On W0 and W2 ø-Symmetric Contact Manifold Admitting Quarter-Symmetric Metric Connection

The paper deals locally W0 and W2 curvature tensor of ø-symmetric K-contact manifolds with quarter-symmetric metric connection and some results are obtained.

Contact Dynamics: Legendrian and Lagrangian Submanifolds

We are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two different generators of the same Legendrian submanifold. A variant of contact Tulczyjew’s triple is constructed for evolution contact dynamics.

From Geometry to Coherent Dissipative Dynamics in Quantum Mechanics

Starting from the geometric description of quantum systems, we propose a novel approach to time-independent dissipative quantum processes according to which energy is dissipated but the coherence of the states is preserved. Our proposal consists of extending the standard symplectic picture of quantum mechanics to a contact manifold and then obtaining dissipation by using appropriate contact Hamiltonian dynamics. We work out the case of finite-level systems for which it is shown, by means of the corresponding contact master equation, that the resulting dynamics constitute a viable alternative candidate for the description of this subclass of dissipative quantum systems. As a concrete application, motivated by recent experimental observations, we describe quantum decays in a 2-level system as coherent and continuous processes.

A New Class of Contact Pseudo Framed Manifolds with Applications

In this paper, we introduce a new class of contact pseudo framed (CPF)-manifolds M , g , f , λ , ξ by a real tensor field f of type 1,1 , a real function λ such that f 3 = λ 2 f where ξ is its characteristic vector field. We prove in our main Theorem 2 that M admits a closed 2-form Ω if λ is constant. In 1976, Blair proved that the vector field ξ of a normal contact manifold is Killing. Contrary to this, we have shown in Theorem 2 that, in general, ξ of 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.

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.

$W_{2}$-CURVATURE TENSOR ON K-CONTACT MANIFOLDS

The object of the present paper is to obtain sufficient conditions for a K-contact manifold to be a Sasakian manifold.

On Lorentezian almost Para-contact Manifold

In the present paper, we focus on Lorentzian almost para-contact manifold and explain their relationship. In section 1, we have introduced the historical background of a contact manifold. Next, in section 2, we have studied the basic formulae of the Lorentzian metric manifold. Further, in section 3, we introduced a new tensor field h as well as calculated some theorems and lemma. Now in section 4, we have investigated curvature properties and their relationship with the Lorentzian almost para-contact manifold. In the end section, we have discussed the entire work

Partially integrable almost CR structures

Let (M, D) be a compact contact manifold with dim R M = 2n ≥ 5. This means that: M is a C ∞ differential manifold with dim R M = 2n ≥ 5. And D is a subbundle of the tangent bundle TM which satisfying; there is a real one form θ such that D = {X : X ∈ TM, θ(X) = 0}, and θ ^ Λ n−1(d ) ≠ 0 at every point of p of M. Especially, we assume that our D admits almost CR structure,(M, S). In this paper, inspired by the work of Matsumoto([M]), we study the difference of partially integrable almost CR structures from actual CR structures. And we discuss partially integrable almost CR structures from the point of view of the deformation theory of CR structures ([A1],[AGL]).

A Boothby–Wang Theorem for Besse Contact Manifolds

A Reeb flow on a contact manifold is called Besse if all its orbits are periodic, possibly with different periods. We characterize contact manifolds whose Reeb flows are Besse as principal $$S^1$$ S 1 -orbibundles over integral symplectic orbifolds satisfying some cohomological condition. Apart from the cohomological condition, this statement appears in the work of Boyer and Galicki in the language of Sasakian geometry (Boyer and Galicki in Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008). We illustrate some non-commonly dealt with perspective on orbifolds in a proof of the above result. More precisely, we work with orbifolds as quotients of manifolds by smooth Lie group actions with finite stabilizer groups. By introducing all relevant orbifold notions in this equivariant way, we avoid patching constructions with orbifold charts. As an application, and building on work by Cristofaro-Gardiner–Mazzucchelli, we deduce a complete classification of closed Besse contact 3-manifolds up to strict contactomorphism.

Contact Dual Pairs

We introduce and study the notion of contact dual pair adopting a line bundle approach to contact and Jacobi geometry. A contact dual pair is a pair of Jacobi morphisms defined on the same contact manifold and satisfying a certain orthogonality condition. Contact groupoids and contact reduction are the main sources of examples. Among other properties, we prove the characteristic leaf correspondence theorem for contact dual pairs that parallels the analogous result of Weinstein for symplectic dual pairs.