A Symplectic Dynamics Proof of the Degree–Genus Formula

We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere (defining the Hopf fibration), with boundaries oriented positively by the flow. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory.

Cartan connections and path structures with large automorphism groups

In this paper, we classify compact manifolds of dimension three equipped with a path structure and a fixed contact form (which we refer to as a strict path structure) under the hypothesis that their automorphism group is non-compact. We use a Cartan connection associated to the structure and show that its curvature is constant.

Analytical solution to contact characteristics for a variable hyperbolic circular-arc-tooth-trace cylindrical gear

The variable hyperbolic circular-arc-tooth-trace (VH-CATT) cylindrical gear is a new type of gear. In order to research the contact characteristics of the VH-CATT cylindrical gear, tooth surface mathematical models of this kind of gear pair are derived based on the forming principle of the rotating double-edged cutting method with great cutter head in this regard. Then, according to the differential geometry theory and Hertz theory, the formula of the induced normal curvature and equation of the contact ellipse are derived based on the condition of continuous tangency of two meshing surfaces, which proves that the contact form of VH-CATT cylindrical gear is point contact. The present work establishes analytical solutions to research the effect of different parameters for the contact characteristic of the VH-CATT cylindrical gear by incorporating elastic deformation on the tooth surface, which have shown that the module, tooth number and cutter radius have a crucial effect on the induced normal curvature and contact ellipse of the VH-CATT cylindrical gear in the direction of tooth trace and tooth profile. Moreover, a theoretical analysis solution, a finite element analysis and the gear tooth contact pattern are carried out to verify the correctness of the computerized model and to investigate the contact type of the gear; it is verified that the contact form on the concave surface of the driving VH-CATT cylindrical gear rotates from dedendum at the heel to the addendum at toe and is an instantaneous oblique ellipse due to elastic deformation of the contact tooth profile, and the connecting line of the ellipse center is the contact trace path. It is indicated that the research results are beneficial for research on tooth break reduction, pitting, wear resistance and fatigue life improvement of the VH-CATT cylindrical gear. The results also have a certain reference value for development of the VH-CATT cylindrical gear.

Sasaki–Ricci Flow and Deformations of Contact Action–Angle Coordinates on Spaces T1,1 and Yp,q

In this paper, we are concerned with completely integrable Hamiltonian systems and generalized action–angle coordinates in the setting of contact geometry. We investigate the deformations of the Sasaki–Einstein structures, keeping the Reeb vector field fixed, but changing the contact form. We examine the modifications of the action–angle coordinates by the Sasaki–Ricci flow. We then pass to the particular cases of the contact structures of the five-dimensional Sasaki–Einstein manifolds T1,1 and Yp,q.

Pseudorotations of the -disc and Reeb flows on the -sphere

We use Lerman’s contact cut construction to find a sufficient condition for Hamiltonian diffeomorphisms of compact surfaces to embed into a closed $3$ -manifold as Poincaré return maps on a global surface of section for a Reeb flow. In particular, we show that the irrational pseudorotations of the $2$ -disc constructed by Fayad and Katok embed into the Reeb flow of a dynamically convex contact form on the $3$ -sphere.

Influence of contact form on mechanical wear characteristics of single diamond scratching on Ta12W

Purpose Mechanical wear is the main wear form of abrasive single crystal diamond (SCD) grit. The mechanical wear of SCD grit has a significant influence on the tool life and machining quality. This paper aims to investigate the influence of grit–workpiece contact form on the mechanical wear characteristics of SCD grit. Design/methodology/approach Three different grit–workpiece contact forms, which involved point/line/face contact forms, are investigated to reveal the wear mechanism of SCD grit scratching on Ta12W. The wear morphology, wear volume and scratching forces are measured, and the stress models of different contact forms are analyzed. Findings The results show that mechanical wear mainly occurs in the grit–workpiece contact area and increases gradually from contact area to entire SCD grit. The scratching forces vary with the mechanical wear progress of SCD grits. The SCD grit with point contact form is the most prone to produce wear. The SCD grit with face contact form can remove more material volume than the other two SCD grits, and it is the most wear resistant. The stress state is closely related with the mechanical wear of SCD grit. The contact form has a significant influence on the mechanical wear of SCD grit. Originality/value The results of this study can provide a theoretical basis for the fabrication of abrasive tools.

Planarity in Higher-Dimensional Contact Manifolds

We obtain several results for (iterated) planar contact manifolds in higher dimensions. (1) Iterated planar contact manifolds are not weakly symplectically co-fillable. This generalizes a 3D result of Etnyre [ 14] to a higher-dimensional setting, where the notion of weak fillability is that due to Massot-Niederkrüger-Wendl [ 38]. (2) They do not arise as nonseparating weak contact-type hypersurfaces in closed symplectic manifolds. This generalizes a result by Albers-Bramham-Wendl [ 4]. (3) They satisfy the Weinstein conjecture, that is, every contact form admits a closed Reeb orbit. This is proved by an alternative approach as that of [ 2] and is a higher-dimensional generalization of a result of Abbas-Cieliebak-Hofer [ 1]. The results follow as applications from a suitable symplectic handle attachment, which bears some independent interest.