach to quantum mechanics simplifies time-dependent problems.

In the paper we develop a framework for the alternative way of the study of a local geometry of almost cosymplectic manifolds with Kahlerian leaves. The main idea is to apply the concept of a geometry and analysis of CR manifolds. Locally the almost cosymplectic manifold is modeled on the 'mixed' space RxCn. There is g…

Classifies homogeneous Pfaffian forms on graded manifolds.

Principal binets generalize curvature line surfaces to square lattices and are a discrete integrable system.

As a step toward proving an index theorem for hypoelliptic operators Heisenberg manifolds, including those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes' tangent groupoid of a manifold $M$. As it is well known for a Heisenberg manifold $(M,H)$ the relevant notion of tangent is…

We consider the space of differential operators $\mathcal{D}_{λμ}$ acting between $λ$- and $μ$-densities defined on $S^{1|2}$ endowed with its standard contact structure. This contact structure allows one to define a filtration on $\mathcal{D}_{λμ}$ which is finer than the classical one, obtained by writting a differen…

Corrected Monti's blow-up analysis for H-minimizing sets in Heisenberg group.

This paper applies differential algebra to study equations in mathematical physics.

Research finds double bubbles in Grushin plane with specific geometric properties.

New examples show contact structures can be isotopic but not contact-isotopic.

The study explores embedding closed contact manifolds in higher dimensions.

Study on contact Hamiltonian functions for singular contact structures.

Contact round surgery proves existence of contact structures on 3-manifolds.

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

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

Simplified construction of contact triad connection for analysis.

The paper examines conditions for contact surgeries on rational homology 3-spheres.