Dynamics of a Contact Structure Along a Vector Field of its Kernel

2008 ◽  
Vol 8 (1) ◽  
Author(s):  
Abbas Bahri ◽  
Yongzhong Xu
Keyword(s):  
Vector Field ◽  
Variational Problem ◽  
Contact Structure ◽  
Contact Form ◽  
Dual Form

AbstractIn this paper we prove that in order to define the homology of [3], the hypothesis that there exists a vector field in the kernel of the contact form which defines a dual form with the same orientation is not essential. The technique is quantitative: as we introduce a large amount of rotation near the zeroes of the vector field in the kernel, we track down the modification of the variational problem and provide bounds on a key quantity (denoted by τ).

scholarly journals Periodic orbits in virtually contact structures

2018 ◽  
Vol 12 (02) ◽  
pp. 371-418
Author(s):  
Youngjin Bae ◽  
Kevin Wiegand ◽  
Kai Zehmisch
Keyword(s):  
Periodic Solutions ◽  
Potential Function ◽  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Contact Structure ◽  
Critical Value ◽  
Holomorphic Curves ◽  
Contact Structures ◽  
Contact Form ◽  
Contact Manifolds

We prove that certain non-exact magnetic Hamiltonian systems on products of closed hyperbolic surfaces and with a potential function of large oscillation admit non-constant contractible periodic solutions of energy below the Mañé critical value. For that we develop a theory of holomorphic curves in symplectizations of non-compact contact manifolds that arise as the covering space of a virtually contact structure whose contact form is bounded with all derivatives up to order three.

scholarly journals Explicit concave fillings of contact three-manifolds

2002 ◽  
Vol 133 (3) ◽  
pp. 431-441 ◽  
Author(s):  
DAVID T. GAY
Keyword(s):  
Vector Field ◽  
Symplectic Manifold ◽  
Contact Manifold ◽  
Contact Form

Suppose that (X,ω) is a symplectic manifold and that there exists a Liouville vector field V defined in a neighbourhood of and transverse to M = ∂X. Then V induces a contact form α = ιVω[mid ]M on M which determines the germ of ω along M. One should think of the contact manifold (M,ξ = ker α) as controlling the behaviour of ω ‘at infinity’. If V points out of X along M then we call (X,ω) a convex filling of (M,ξ), and if V points into X along M then we call (X,ω) a concave filling of (M,ξ).

scholarly journals Foliations by minimal surfaces and contact structures on certain closed3-manifolds

2003 ◽  
Vol 2003 (21) ◽  
pp. 1323-1330
Author(s):  
Richard H. Escobales
Keyword(s):  
Vector Field ◽  
Fundamental Group ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Vector Fields ◽  
Contact Structure ◽  
Contact Structures ◽  
Volume Form ◽  
Basic Vector ◽  
Transverse Volume

Let(M,g)be a closed, connected, orientedC∞Riemannian 3-manifold with tangentially oriented flowF. Suppose thatFadmits a basic transverse volume formμand mean curvature one-formκwhich is horizontally closed. Let{X,Y}be any pair of basic vector fields, soμ(X,Y)=1. Suppose further that the globally defined vector𝒱[X,Y]tangent to the flow satisfies[Z.𝒱[X,Y]]=fZ𝒱[X,Y]for any basic vector fieldZand for some functionfZdepending onZ. Then,𝒱[X,Y]is either always zero andH, the distribution orthogonal to the flow inT(M), is integrable with minimal leaves, or𝒱[X,Y]never vanishes andHis a contact structure. If additionally,Mhas a finite-fundamental group, then𝒱[X,Y]never vanishes onM, by the above together with a theorem of Sullivan (1979). In this caseHis always a contact structure. We conclude with some simple examples.

Dynamics of the Third Exotic Contact Form on the Sphere Along a Vector Field in its Kernel

2016 ◽  
Vol 16 (2) ◽  
Author(s):  
Ali Maalaoui
Keyword(s):  
Vector Field ◽  
Contact Form ◽  
The Third

AbstractIn this paper we study the dynamics of the third exotic contact form of Gonzalo and Varela [

DETERMINATION OF THE FASTEST TRAJECTORIES OF MATERIAL POINT MOTION IN A HORIZONTAL VECTOR FIELD

2021 ◽  
Vol 4 ◽  
pp. 19-27
Author(s):  
Victor Legeza ◽  
◽  
Alexander Neshchadym ◽  
Keyword(s):  
Optimal Control ◽  
Vector Field ◽  
Variational Problem ◽  
Material Point ◽  
Minimum Time ◽  
Algebraic Equations ◽  
Starting Point ◽  
Point Motion ◽  
Oscillatory Character ◽  
Pass Through

The article proposes a solution to the well-known Zermelo navigation problem by classical variational methods. The classical Zermelo problem within the framework of optimal control theory is formulated as follows. The ship must pass through the region of strong currents, the magnitude and direction of the current velocity are set as functions of phase variables. In this case, the relative speed of the ship is set, the module of which remains constant during movement. It is necessary to find such an optimal control that ensures the arrival of the ship at a given point in the minimum time, i.e. control of the ship by fast-action should be determined. In this paper, we consider the brachistochronic motion of a material point in a plane vector field of a mobile fluid, for which the classical variational problem of finding extreme trajectories is formulated. The aim of the study is to obtain equations of extreme trajectories along which a material point moves from a given starting point to a given finish point in the least amount of time. The solution to the problem was carried out using the classical methods of the theory of the calculus of variations. For a given variant of the boundary conditions, algebraic equations of extremals of motion of a material point were established in the form of segments of a power series. A comparative analysis of the fast-action was carried out both along extreme trajectories and along an alternative path — along a straight line that connects two given start and finish points. Analysis of the results showed that the considered variational problem has two solutions, which differ only in sign. However, only one solution provides the minimum time for moving a material point between two given points. It was also found that the extreme trajectory of the brachistochronic motion of a point is not straight, but has an oscillatory character.

scholarly journals Critical associated metrics on contact manifolds III

1991 ◽  
Vol 50 (2) ◽  
pp. 189-196 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Vector Field ◽  
Contact Structure ◽  
Contact Manifold ◽  
Characteristic Vector ◽  
Sphere Bundle ◽  
Contact Manifolds ◽  
Parameter Group ◽  
Group Of Isometries ◽  
Characteristic Vector Field ◽  
Regular Contact

AbstractIn the first paper of this series we studied on a compact regular contact manifold the integral of the Ricci curvature in the direction of the characteristic vector field considered as a functional on the set of all associated metrics. We showed that the critical points of this functional are the metrics for which the characteristic vector field generates a 1-parameter group of isometries and conjectured that the result might be true without the regularity of the contact structure. In the present paper we show that this conjecture is false by studying this problem on the tangent sphere bundle of a Riemannian manifold. In particular the standard associated metric is a critical point if and only if the base manifold is of constant curvature +1 or −1; in the latter case the characteristic vector field does not generate a 1-parameter group of isometries.

scholarly journals On local equivalence for vector field systems

1990 ◽  
Vol 42 (2) ◽  
pp. 215-229 ◽  
Author(s):  
P.J. Vassiliou
Keyword(s):  
Vector Field ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Contact Structure ◽  
Sufficient Conditions ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
General Integral ◽  
Field Systems ◽  
Local Equivalence

We give sufficient conditions for C∞ vector field systems on Rn with genus g = 1 to be diffeomorphic to a contact structure. The diffeomorphism is explicitly constructed and used to give the most general integral submanifolds for the systems. Finally the implications of these results for integrable hyperbolic partial differential equations in the plane is discussed.

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

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 591
Author(s):  
Mihai Visinescu
Keyword(s):  
Vector Field ◽  
Hamiltonian Systems ◽  
Ricci Flow ◽  
Contact Structures ◽  
Contact Form ◽  
Einstein Manifolds ◽  
Integrable Hamiltonian Systems ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Completely Integrable Hamiltonian Systems

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.

On a type of K-contact Riemannian manifold

1972 ◽  
Vol 13 (4) ◽  
pp. 447-450 ◽  
Author(s):  
M. C. Chaki ◽  
D. Ghosh
Keyword(s):  
Vector Field ◽  
Contact Structure ◽  
Contact Manifold ◽  
Riemannian Metric ◽  
Positive Definite ◽  
Almost Contact Metric Structure ◽  
Almost Contact Structure ◽  
Almost Contact Metric ◽  
Contact Metric Structure ◽  
Image Position

Let M be an n-dimensional (n = 2m + 1, m ≦ 1) real differentiable manifold. if on M there exist a tensor field , a contravariant vector field ξi and a convariant vector field ηi such that then M is said to have an almost contact structure with the structure tensors (φ,ξ, η) [1], [2]. Further, if a positive definite Riemannian metric g satisfies the conditions then g is called an associated Riemannian metric to the almost contact structure and M is then said to have an almost contact metric structure. On the other hand, M is said to have a contact structure [2], [4] if there exists a 1-form η over M such that η ∧ (dη)m ≠ 0 everywhere over M where dη means the exterior derivation of η and the symbol ∧ means the exterior multiplication. In this case M is said to be a contact manifold with contact form η. It is known [2, Th. 3,1] that if η = ηidxi is a 1-form defining a contact structure, then there exists a positive definite Riemannian metric in gij such that and define an almost contact metric structure with and ηi where the symbol ∂i standing for ∂/∂xi.

SASAKIAN 3-MANIFOLD AS A RICCI SOLITON REPRESENTS THE HEISENBERG GROUP

2011 ◽  
Vol 08 (01) ◽  
pp. 149-154 ◽  
Author(s):  
RAMESH SHARMA ◽  
AMALENDU GHOSH
Keyword(s):  
Vector Field ◽  
Heisenberg Group ◽  
Contact Structure ◽  
Ricci Soliton ◽  
Potential Vector ◽  
3 Dimensional ◽  
Sasakian Metric ◽  
Potential Vector Field

We show that, if a 3-dimensional Sasakian metric is a non-trivial Ricci soliton, then it is expanding and homothetic to the standard Sasakian metric on the Heisenberg group nil3. We have also discussed properties of the Ricci soliton potential vector field that relate to the underlying contact structure.

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