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In this paper, we propose a discrete Hamilton–Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton–Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton–Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton–Jacobi equation is related to the continuous Hamilton–Jacobi theory presented in this work. Then, we propose geometric foundations of the discrete Hamilton–Jacobi equations on contact manifolds in terms of discrete contact flows. At the end of the paper, we provide a numerical example to test the theory. [ABSTRACT FROM AUTHOR]

The generalization of the virial theorem, introduced by Clausius in statistical mechanics, has recently been carried out in the framework of geometric approaches to Hamiltonian and Lagrangian theories and it has been formulated in an intrinsic way. It is here revisited not only in its more general situation of a generic vector field but mainly from the perspective of contact and cosymplectic geometry. The previous generalizations allowing virial like relations from one-parameter groups of non-strictly canonical transformations and the rôle of Killing and conformal Killing vector fields for Lagrangians of a mechanical type are here completed with the theory for contact Hamiltonian, as well as gradient and evolution, vector fields. The corresponding theories in the framework of cosymplectic geometry and the particular case of time-dependent vector fields are also developed. [ABSTRACT FROM AUTHOR]

In this paper, we establish nonlinear ellipticity of the equation of contact instantons with Legendrian boundary condition on punctured Riemann surfaces by proving the a priori elliptic coercive estimates for the contact instantons with Legendrian boundary condition, and prove an asymptotic exponential C ∞ -convergence result at a puncture under the uniform C 1 bound. We prove that the asymptotic charge of contact instantons at the punctures under the Legendrian boundary condition vanishes. This eliminates the phenomenon of the appearance of spiraling cusp instanton along a Reeb core, which removes the only remaining obstacle towards the compactification and the Fredholm theory of the moduli space of contact instantons in the open string case, which plagues the closed string case. Leaving the study of C 1 -estimates and details of Gromov-Floer-Hofer style compactification of contact instantons to [27], we also derive an index formula which computes the virtual dimension of the moduli space. These results are the analytic basis for the sequels [27–29] and [36] containing applications to contact topology and contact Hamiltonian dynamics. [ABSTRACT FROM AUTHOR]

In this paper we study coisotropic reduction in different types of dynamics according to the geometry of the corresponding phase space. The relevance of coisotropic reduction is motivated by the fact that these dynamics can always be interpreted as Lagrangian or Legendrian submanifolds. Furthermore, Lagrangian or Legendrian submanifolds can be reduced by a coisotropic one. [ABSTRACT FROM AUTHOR]

The hard Lefschetz property (HLP) is an important property which has been studied in several categories of the symplectic world. For Sasakian manifolds, this duality is satisfied by the basic cohomology (so, it is a transverse property), but a new version of the HLP has been recently given in terms of duality of the cohomology of the manifold itself in [1]. Both properties were proved to be equivalent (see [2]) in the case of K-contact flows. In this paper, we extend both versions of the HLP (transverse and not) to the more general category of isometric flows, and show that they are equivalent. We also give some explicit examples which illustrate the categories where the HLP could be considered. [ABSTRACT FROM AUTHOR]

In this paper, we propose a discrete Hamilton–Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton–Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton–Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton–Jacobi equation is related to the continuous Hamilton–Jacobi theory presented in this work. Then, we propose geometric foundations of the discrete Hamilton–Jacobi equations on contact manifolds in terms of discrete contact flows. At the end of the paper, we provide a numerical example to test the theory.

In this article, we study contact metric 3-manifolds with pseudo-parallel characteristic Jacobi operator. Contact metric 3-manifolds with pseudo-parallel characteristic Jacobi operator are M ℓ -manifolds (contact metric 3-manifolds with vanishing characteristic Jacobi operator) or generalized contact (κ , μ , ν) -spaces. Moreover, we prove that contact metric 3-manifolds with pseudo-parallel characteristic Jacobi operator are classified into four classes. In particular, we give a complete classification of homogeneous contact metric 3-manifolds with proper pseudo-parallel characteristic Jacobi operator. [ABSTRACT FROM AUTHOR]

In this article, we establish a Gaffney type inequality, in Wℓ,p{W}^{\ell ,p}-Sobolev spaces, for differential forms on sub-Riemannian contact manifolds without boundary, having bounded geometry (hence, in particular, we have in mind noncompact manifolds). Here, p∈]1,∞[p\in ]1,\infty {[} and ℓ=1,2\ell =1,2 depending on the order of the differential form we are considering. The proof relies on the structure of the Rumin’s complex of differential forms in contact manifolds, on a Sobolev-Gaffney inequality proved by Baldi-Franchi in the setting of the Heisenberg groups and on some geometric properties that can be proved for sub-Riemannian contact manifolds with bounded geometry.

In this article, we introduce conformal anti-invariant Riemannian maps from or to Sasakian manifolds to or from Riemannian manifolds. We construct examples of such maps for justification. We study the geometry of foliations and derive the conditions for horizontal and vertical distributions to be totally geodesic. Also, we derive results on homotheticity of these maps. [ABSTRACT FROM AUTHOR]

In [Y.-G. Oh and R. Wang, Analysis of contact Cauchy-Riemann maps I: A priori Ck estimates and asymptotic convergence, Osaka J. Math. 55(4) (2018) 647-679; Y.-G. Oh and R. Wang, Analysis of contact Cauchy-Riemann maps II: Canonical neighborhoods and exponential convergence for the Morse-Bott case, Nagoya Math. J. 231 (2018) 128-223], the authors studied the nonlinear elliptic system ∂̄ πw=0,d(w∗λ∘j)=0 without involving symplectization for each given contact triad (Q,λ,J), and established the a prioriWk, 2 elliptic estimates and proved the asymptotic (subsequence) convergence of the map w:Σ̇ →Q for any solution, called a contact instanton, on Σ̇ under the hypothesis ∥w∗λ∥C0<∞ and dπw∈L2∩L4. The asymptotic limit of a contact instanton is a 'spiraling' instanton along a 'rotating' Reeb orbit near each puncture on a punctured Riemann surface Σ̇. Each limiting Reeb orbit carries a 'charge' arising from the integral of w∗λ∘j. In this paper, we further develop analysis of contact instantons, especially the W1,p estimate for p>2 (or the C1 -estimate), which is essential for the study of compactification of the moduli space and the relevant Fredholm theory for contact instantons. In particular, we define a Hofer-type off-shell energy Eλ(j,w) for any pair (j,w) with a smooth map w satisfying d(w∗λ∘j)=0, and develop the bubbling-off analysis and prove an ϵ -regularity result. We also develop the relevant Fredholm theory and carry out index calculations (for the case of vanishing charge). [ABSTRACT FROM AUTHOR]

We consider smooth embedded surfaces in a 3D contact sub-Riemannian manifold and the problem of the finiteness of the induced distance (i.e., the infimum of the length of horizontal curves that belong to the surface). Recently it has been proved that for a surface having the topology of a sphere embedded in a tight co-orientable structure, the distance is always finite. In this paper we study closed surfaces of genus larger than 1, proving that such surfaces can be embedded in such a way that the induced distance is finite or infinite. We then study the structural stability of the fmiteness/not-finiteness of the distance. [ABSTRACT FROM AUTHOR]

We extend Fedosov deformation quantization to general contact manifolds. Unlike the case of symplectic manifolds, not every classical observable on a contact manifold is generally quantized. On examination of possible obstructions to quantization, we obtain some new invariants of contact manifolds. [ABSTRACT FROM AUTHOR]

Let (M; α, ω) be a contact manifold in the sense of Okassa. We consider Marsden-Weinstein reduction theorem which induces Jacobi- Liouville theorem as special case. The aim of this paper is to describe the Hamiltonian dynamics, integrability of characteristic distribution on a contact manifold and study the reduction. [ABSTRACT FROM AUTHOR]

The aim of this paper is to characterize K-contact and Sasakian manifolds whose metrics are generalized quasi-Einstein metric. It is proven that if the metric of a K-contact manifold is generalized quasi-Einstein metric, then the manifold is of constant scalar curvature and in the case of a Sasakian manifold the metric becomes Einstein under certain restriction on the potential function. Several corollaries have been provided. Finally, we consider Sasakian 3-manifold whose metric is generalized quasi-Einstein metric. [ABSTRACT FROM AUTHOR]

We study the global topology of the space L of loops of contactomorphisms of a non-orderable closed contact manifold (M 2 n + 1 , ξ). We filter L by a quantitative measure of the "positivity" of the loops and describe the topology of L in terms of the subspaces of the filtration. In particular, we show that the homotopy groups of L are subgroups of the homotopy groups of the subspace of positive loops L +. We obtain analogous results for the space of loops of Legendrian submanifolds in (M 2 n + 1 , ξ). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the behavior of semi-invariant Riemannian maps taking Sasakian structure as total manifolds satisfying Ricci soliton equation, to Riemannian manifolds. We establish necessary and sufficient conditions for the cases when fibers and r a n g e F ⁎ are Einstein. Further, we calculate scalar curvature for r a n g e F ⁎ , fibers and total manifolds. Also, we derive some inequalities for semi-invariant Riemannian maps from Sasakian space forms satisfying Ricci soliton equation, to Riemannian manifolds. We construct some examples in support of assumed maps. [ABSTRACT FROM AUTHOR]

We study contact 3-manifolds using the techniques of sub-Riemannian geometry and geometric measure theory, in particular establishing properties of their Lipschitz homotopy groups. We prove a biLipschitz version of the Theorem of Darboux: a contact (2 n + 1) -manifold endowed with a sub-Riemannian structure is locally biLipschitz equivalent to the Heisenberg group H n with its Carnot-Carathéodory metric. Then each contact (2 n + 1) -manifold endowed with a sub-Riemannian structure is purely k -unrectifiable for k > n. We then extend results of Dejarnette et al. [5] and Wenger and Young [20] on the Lipschitz homotopy groups of H 1 to an arbitrary contact 3-manifold endowed with a Carnot-Carathéodory metric, namely that for any contact 3-manifold the first Lipschitz homotopy group is uncountably generated and all higher Lipschitz homotopy groups are trivial. Therefore, in the sense of Lipschitz homotopy groups, a contact 3-manifold is a K (π, 1) -space with an uncountably generated first homotopy group. Along the way, we prove that each open distributional embedding between purely 2-unrectifiable sub-Riemannian manifolds induces an injective map on the associated first Lipschitz homotopy groups. Therefore, each open subset of a contact 3-manifold determines an uncountable subgroup of the first Lipschitz homotopy group of the contact 3-manifold. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study ∗-Weyl curvature tensor within the framework of Sasakian and (κ, μ)-contact manifolds. [ABSTRACT FROM AUTHOR]

The characterization of curves plays an important role in both geometry and topology of almost contact manifolds. Olszak found the equation 2 α β + ξ [ β ] = 0 on normal almost contact manifolds. The pair (α , β) denotes the type of these manifolds. In this study, we obtained the curvatures of non-geodesic Frenet curves on 3 -dimensional normal almost contact manifolds without neglecting α and β , and provided the results of their characterization. We exemplified these results with examples. [ABSTRACT FROM AUTHOR]

We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of contact autonomous mechanical systems, which is based on the approach of the pioneering work of R. Skinner and R. Rusk. This framework permits to skip the second order differential equation problem, which is obtained as a part of the constraint algorithm (for singular or regular Lagrangians), and is especially useful to describe singular Lagrangian systems. Some examples are also discussed to illustrate the method. A unified geometric framework is presented for describing both the Lagrangian and Hamiltonian formalisms of contact autonomous mechanical systems, which is based on the approach of the pioneering work of R. Skinner and R. Rusk. This framework permits to skip the second order differential equation problem, which is obtained as a part of the constraint algorithm (for singular or regular Lagrangians), and is especially useful to describe singular Lagrangian systems. Some examples are also discussed to illustrate the method. [ABSTRACT FROM AUTHOR]

The analysis of the covariant brackets on the space of functions on the solutions to a variational problem in the framework of contact geometry initiated in the companion letter 1 9 is extended to the case of the multisymplectic formulation of the free Klein–Gordon theory and of the free Schrödinger equation. [ABSTRACT FROM AUTHOR]

The formulation of covariant brackets on the space of solutions to a variational problem is analyzed in the framework of contact geometry. It is argued that the Poisson algebra on the space of functionals on fields should be read as a Poisson subalgebra within an algebra of functions equipped with a Jacobi bracket on a suitable contact manifold. [ABSTRACT FROM AUTHOR]

We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) manifold of the first kind. We prove that the two notions are equivalent if there exists a Riemannian metric such that the Lee vector field is unitary and parallel and its metric dual 1-form coincides with the Lee 1-form. Finally, we discuss several examples of compact l.c.s. manifolds of the first kind which do not admit compatible Vaisman metrics. [ABSTRACT FROM AUTHOR]

In this paper, we analyze in detail a collection of motivating examples to consider b m -symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every b m -symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to b -symplectic manifolds: b -contact manifolds. [ABSTRACT FROM AUTHOR]

Abstract The dynamic analysis of planar mechanisms with multiple revolute clearance joints is performed using the isogeometric analysis for the first time. A general multiple patches based revolute clearance joint model, which can be integrated into other parts or structures without changing the procedure of contact treatment, is proposed. A stable and optimized contact detection algorithm that accompanies the joint model is also introduced. Benefit from the high order NURBS basis functions, the geometrically exact covariant formulations of contact geometry, contact force and its Jacobian matrix are implemented. The implicit generalized- α method is employed to solve the time dependent nonlinear system equations. The effect of joint clearance on rigid and flexible mechanisms is studied through numerical examples. The accuracy and efficiency of the proposed approach are verified through a comparison with other conventional methods. The proposed approach exhibits great stability in long-term simulation regardless of the number of clearance joints, the size of clearance, the flexibility of the link and bearing. Results show that isogeometric analysis can provide a unified framework for joint clearance analysis of rigid and flexible systems. In the field of analyzing the combined effect of joint clearance and bodies' large flexibility, the proposed method has significant advantages. [ABSTRACT FROM AUTHOR]

A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisman structure, that is, a LCK structure whose associated Lee form is parallel. [ABSTRACT FROM AUTHOR]

We discuss the classification of simply connected, complete (κ, μ)-spaces from the point of view of homogeneous spaces. In particular, we exhibit new models of (κ, μ)-spaces having Boeckx invariant -1. Finally, we prove that the number (n+1)(n+2)/2 is the maximum dimension of the automorphism group of a contact metric manifold of dimension 2n + 1, n ≥ 2, whose symmetric operator h has rank at least 3 at some point; if this dimension is attained, and the dimension of the manifold is not 7, it must be a (κ, μ)-space. The same conclusion holds also in dimension 7 provided the almost CR structure of the contact metric manifold under consideration is integrable. [ABSTRACT FROM AUTHOR]

In this note we show families of homogeneous Sasakian manifolds G/H which are nonformal. The non-formality condition is expressed in terms of characters of a maximal torus in G. [ABSTRACT FROM AUTHOR]

These are notes based on a mini-course at the conference RIEMain in Contact, held in Cagliari, Sardinia, in June 2018. The main theme is the connection between Reeb dynamics and topology. Topics discussed include traps for Reeb flows, plugs for Hamiltonian flows, the Weinstein conjecture, Reeb flows with finite numbers of periodic orbits, and global surfaces of section for Reeb flows. The emphasis is on methods of construction, e.g. contact cuts and lifting group actions in Boothby-Wang bundles, that might be useful for other applications in contact topology. [ABSTRACT FROM AUTHOR]

It is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure. We identify the necessary and sufficient condition for its associated cohomology to be isomorphic to the cohomology associated to trivial (zero) holomorphic Poisson structure. We also identify a sufficient condition for this isomorphism to be at the level of Gerstenhaber algebras. [ABSTRACT FROM AUTHOR]

This survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference "RIEMain in Contact", 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers' kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense "tighten up" the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of "tight" vs. "overtwisted" or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture. [ABSTRACT FROM AUTHOR]

In this paper, we discuss about almost complex structures and complex structures on Riemannian manifolds, symplectic manifolds and contact manifolds. We have also shown a special comparison between complex symplectic geometry and complex contact geometry. Also, the existence of a complex submanifold of n-dimensional complex manifold which intersects a real submanifold. [ABSTRACT FROM AUTHOR]

A complex contact structure γ is defined by a system of holomorphic local 1-forms satisfying the completely non-integrability condition. The contact structure induces a subbundle Ker   γ of the tangent bundle and a line bundle L. In this paper, we prove that the sheaf of holomorphic k -vectors on a complex contact manifold splits into the sum of 𝒪 (∧ k Ker   γ) and 𝒪 (L ⊗ ∧ k − 1 Ker   γ) as sheaves of ℂ -module. The theorem induces the short exact sequence of cohomology of holomorphic k -vectors, and we obtain vanishing theorems for the cohomology of 𝒪 (∧ k Ker   γ). [ABSTRACT FROM AUTHOR]

In this paper, we introduce implicit Hamiltonian dynamics in the framework of contact geometry in two different ways: first, we introduce classical implicit Hamiltonian dynamics on a contact manifold, followed by evolution Hamiltonian dynamics. In the first case, implicit contact Hamiltonian dynamics is defined as a Legendrian submanifold of a tangent contact space, whilst the implicit evolution dynamic is understood as a Lagrangian submanifold of a certain symplectic space embedded into the tangent contact space. To conclude, we propose a geometric Hamilton-Jacobi theory for both of these formulations. [ABSTRACT FROM AUTHOR]

Abstract Let ( X m + 1 , g ) be an ( m + 1 ) -dimensional globally hyperbolic spacetime with Cauchy surface M m , and let M ˜ m be the universal cover of the Cauchy surface. Let N X be the contact manifold of all future directed unparameterized light rays in X that we identify with the spherical cotangent bundle S T ∗ M. Jointly with Stefan Nemirovski we showed that when M ˜ m is not a compact manifold, then two points x , y ∈ X are causally related if and only if the Legendrian spheres S x , S y of all light rays through x and y are linked in N X. In this short note we use the contact Bott–Samelson theorem of Frauenfelder, Labrousse and Schlenk to show that the same statement is true for all X for which the integral cohomology ring of a closed M ˜ is not the one of the CROSS (compact rank one symmetric space). If M admits a Riemann metric g ¯ , a point x and a number ℓ > 0 such that all unit speed geodesics starting from x return back to x in time ℓ , then ( M , g ¯ ) is called a Y ℓ x manifold. Jointly with Stefan Nemirovski we observed that causality in ( M × R , g ¯ ⊕ − t 2 ) is not equivalent to Legendrian linking. Every Y ℓ x -Riemann manifold has compact universal cover and its integral cohomology ring is the one of a CROSS. So we conjecture that Legendrian linking is equivalent to causality if and only if one can not put a Y ℓ x Riemann metric on a Cauchy surface M. [ABSTRACT FROM AUTHOR]

A foliation is said to admit a foliated contact structure if there is a codimension $$1$$ distribution in the tangent space of the foliation such that the restriction to any leaf is contact. We prove a version of the Weinstein conjecture for Reeb vector fields in the presence of an overtwisted leaf. The result is shown to be sharp. [ABSTRACT FROM AUTHOR]

Hierarchical MAX model (HMAX) is a bio-inspired model mimicking the visual information processing of visual cortex. However, the visual processing of lower level, such as retina and lateral geniculate nucleus (LGN), is not concerned, and the properties of higher-level neurons are not sufficiently specified. Given that, we develop an extended HMAX model, denoted as E-HMAX, by the following biologically plausible ways. First, contrast normalization is conducted on the input image to simulate the processing of human retina and LGN. Second, log-polar Gabor (GLoP) filters are used to simulate the properties of V1 simple cells instead of Gabor filters. Then, sparse coding on multi-manifolds is modeled to compute the V4 simple cell response instead of Euclidean distance. Meanwhile, a template learning method based on dictionary learning on multi-manifolds is proposed to select informative templates during template learning stage. Experimental results demonstrate that the proposed model has greatly outperformed the standard HMAX model. It is also comparable to some state-of-the-art approaches such as EBIM and OGHM-HMAX. [ABSTRACT FROM AUTHOR]

For any integer n ≥ 2, we construct an infinite family of (4n-1)-dimensional contact manifolds, each of which admits infinitely many pairwise homotopy inequivalent Stein fillings. [ABSTRACT FROM AUTHOR]