scholarly journals COMBED 3-MANIFOLDS WITH CONCAVE BOUNDARY, FRAMED LINKS, AND PSEUDO-LEGENDRIAN LINKS

2001 ◽  
Vol 10 (01) ◽  
pp. 1-35 ◽  
Author(s):  
RICCARDO BENEDETTI ◽  
CARLO PETRONIO
Keyword(s):  
Vector Field ◽  
Finite Order ◽  
Singular Vector ◽  
Contact Structures ◽  
Winding Number ◽  
Legendrian Links ◽  
Concave Boundary

We provide combinatorial realizations, according to the usual objects/moves scheme, of the following three topological categories: (1) pairs (M, v) where M is a 3-manifold (up to diffeomorphism) and v is a (non-singular vector) field, up to homotopy; here possibly ∂M≠∅, and v may be tangent to ∂M, but only in a concave fashion, and homotopy should preserve tangency type; (2) framed links L in M, up to framed isotopy; (3) triples (M, v, L), with (M, v) as above and L transversal to v, up to pseudo-Legendrian isotopy (transverality-preserving simultaneous homotopy of v and isotopy of L). All realizations are based on the notion of branched standard spine, and build on results previously obtained, Links are encoded by means of diagrams on branched spines, where the diagram is C 1 with respect to the branching. Several motivations for being interested in combinatorial realizations of the topological categories considered in this paper are given in the introduction. The encoding of links is suitable for the comparison of the framed and the pseudo-Legendrian categories, and some applications are given in connection with contact structures, torsion and finite-order invariants. An estension of Trace's notion of winding number of a knot diagram is introduced and discussed.

A nonlinear system with singular vector field near equilibria

1981 ◽  
Vol 12 (1) ◽  
pp. 57-71
Author(s):  
Stephen L. Campbell ◽  
Jack W. Silverstein
Keyword(s):  
Vector Field ◽  
Nonlinear System ◽  
Singular Vector

Finite order corks

2017 ◽  
Vol 28 (06) ◽  
pp. 1750034 ◽  
Author(s):  
Motoo Tange
Keyword(s):  
Positive Integer ◽  
Finite Order ◽  
Stein Manifold ◽  
Contact Structures ◽  
Arbitrary Positive Integer ◽  
Branched Cover

We show that for any positive integer [Formula: see text], there exist pairs of compact, contractible, Stein 4-manifolds and order [Formula: see text] self-diffeomorphisms of the boundaries that do not extend to the full manifolds. Each boundary of the Stein 4-manifolds is a cyclic branched cover along a slice knot embedded in the boundary of a contractible 4-manifold. Each pair is called a finite order cork, we give a method producing examples of many finite order corks, which are possibly not a Stein manifold. The example of the Stein cork gives a diffeomorphism generating [Formula: see text] homotopic but non-isotopic Stein fillable contact structures for an arbitrary positive integer [Formula: see text].

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.

On a Transformation Group

1969 ◽  
Vol 21 ◽  
pp. 935-941 ◽  
Author(s):  
S. K. Kaul
Keyword(s):  
Finite Order ◽  
Metric Space ◽  
Transformation Group ◽  
Real Numbers ◽  
Compact Metric Space ◽  
Topological Transformation ◽  
Linear Fractional Transformations ◽  
Group Of Homeomorphisms

0. Let Γ denote a group of real linear fractional transformations (the constants defining any element of Γ are real numbers); see (3, § 2, p. 10). Then it is known that Γ is discontinuous if and only if it is discrete (3, Theorem 2F, p. 13).Now Γ may also be regarded, equivalently, as a group of homeomorphisms of a disc D onto itself; and if Γ is discrete, then, except for elements of finite order, each element of Γ is either of type 1 or type 2 (see Definitions 0.1 and 0.2 below).We wish to generalize the result quoted above in purely topological terms. Thus, throughout this paper we denote by X a compact metric space with metric d, and by G a topological transformation group on X each element of which, except the identity e, is either of type 1 or type 2. Let L = ﹛a ∈ X: g(a) = a for some g in G — e﹜, and . We assume furthermore that 0 is non-empty.

scholarly journals Extended Fuller index, sky catastrophes and the Seifert conjecture

2018 ◽  
Vol 29 (13) ◽  
pp. 1850096 ◽  
Author(s):  
Yasha Savelyev
Keyword(s):  
Vector Field ◽  
Periodic Orbits ◽  
General Class ◽  
Smooth Manifold ◽  
Vector Fields ◽  
Singular Vector ◽  
Natural Question ◽  
Reeb Vector ◽  
Fuller Index

We extend the classical Fuller index, and use this to prove that for a certain general class of vector fields [Formula: see text] on a compact smooth manifold, if a homotopy of smooth non-singular vector fields starting at [Formula: see text] has no sky catastrophes as defined by the paper, then the time 1 limit of the homotopy has periodic orbits. This class of vector fields includes the Hopf vector field on [Formula: see text]. A sky catastrophe is a kind of bifurcation originally discovered by Fuller. This answers a natural question that existed since the time of Fuller’s foundational papers. We also put strong constraints on the kind of sky-catastrophes that may appear for homotopies of Reeb vector fields.

scholarly journals ENERGY OF GLOBAL FRAMES

2008 ◽  
Vol 84 (2) ◽  
pp. 155-162
Author(s):  
FABIANO G. B. BRITO ◽  
PABLO M. CHACÓN
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Lower Bound ◽  
Vector Fields ◽  
Singular Vector ◽  
Unit Vector ◽  
Energy Functional ◽  
Closed Riemannian Manifold ◽  
Total Scalar Curvature ◽  
Dimension 2

AbstractThe energy of a unit vector field X on a closed Riemannian manifold M is defined as the energy of the section into T1M determined by X. For odd-dimensional spheres, the energy functional has an infimum for each dimension 2k+1 which is not attained by any non-singular vector field for k>1. For k=1, Hopf vector fields are the unique minima. In this paper we show that for any closed Riemannian manifold, the energy of a frame defined on the manifold, possibly except on a finite subset, admits a lower bound in terms of the total scalar curvature of the manifold. In particular, for odd-dimensional spheres this lower bound is attained by a family of frames defined on the sphere minus one point and consisting of vector fields parallel along geodesics.

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.

Local Bifurcation of Critical Periods in Vector Fields With Homogeneous Nonlinearities of the Third Degree

1993 ◽  
Vol 36 (4) ◽  
pp. 473-484 ◽  
Author(s):  
C. Rousseau ◽  
B. Toni
Keyword(s):  
Vector Field ◽  
Finite Order ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Local Bifurcation ◽  
Critical Periods ◽  
The Third ◽  
Bifurcation Of Critical Periods

AbstractIn this paper we study the local bifurcation of critical periods of periodic orbits in the neighborhood of a nondegenerate centre of a vector field with a homogeneous nonlinearity of the third degree. We show that at most three local critical periods bifurcate from a weak linear centre of finite order or from the linear isochrone and at most two local critical periods from the nonlinear isochrone. Moreover, in both cases, there are perturbations with the maximum number of critical periods.

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