Geodesic vector fields, induced contact structures and tightness in dimension three

Geodesic and Conformally Reeb Vector Fields on Flat 3-Manifolds

Abstract. In the present paper, we prove three fundamental results concerning almost ∗-Ricci soliton in the framework of para-Sasakian manifold. The paper is organised as follows:• If a para-Sasakian metric g represents an almost ∗-Ricci soliton with potential vector field V is Jacobi along Reeb vector field ξ, then g becomes a ∗-Ricci soliton.• If a para-Sasakian metric g represents an almost ∗-Ricci soliton with potential vector field V as infinitesimal paracontact transformation, then V is killing and g is η-Einstein.• If a para-Sasakian metric g represents an almost ∗-Ricci soliton with potential vector field V is collinear with the Reeb vector field ξ, then λ = 0, V is strict and g is η-Einstein.

In this paper, we use incompressible vector fields for characterizing Killing vector fields. We show that on a compact Riemannian manifold, a nontrivial incompressible vector field has a certain lower bound on the integral of the Ricci curvature in the direction of the incompressible vector field if, and only if, the vector field ξ is Killing. We also show that a nontrivial incompressible vector field ξ on a compact Riemannian manifold is a Jacobi-type vector field if, and only if, ξ is Killing. Finally, we show that a nontrivial incompressible vector field ξ on a connected Riemannian manifold has a certain lower bound on the Ricci curvature in the direction of ξ, and if ξ is also a geodesic vector field, it necessarily implies that ξ is Killing.

We introduce the concept of $\varepsilon\,$-contact metric structures on oriented (pseudo-)Riemannian three-manifolds, which encompasses the usual Riemannian contact metric, Lorentzian contact metric and para-contact metric structures, but which also allows the possibility for the Reeb vector field to be null. We investigate in more detail this latter case, which we call null contact structure. We observe that it is possible to extend in a natural and meaningful way both the Sasaki and K-contact conditions for null-contact structures, but we find that they are not equivalent conditions, in contradistinction to the situation for non-lightlike Reeb vector fields. Finally, we define the notion of $\varepsilon\eta\,$-Einstein structures and we discover that appropriate direct products of these structures produce solutions of six-dimensional minimal supergravity coupled to a tensor multiplet with constant dilaton.

A contact structure in a 3-dimensional manifold is a completely nonintegrable 2-dimensional distribution: if the distribution is the kernel of a (locally defined) 1-form λ then λ∧dλ � 0everywhere. The standard contact structure in 3-space, arising from the identification of R 3 with the space of 1-jets of functions on the line, is given by the contact form λ = dz − yd x; here x is a point on the line, z is the value of a function and y the value of the derivative at x. The same formula defines the standard contact structure in the space of 1-jets of functions on the circle (x being the cyclic coordinate), the space which is topologically the solid torus. In this paper we will be concerned only with these two contact manifolds. In a contact 3-dimensional manifold one considers two classes of knots and links: the transverse and the Legendrian ones. The former are everywhere transverse to the contact distribution, and the latter are everywhere tangent to it. Every topological isotopy class of knots in a contact 3-fold contains a transverse and a Legendrian knot. The main problem of contact knot theory (which has made, so far, only a few first steps) is to classify, up to contact isotopies, transverse and Legendrian knots within each topological isotopy class. In particular, one would like to have specifically contact invariants of transverse and Legendrian knots and links. One such invariant is easily defined. Legendrian and transverse knots in the standard contact space have natural framings given by the vector fields ∂z and ∂y, respectively. The corresponding self-linking number is called the Bennequin number; it is denoted by β(K) where K is a Legendrian or a transverse knot 1 . Similarly one defines the Bennequin number of an oriented Legendrian or transverse link. The study of knots in contact 3-dimensional manifolds was put forward by the seminal paper by D. Bennequin [Be]. One of the main results of this paper

Let α be a contact form on S3, let ξ be its Reeb vector-field and let v be a non-singular vector-field in ker α. Let Cβ be the space of curves x on S3 such ẋ = aξ + bv, ȧ = 0, a ≩ 0. Let L+, respectively L−, be the set of curves in Cβ such that b ≥ 0, respectively b ≤ 0. Let, for x ∈ Cβ, J(x) = ∫0 1 αx(ẋ)dt. The framework of the present paper has been introduced previously in eg [3]. We establish in this paper that some cycles (an infinite number of them, indexed by odd integers, tending to ∞) in the S 1-equivariant homology of Cβ, relative to L+ ∪ L− and to some specially designed ”bottom set”, see section 4, are achieved in the Morse complex of (J,Cβ) by unions of unstable manifolds of critical points (at infinity)which must include periodic orbits of ξ; ie unions of unstable manifolds of critical points at infinity alone cannot achieve these cycles. At the odd indexes (2k−1) = 1+(2k−2), 1 for the linking, (2k−2) for the S1-equivariance, we find that the equivariant contributions of a critical point at infinity to L+ and to L− are fundamentally asymmetric when compared to those of a periodic orbit [5]. The topological argument of existence of a periodic orbit for ξ turns out therefore to be surprisingly close, in spirit, to the linking/equivariant argument of P. Rabinowitz in [12]; e.g. the definition of the ”bottom sets” of section 4 can be related in part to the linking part in the argument of [12]. The objects and the frameworks are strikingly different, but the original proof of [12] can be recognized in our proof, which uses degree theory, the Fadell-Rabinowitz index [8] and the fact that πn+1(Sn) = Z2, n ≥ 3. We need of course to prove, in our framework, that these topological classes cannot be achieved by critical points at infinity only, periodic orbits of ξ excluded, and this is the fundamental difficulty. The arguments hold under the basic assumption that no periodic orbit of index 1 connects L+ and L−. It therefore follows from the present work that either a periodic orbit of index 1 connects L+ and L− (as is probably the case for all three dimensional over-twisted [8] contact forms, see the work of H. Hofer [10], the periodic orbit found in [10] should be of index 1 in the present framework); or (with a flavor of exclusion in either/or) a linking/ equivariant variational argument a la P. Rabinowitz [12] can be put to work. Existence of (possibly multiple) periodic orbits of ξ, maybe of high Morse index, follows then. Therefore, to a certain extent, the present result runs, especially in the case of threedimensional over-twisted [8] contact forms, against the existence of non-trivial algebraic invariants defined by the periodic orbits of ξ and independent of what ker α and/or α are.

It is well known that the sphere S6(1) admits an almost complex structure J which is nearly Kähler. If M is a hypersurface of an almost Hermitian manifold with a unit normal vector field N, the tangent vector field ξ=−JN is said to be characteristic or the Reeb vector field. The Jacobi operator with respect to ξ is called the structure Jacobi operator, and is denoted by l=R(·,ξ)ξ, where R is the curvature tensor on M. The study of Riemannian submanifolds in different ambient spaces by means of their Jacobi operators has been highly active in recent years. In particular, many recent results deal with questions around the existence of hypersurfaces with a structure Jacobi operator that satisfies conditions related to their parallelism. In the present paper, we study the parallelism of the structure Jacobi operator of real hypersurfaces in the nearly Kähler sphere S6(1). More precisely, we prove that such real hypersurfaces do not exist.

In this paper, we give some characterizations by considering ∗-Ricci soliton as a Kenmotsu metric. We prove that if a Kenmotsu manifold represents an almost ∗-Ricci soliton and the potential vector field [Formula: see text] is a Jacobi along the Reeb vector field, then it is a steady ∗-Ricci soliton. Next, we show that a Kenmotsu matric endowed an almost ∗-Ricci soliton which is Einstein metric if it is [Formula: see text]-Einstein or the potential vector field [Formula: see text] is collinear to the Reeb vector field or [Formula: see text] is an infinitesimal contact transformation.

We show that a compact contact Ricci soliton with a potential vector field V collinear with the Reeb vector field, is Einstein. We also show that a homogeneous H-contact gradient Ricci soliton is locally isometric to En+1 × Sn(4). Finally we obtain conditions so that the horizontal and tangential lifts of a vector field on the base manifold may be potential vector fields of a Ricci soliton on the unit tangent bundle.

Does every smooth vector field on a closed three-manifold, for example the three-sphere, have a closed orbit? The answer is no, according to counterexamples by K. Kuperberg and others. On the other hand, there is a special class of vector fields, called Reeb vector fields, which are associated to contact forms. The three-dimensional case of the Weinstein conjecture asserts that every Reeb vector field on a closed oriented three-manifold has a closed orbit. This conjecture was recently proved by Taubes using Seiberg-Witten theory. We give an introduction to the Weinstein conjecture, the main ideas in Taubes’s proof, and the bigger picture into which it fits.

Given a vector field on a space, one natural question is how long its flow is defined. In this paper we study time dependant holomorphic vector fields X on C n with holomorphic coefficients. We say that the orbit of p c C n explodes if the integral curve o f p is unbounded on some finite time interval [0, to > C ~÷. Our main question then is whether, in a given class of time dependant holomorphic vector fields, there is a dense subclass of vector fields for each of which a dense set of orbits explode. We study here the cases of time dependant holomorphic Hamiltonian vector fields and holomorphic Reeb vector fields (the time independant case was discussed in ([FG1]) and ([FG2]). In the Hamiltonian case, our main result is Theorem 2.1 in which we obtain density of exploding orbits. To the contrary, in the second case, exploding orbits are not dense. We give two examples: one with a fixed contact form and one with a time dependent contact form. In both these examples, the velocity varies with time.

In this paper, we first find the properties of the generalized Tanaka–Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the ∇ ^ -geodesic is a magnetic curve (for ∇) along slant curves. Finally, we prove that when c ≤ 0 , there does not exist a non-geodesic slant Frenet curve satisfying the ∇ ^ -Jacobi equations for the ∇ ^ -geodesic vector fields in M. Thus, we construct the explicit parametric equations of pseudo-Hermitian pseudo-helices in Lorentzian space forms M 1 3 ( H ^ ) for H ^ = 2 c > 0 .

We establish a relation between the growth of the cylindrical contact homology of a contact manifold and the topological entropy of Reeb flows on this manifold. We show that if a contact manifold $(M,\xi)$ admits a hypertight contact form $\lambda_0$ for which the cylindrical contact homology has exponential homotopical growth rate, then the Reeb flow of every contact form on $(M,\xi)$ has positive topological entropy. Using this result, we provide numerous new examples of contact 3-manifolds on which every Reeb flow has positive topological entropy.

A contact structure on a closed oriented 3-manifold 3 is a completely nonintegrable plane field on it. The existence of a contact structure on 3 was first proved by Martinet [14] and then Lutz [12], [13] for each homotopy class of plane fields. A co-orientable contact structure is given as the kernel of a 1-form α with α ∧ α > 0 or < 0, which is called a positive or negative contact form respectively. Thurston and Winkelnkemper [17] deduced the existence of a contact form in an elegant way from Alexander’s theorem [1] on open-book decompositions. Eliashberg [2] showed that the most of these contact structures are, however, too flexible for geometrical interest (see §2). On the other hand, he completely characterized more historic examples related closely to the complex analysis in several variables, namely, the strictly pseudo-convex boundary of compact Stein surfaces ([4]). The symplectic fillability is one of its significant generalizations. Recently Loi and Piergallini [11] translated Eliashberg’s characterization of compact Stein surfaces into the language of Lefschetz fibration by using Gompf’s method [8]. They also showed that 3 is realizable as the boundary of a Stein surface if and only if it admits an open-book decomposition whose monodromy map is a composition of right-handed Dehn-twists. In this note, we improve Thurston-Winkelnkemper’s construction so that we obtain a symplectically fillable contact structure in the case where the monodromy map of a given open-book decomposition is a composition of right-handed Dehn-twists along mutually disjoint curves (Theorem 2). An interesting by-product of our construction is Theorem 3 in §4 which gives a deformation of symplectically fillable contact structures into a foliation with a Reeb component. Note that a foliation admitting a Reeb component itself is not symplectically fillable. I would like to thank the referee for noticing the importance of Theorem 3.

We prove that if contact strongly pseudo-convex integrable CR-manifold admits a *-Ricci soliton where the soliton vector Z is contact, then the Reeb vector field ? is an eigenvector of the Ricci operator at each point if and only if ? is constant. Then we study contact strongly pseudo-convex integrable CR-manifold such that 1 is a almost *-Ricci soliton with potential vector field Z collinear with ?. To this end, we prove that if a 3-dimensional contact metric manifoldMwith Q? = ?Q which admits a gradient almost *-Ricci soliton, then either M is flat or f is constant.