Narrow Results

We define a ${\mathbb{Z}}_k$-equivariant version of the cylindrical contact homology used by Eliashberg–Kim–Polterovich [11] to prove contact non-squeezing for prequantized integer-capacity balls $B(R)\times S^1\subset {ℝ}^{2n}\times S^1$, $R\in \mathbb{N}$ and we use it to extend their result to all $R\ge 1$. Specifically, we prove if $R\ge 1$ there is no $\psi \in \text{Cont}({ℝ}^{2n}\times S^1)$, the group of compactly supported contactomorphisms of ${ℝ}^{2n}\times S^1$ which squeezes $\widehat{B}(R)=B(R)\times S^1$ into itself, i.e. maps the closure of $\widehat{B}(R)$ into $\widehat{B}(R)$. A sheaf theoretic proof of non-existence of corresponding $\psi \in {\text{Cont}}_0({ℝ}^{2n}\times S^1)$, the identity component of $\text{Cont}({ℝ}^{2n}\times S^1)$, is due to Chiu [7] it is not known if this is strictly weaker. Our construction has the advantage of retaining the contact homological viewpoint of Eliashberg–Kim–Polterovich and its potential for application in prequantizations of other Liouville manifolds. It makes use of the ${\mathbb{Z}}_k$-action generated by a vertical $1/k$-shift but can also be related, for prequantized balls, to the ${\mathbb{Z}}_k$-equivariant contact homology developed by Milin [16] in her proof of orderability of lens spaces.

We analyze the transverse Kähler–Ricci flow equation on Sasaki-Einstein space $Y^{p,q}$. Explicit solutions are produced representing new five-dimensional Sasaki structures. Solutions which do not modify the transverse metric preserve the Sasaki–Einstein feature of the contact structure. If the transverse metric is altered, the deformed metrics remain Sasaki, but not Einstein.

We study the transverse Kähler structure of the Sasaki–Einstein space $T^{1,1}$. A set of local holomorphic coordinates is introduced and a Sasakian analogue of the Kähler potential is given. We investigate deformations of the Sasaki–Einstein structure preserving the Reeb vector field, but modifying the contact form. For this kind of deformations, we consider the Sasaki–Ricci flow which converges in a suitable sense to a Sasaki–Ricci soliton. Finally, it is described the constructions of Hamiltonian holomorphic vector fields and Hamiltonian function on the $T^{1,1}$ manifold.

In this paper we examine the relationship between various types of positivity for knots and the concordance invariant τ discovered by Ozsváth and Szabó and independently by Rasmussen. The main result shows that, for fibered knots, τ characterizes strong quasipositivity. This is quantified by the statement that for K fibered, τ(K) = g(K) if and only if K is strongly quasipositive. A corollary is that any knot admitting a lens space surgery or, more generally, an L-space surgery, is strongly quasipositive. In addition, we survey existing results regarding τ and forms of positivity and highlight several consequences concerning the types of knots which are (strongly) (quasi) positive.

We show that a decorated knot concordance ${𝒞}$ from $K_0$ to $K_1$ induces an $𝔽[U]$-module homomorphism

We prove a Steiner formula for regular surfaces with no characteristic points in 3D contact sub-Riemannian manifolds endowed with an arbitrary smooth volume. The formula we obtain, which is equivalent to a half-tube formula, is of local nature. It can thus be applied to any surface in a region not containing characteristic points. We provide a geometrical interpretation of the coefficients appearing in the expansion, and compute them on some relevant examples in three-dimensional sub-Riemannian model spaces. These results generalize those obtained in [Z. M. Balogh, F. Ferrari, B. Franchi, E. Vecchi and K. Wildrick, Steiner’s formula in the Heisenberg group, Nonlinear Anal. 126 (2015) 201–217; M. Ritoré, Tubular neighborhoods in the sub-Riemannian Heisenberg groups, Adv. Calc. Var. 14(1) (2021) 1–36] for the Heisenberg group.

We compute the first cohomology of the affine Lie superalgebra $𝔞𝔣𝔣(2|1)$ on the (1,2)-dimensional real superspace with coefficients in the superspace ${𝔇}_{\lambda ;\mu }^2$ of linear differential operators acting on weighted densities. We also compute the same, but $𝔞𝔣𝔣(1|1)$-relative, cohomology. We explicitly give $1$-cocycles spanning these cohomology.

These are the lecture notes for the course given at the “XXVII International Fall Workshop on Geometry and Physics” held in Seville (Spain) in September 2018. We review the geometric formulation of equilibrium thermodynamics by means of contact geometry, together with the associated metric structures arising from thermodynamic fluctuation theory and the use of Hamiltonian flows to describe thermodynamic processes. Finally, we discuss the state of the art of the subject, its connection with other areas of physics, and suggest several possible further directions.

We present a systematic treatment of line bundle geometry and Jacobi manifolds with an application to geometric mechanics that has not been noted in the literature. We precisely identify categories that generalize the ordinary categories of smooth manifolds and vector bundles to account for a lack of choice of a preferred unit, which in standard differential geometry is always given by the global constant function $1$. This is what we call the “unit-free” approach. After giving a characterization of local Lie brackets via their symbol maps, we apply our novel categorical language to review Jacobi manifolds and related notions such as Lichnerowicz brackets and Jacobi algebroids. The main advantage of our approach is that Jacobi geometry is recovered as the direct unit-free generalization of Poisson geometry, with all the familiar notions translating in a straightforward manner. We then apply this formalism to the question of whether there is a unit-free generalization of Hamiltonian mechanics. We identify the basic categorical structure of ordinary Hamiltonian mechanics to argue that it is indeed possible to find a unit-free analogue. This paper serves as a prelude to the investigation of dimensioned structures, an attempt at a general mathematical framework for the formal treatment of physical quantities and dimensional analysis.

This work applies the contact formalism of classical mechanics and classical field theory, introduced by Herglotz and later developed in the context of contact geometry, to describe electromagnetic systems with dissipation. In particular, we study an electron in a non-perfect conductor and a variation of the cyclotron radiation. In order to apply the contact formalism to a system governed by the Lorentz force, it is necessary to generalize the classical electromagnetic gauge and add a new term in the Lagrangian. We also apply the k-contact theory for classical fields to model the behavior of electromagnetic fields themselves under external damping. In particular, we show how the theory describes the evolution of electromagnetic fields in media under some circumstances. The corresponding Poynting theorem is derived. We discuss its applicability to the Lorentz dipole model and to a highly resistive dielectric.

The contact geometric structure of the thermodynamic phase space is used to introduce a novel symplectic structure on the tangent bundle of the equilibrium space. Moreover, it turns out that the equilibrium space can be interpreted as a Lagrange submanifold of the corresponding tangent bundle, if the fundamental equation is known explicitly. As a consequence, Hamiltonians can be defined that describe thermodynamic processes.

In this paper, we investigate various types of symmetries and their mutual relationships in Hamiltonian systems defined on manifolds with different geometric structures: symplectic, cosymplectic, contact and cocontact. In each case, we pay special attention to non-standard (non-canonical) symmetries, in particular scaling symmetries and canonoid transformations, as they provide new interesting tools for the qualitative study of these systems. Our main results are the characterizations of these non-standard symmetries and the analysis of their relation with conserved (or dissipated) quantities.

We give a self-contained and geometric account of a recent approach to the Einstein field equations of general relativity, based on families of null foliations of space–time. We then use exterior differential systems to make explicit the correspondence between conformal Lorentzian geometry in dimensions three and four and the contact geometry of special classes of differential systems.

Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra of contact vector fields on the (1, 1)-dimensional real or complex superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute the same, but 𝔬𝔰𝔭(1|2)-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal 𝔬𝔰𝔭(1|2)-trivial deformations of the -module structure on the superspaces of symbols of differential operators. We prove that any generic formal 𝔬𝔰𝔭(1|2)-trivial deformation of this -module is equivalent to a polynomial one of degree ≤ 4. This work is the simplest superization of a result by Bouarroudj [On 𝔰𝔩(2)-relative cohomology of the Lie algebra of vector fields and differential operators, J. Nonlinear Math. Phys. No. 1 (2007) 112–127]. Further superizations correspond to 𝔬𝔰𝔭(N|2)-relative cohomology of the Lie superalgebras of contact vector fields on 1|N-dimensional superspace.

Following a line of reasoning suggested by Eliashberg, we prove Cerf's theorem that any diffeomorphism of the 3-sphere extends over the 4-ball. To this end we develop a moduli-theoretic version of Eliashberg's filling-with-holomorphic-discs method.

An important tool to analyse the causal structure of a Lorentzian manifold is given by the Lorentzian distance function. We define a class of Lorentzian distance functions on the group of contactomorphisms of a closed contact manifold depending on the choice of a contact form. These distance functions are continuous with respect to the Hofer norm for contactomorphisms defined by Shelukhin [The Hofer norm of a contactomorphism, J. Symplectic Geom. 15 (2017) 1173–1208] and finite if and only if the group of contactomorphisms is orderable. To prove this, we show that intervals defined by the positivity relation are open with respect to the topology induced by the Hofer norm. For orderable Legendrian isotopy classes we show that the Chekanov-type metric defined in [D. Rosen and J. Zhang, Chekanov’s dichotomy in contact topology, Math. Res. Lett. 27 (2020) 1165–1194] is nondegenerate. In this case, similar results hold for a Lorentzian distance functions on Legendrian isotopy classes. This leads to a natural class of metrics associated to a globally hyperbolic Lorentzian manifold such that its Cauchy hypersurface has a unit co-tangent bundle with orderable isotopy class of the fibres.

In this work, we use the fact that kinematics of light propagation in a non-dispersive medium associated with a bi-metric spacetime is expressed by means of a 1-parameter family of contact transformations. We present a general technique to find such transformations and explore some explicit examples for Minkowski and anti-deSitter spacetimes geometries.