Contact Manifolds Research Articles

Contact forms with large systolic ratio in arbitrary dimensions

If a contact form on a (2n+1)-dimensional closed contact manifold admits closed Reeb orbits, then its systolic ration is defined to be the quotient of (n+1)th power of the shortest period of Reeb orbits by the contact volume. We prove that every co-orientable contact structure on any closed contact manifold admits a contact form with arbitrarily systolic ratio. This statement generalizes the recent result of Abbondandolo et al. in dimension three to higher dimensions. We extend the plug construction of Abbondandolo et. al. to any dimension, by means of generalizing the hamiltonian disc maps studied by the authors to the symplectic ball of any dimension. The plug is a mapping torus and it is equipped with a special contact form so that one can use it to modify a given contact form if the Reeb flow leads to a circle bundle on a large portion of the given contact manifold. Inserting the plug sucks up the contact volume while the minimal period remains the same. Following the ideas of Abbondandolo et al. and using Giroux's theory of Liouville open books, we show that any co-orientable contact structure is defined by a contact form, which is suitable to be modified via inserting plugs.

A New Class of Contact Pseudo Framed Manifolds with Applications

In this paper, we introduce a new class of contact pseudo framed (CPF)-manifoldsM,g,f,λ,ξby a real tensor fieldfof type1,1, a real functionλsuch thatf3=λ2fwhereξis its characteristic vector field. We prove in our main Theorem 2 thatMadmits a closed 2-formΩifλis constant. In 1976, Blair proved that the vector fieldξof a normal contact manifold is Killing. Contrary to this, we have shown in Theorem 2 that, in general,ξof a normal CPF-manifold is non-Killing. We also have established a link of CPF-hypersurfaces with curvature, affine, conformal collineations symmetries, and almost Ricci soliton manifolds, supported by three applications. Contrary to the odd-dimensional contact manifolds, we construct several examples of even- and odd-dimensional semi-Riemannian and lightlike CPF-manifolds and propose two problems for further consideration.

Schatten-von Neumann classes of integral operators

In this work we establish sharp kernel conditions ensuring that the corresponding integral operators belong to Schatten-von Neumann classes. The conditions are given in terms of the spectral properties of operators acting on the kernel. As applications we establish several criteria in terms of different types of differential operators and their spectral asymptotics in different settings: compact manifolds, operators on lattices, domains in Rn of finite measure, and conditions for operators on Rn given in terms of anharmonic oscillators. We also give examples in the settings of compact sub-Riemannian manifolds, contact manifolds, strictly pseudo-convex CR manifolds, and (sub-)Laplacians on compact Lie groups.

Classical boundary field theory of Jacobi sigma models by Poissonization

In this paper, we are going to construct the classical field theory on the boundary of the embedding of \mathbb{R} \times S^{1}ℝ×S1 into the manifold MM by the Jacobi sigma model. By applying the poissonization procedure and by generalizing the known method for Poisson sigma models, we express the fields of the model as perturbative expansions in terms of the reduced phase space of the boundary. We calculate these fields up to the second order and illustrate the procedure for contact manifolds.

NSPG: An Efficient Posture Generator Based on Null-Space Alteration and Kinetostatics Constraints.

Most of the locomotion and contact planners for multi-limbed robots rely on a reduction of the search space to improve the performance of their algorithm. Posture generation plays a fundamental role in these types of planners providing a collision-free, statically stable whole-body posture, projected onto the planned contacts. However, posture generation becomes particularly tedious for complex robots moving in cluttered environments, in which feasibility can be hard to accomplish. In this work, we take advantage of the kinematic structure of a multi-limbed robot to present a posture generator based on hierarchical inverse kinematics and contact force optimization, called the null-space posture generator (NSPG), able to efficiently satisfy the aforementioned requisites in short times. A new configuration of the robot is produced through conservatively altering a given nominal posture exploiting the null-space of the contact manifold, satisfying geometrical and kinetostatics constraints. This is achieved through an adaptive random velocity vector generator that lets the robot explore its workspace. To prove the validity and generality of the proposed method, simulations in multiple scenarios are reported employing different robots: a wheeled-legged quadruped and a biped. Specifically, it is shown that the NSPG is particularly suited in complex cluttered scenarios, in which linear collision avoidance and stability constraints may be inefficient due to the high computational cost. In particular, we show an improvement of performances being our method able to generate twice feasible configurations in the same period. A comparison with previous methods has been carried out collecting the obtained results which highlight the benefits of the NSPG. Finally, experiments with the CENTAURO platform, developed at Istituto Italiano di Tecnologia, are carried out showing the applicability of the proposed method to a real corridor scenario.

On Einstein-type contact metric manifolds

In this paper, we consider an Einstein-type equation on contact metric manifolds. First, we prove that any K-contact manifold admitting an Einstein-type metric is isometric to a unit sphere. Similar conclusion also holds for a contact metric manifold admitting an Einstein type metric with zero radial Weyl curvature and satisfying a commutativity condition. Finally, we study an Einstein-type equation on (k,μ)-contact manifold.

Jacobi Manifolds, Contact Manifolds and Contactomorphism

Let M be a smooth manifold and let D(M) be the module of first order differential operators on M. In this work, we give a link between Jacobi manifolds and Contact manifolds. We also generalize the notion of contactomorphism on M and
 thus, we characterize the Contact diffeomorphisms.

The singular Weinstein conjecture

In this article, we investigate Reeb dynamics on bm-contact manifolds, previously introduced in [37], which are contact away from a hypersurface Z but satisfy certain transversality conditions on Z. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the well-known Weinstein conjecture. Contrary to the initial expectations, examples of compact bm-contact manifolds without periodic Reeb orbits outside Z are provided. Furthermore, we prove that in dimension 3, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the bm-Reeb flow exist in any dimension. This investigation goes hand-in-hand with the Weinstein conjecture on non-compact manifolds having compact ends of convex type. In particular, we extend Hofer's arguments to open overtwisted contact manifolds that are R+-invariant in the open ends, obtaining as a corollary the existence of periodic bm-Reeb orbits away from the critical set. The study of bm-Reeb dynamics is motivated by well-known problems in fluid dynamics and celestial mechanics, where those geometric structures naturally appear. In particular, we prove that the dynamics on positive energy level-sets in the restricted planar circular three body problem are described by the Reeb vector field of a b3-contact form that admits an infinite number of periodic orbits at the critical set.

Relative growth rate and contact Banach–Mazur distance

In this paper, we define non-linear versions of Banach–Mazur distance in the contact geometry set-up, called contact Banach–Mazur distances and denoted by \(d_\mathrm{CBM}\). Explicitly, we consider the following two set-ups, either on a contact manifold \(W \times S^1\) where W is a Liouville manifold, or a closed Liouville-fillable contact manifold M. The inputs of \(d_\mathrm{CBM}\) are different in these two cases. In the former case the inputs are (contact) star-shaped domains of \(W \times S^1\) which correspond to the homotopy classes of positive contact isotopies, and in the latter case the inputs are contact 1-forms of M inducing the same contact structure. In particular, the contact Banach–Mazur distance \(d_\mathrm{CBM}\) defined in the former case is motivated by the concept, relative growth rate, which was originally defined and studied by Eliashberg–Polterovich. The main results are the large-scale geometric properties in terms of \(d_\mathrm{CBM}\). In addition, we propose a quantitative comparison between elements in a certain subcategory of the derived categories of sheaves of modules (over certain topological spaces). This is based on several important properties of the singular support of sheaves and Guillermou–Kashiwara–Schapira’s sheaf quantization.

Geometrical particles and dualities related to curves in null de Sitter 3-sphere

In this paper, we confine the trajectory of geometrical particles to the de Sitter three-dimensional space–time, we model geometrical particles as spacelike tachyons. Using the Legendrian duality theory of pseudo-spheres and contact manifolds theory we establish the dual relationships between spacelike moving trajectory of geometrical particles and future nullcone hypersurfaces.

Topological and Dynamical Aspects of Jacobi Sigma Models

The geometric properties of sigma models with target space a Jacobi manifold are investigated. In their basic formulation, these are topological field theories—recently introduced by the authors—which share and generalise relevant features of Poisson sigma models, such as gauge invariance under diffeomorphisms and finite dimension of the reduced phase space. After reviewing the main novelties and peculiarities of these models, we perform a detailed analysis of constraints and ensuing gauge symmetries in the Hamiltonian approach. Contact manifolds as well as locally conformal symplectic manifolds are discussed, as main instances of Jacobi manifolds.

Morse-Smale characteristic foliations and convexity in contact manifolds

We generalize a result of Giroux which says that a closed surface in a contact 3 3 -manifold with Morse-Smale characteristic foliation is convex. Specifically, we show that the result holds in contact manifolds of arbitrary dimension. As an application, we show that a particular closed hypersurface introduced by A. Mori is C ∞ C^{\infty } -close to a convex hypersurface.

Geometry of almost Ricci solitons on paracontact metric manifolds

The purpose of this article is to investigate almost Ricci solitons on para- contact manifolds. We demonstrate that a gradient almost Ricci soliton whose metric is para-sasakian turns into Einstein with a constant scalar curvature −2n(2n + 1). Next, we get a few relationships between almost Ricci solitons and Ricci solitons on a K-paracontact manifold and its generalizations. Finally, some findings on para- contact manifolds and H-paracontact manifolds admitting an almost Ricci soliton with a potential vector field that is a pointwise collinear with the Reeb vector field are discovered.

Quantum Darboux theorem

The problem of computing quantum mechanical propagators can be recast as a computation of a Wilson line operator for parallel transport by a flat connection acting on a vector bundle of wavefunctions. In this picture the base manifold is an odd dimensional symplectic geometry, or quite generically a contact manifold that can be viewed as a "phase-spacetime", while the fibers are Hilbert spaces. This approach enjoys a "quantum Darboux theorem" that parallels the Darboux theorem on contact manifolds which turns local classical dynamics into straight lines. We detail how the quantum Darboux theorem works for anharmonic quantum potentials. In particular, we develop a novel diagrammatic approach for computing the asymptotics of a gauge transformation that locally makes complicated quantum dynamics trivial.

Four-dimensional aspects of tight contact 3-manifolds

We conjecture a four-dimensional characterization of tightness: A contact structure on a 3-manifold Y is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in [Formula: see text] An affirmative answer to our conjecture would imply an analogue of the Milnor conjecture for torus knots: If a fibered link L induces a tight contact structure on Y, then its fiber surface maximizes the Euler characteristic among all surfaces in [Formula: see text] with boundary L. We provide evidence for both conjectures by proving them for contact structures with nonvanishing Ozsváth-Szabó contact invariant.

On the equivalence of contact invariants in sutured Floer homology theories

We recently defined an invariant of contact manifolds with convex boundary in Kronheimer and Mrowka's sutured monopole Floer homology theory. Here, we prove that there is an isomorphism between sutured monopole Floer homology and sutured Heegaard Floer homology which identifies our invariant with the contact class defined by Honda, Kazez and Mati\'c in the latter theory. One consequence is that the Legendrian invariants in knot Floer homology behave functorially with respect to Lagrangian concordance. In particular, these invariants provide computable and effective obstructions to the existence of such concordances. Our work also provides the first proof which does not rely on the relative Giroux correspondence that the vanishing or non-vanishing of Honda, Kazez and Mati\'c's contact class is a well-defined invariant of contact manifolds.

Ricci Curvature, Reeb Flows and Contact 3-Manifolds

Given a contact 3-manifold we consider the problem of when a given function can be realized as the Ricci curvature of a Reeb vector field for the contact structure. We will use topological tools to show that every admissible function can be realized as such Ricci curvature for a singular metric which is an honest compatible metric away from a measure zero set. However, we will see that resolving such singularities depends on contact topological data and is yet to be fully understood.

Heegaard Floer invariants of contact structures on links of surface singularities

Let a contact 3-manifold (Y, \xi_0) be the link of a normal surface singularity equipped with its canonical contact structure \xi_0 . We prove a special property of such contact 3-manifolds of "algebraic" origin: the Heegaard Floer invariant c^+(\xi_0)\in \mathrm {HF}^+(-Y) cannot lie in the image of the U -action on \mathrm {HF}^+(-Y) . It follows that Karakurt's "height of U -tower" invariants are always 0 for canonical contact structures on singularity links, which contrasts the fact that the height of U -tower can be arbitrary for general fillable contact structures. Our proof uses the interplay between the Heegaard Floer homology and Némethi's lattice cohomology.

The Yamabe operator and invariants on octonionic contact manifolds and convex cocompact subgroups of F4(−20)

An octonionic contact (OC) manifold is always spherical. We construct the OC Yamabe operator on an OC manifold and prove its transformation formula under conformal OC transformations. An OC manifold is scalar positive, negative or vanishing if and only if its OC Yamabe invariant is positive, negative or zero, respectively. On a scalar positive OC manifold, we can construct the Green function of the OC Yamabe operator and apply it to construct a conformally invariant tensor. It becomes an OC metric if the OC positive mass conjecture is true. We also show the connected sum of two scalar positive OC manifolds to be scalar positive if the neck is sufficiently long. On the OC manifold constructed from a convex cocompact subgroup of F4(−20), we construct a Nayatani-type Carnot–Caratheodory metric. As a corollary, such an OC manifold is scalar positive, negative or vanishing if and only if the Poincare critical exponent of the subgroup is less than, greater than or equal to 10, respectively.

On non-gradient $$(m,\rho )$$-quasi-Einstein contact metric manifolds

Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the $$(m,\rho )$$ -quasi-Einstein structure on a contact metric manifold. First, we prove that if a K-contact or Sasakian manifold $$M^{2n+1}$$ admits a closed $$(m,\rho )$$ -quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature $$2n(2n+1)$$ , and for the particular case—a non-Sasakian $$(k,\mu )$$ -contact structure—it is locally isometric to the product of a Euclidean space $${\mathbb {R}}^{n+1}$$ and a sphere $$S^n$$ of constant curvature 4. Next, we prove that if a compact contact or H-contact metric manifold admits an $$(m,\rho )$$ -quasi-Einstein structure, whose potential vector field V is collinear to the Reeb vector field, then it is a K-contact $$\eta $$ -Einstein manifold.