Geodesic and Conformally Reeb Vector Fields on Flat 3-Manifolds

In this paper, we provide new and simpler proofs of two theorems of Gluck and Harrison on contact structures induced by great circle or line fibrations. Furthermore, we prove that a geodesic vector field whose Jacobi tensor is parallel along flow lines (e.g. if the underlying manifold is locally symmetric) induces a contact structure if the ‘mixed’ sectional curvatures are nonnegative, and if a certain nondegeneracy condition holds. Additionally, we prove that in dimension three, contact structures admitting a Reeb flow which is either periodic, isometric, or free and proper, must be universally tight. In particular, we generalise an earlier result of Geiges and the author, by showing that every contact form on R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document} whose Reeb vector field spans a line fibration is necessarily tight. Furthermore, we provide a characterisation of isometric Reeb vector fields. As an application, we recover a result of Kegel and Lange on Seifert fibrations spanned by Reeb vector fields, and we classify closed contact 3-manifolds with isometric Reeb flows (also known as R-contact manifolds) up to diffeomorphism.

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.

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.

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.

At this time the Cheshire Cat vanished quite slowly, beginning with the end of the tail, ending with a grin, which remained some time after the rest of it had gone, [Ca].0. Introduction.Morse theory for distance functions was initiated by Grove-Shiohama [GS] and Gromov's [Gl] paper where basic notions of the theory were formulated; even this very initial level of the theory leads to important geometric applications.Grove and Shiohama [GS] established a generalized sphere theorem by constructing a vector field on a Riemannian manifold, with the property that the distance function had no stationary points along the integral curves at non-singular points of the vector field.They showed that this vector field had exactly two singular points and hence the manifold was homeomorphic to a sphere.Later Gromov [Gl] was able to bound the sum of the Betti numbers of a positively curved Riemannian manifold.He controlled the location of critical points of a Riemannian distance function on a positively curved manifold using Toponogov's theorem and then was able to bound the number of critical points of this function and hence the homology of the manifold, using a spectral sequence argument.Morse theory for Riemannian distance functions was discussed in [Gr], [L] and other papers, which explained its importance for geometric applications rather than developed the theory itself.Even a suitable concept of the index of a critical point has not been developed.As a result, the most powerful tools of the classical Morse theory such as Morse inequalities and the correspondence between critical points and the handle decomposition of the manifold cannot be used.The relationship with the classical Morse theory has not been investigated either; in particular, the connection between notions of the critical points in both theories is not obvious.In a different direction, the structure of Alexandrov spaces with curvatures bounded below was investigated using distance functions in several papers starting with [BGP] and continuing with [PI], [P2].A key result obtained is a canonical stratification of such Alexandrov spaces into topological manifolds and again the technique is a type of Morse theory, using the distance function.As Alexandrov spaces do not have as much structure as Riemannian manifolds, our theory gives more detailed information on the nature of critical points and index.M. Gromov pointed out in [Gl] that the Morse theory for Riemannian distance functions can be developed by analogy with the classical Morse theory.The aim of this paper is to construct a Morse theory for functions which are minima of finite families of smooth functions and clarify the connection with Riemannian distance functions for non-positively curved manifolds.We develop the theory in the classical style, including the notion of the index for critical points, and clarify relations with the Grove-Shiohama-Gromov approach.

In this article, we prove integral formulas for a Riemannian manifold equipped with a foliation F and a unit vector field N orthogonal to F, and generalize known integral formulas (due to Brito-Langevin-Rosenberg and Andrzejewski-Walczak) for foliations of codimension one. Our integral formulas involve Newton transformations of the shape operator of F with respect to N and the curvature tensor of the induced connection on the distribution D=TF⊕span(N), and this decomposition of D can be regarded as a codimension-one foliation of a sub-Riemannian manifold. We apply our formulas to foliated (sub-)Riemannian manifolds with restrictions on the curvature and extrinsic geometry of the foliation.

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 .

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 study Riemann solitons on Sasakian 3-manifolds. We prove that if a Sasakian 3-manifold (M,g) admits a Riemann soliton with a potential vector field V where divV is constant, then g is homothetic to a Berger sphere. It is also shown that any Sasakian 3-manifold (M,g) that admits a Riemann soliton with potential vector field ??, where ? is Reeb vector field and ? is a smooth function on M, is an Einstein manifold. Also, we prove that if a Sasakian 3-manifold (M,g) is is Einstein, Einstein-semisymmetric, projectively flat, or ?-projectively flat manifold then (M,g) satisfies the Riemann soliton equation. Finally, we prove that if a Sasakian 3-manifold (M,g) has a gradient Riemann soliton with potential vector field ?f, then ? must be constant. Additionally, if a Sasakian 3-manifold (M,g) admits a Riemann soliton (M,g,V,?) such that V is an infinitesimal contact transformation, then the transverse geometry of M is Fano and V is a harmonic infinitesimal automorphism of the contact metric structure.

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.

We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two nonequivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assumption on the solution needed for the uniqueness within the DiPerna-Lions theory of continuity equations in the case of vector fields in the Sobolev space where p is larger than the space dimension, under the assumption that the so-called “forward-backward integral curves” associated to the vector field are trivial for almost every starting point. More precisely, for such vector fields we prove uniqueness and Lagrangianity for weak solutions of the continuity equation that are just locally integrable. Additionally, for such vector fields it is possible to prove almost everywhere uniqueness of (standard) integral curves, which also implies uniqueness of positive measure solutions to the continuity equation with absolutely continuous initial datum.

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.

In this paper, we extend earlier work on motion planning and coordination of multiple agents, to environments of arbitrary polygonal obstacles, using non-gradient vector fields to steer each agent towards their goal configurations while avoiding collisions. We formulate the vector fields so that the pattern of their integral curves depends on a parameter λ. By manipulating the value of λ, we obtain a set of vector fields whose integral curves define the flow lines for an a priori known obstacle environment. We use the vector field design in tandem with model predictive control to compute safe trajectories for multi-agent systems. The competence of the proposed methodology is demonstrated for both static and dynamic environment via simulation results. The efficacy of model predictive control in achieving control trajectories, free of chattering, for multi-agent coordination is validated through comparison to a state feedback coordination and control protocol.