Tightness in contact metric 3-manifolds

The purpose of the present note is to prove the following results. Let M be a connected differentiable manifold which satisfies the second axiom of countability. Then (i) M admits a complete Riemannian metric; (ii) If every Riemannian metric on M is complete, M must be compact. In fact, somewhat stronger results will be given as Theorems 1 and 2 below. Let M be a connected differentiable manifold. It is known that if M satisfies the second axiom of countability, then M admits a Riemannian metric. Conversely, it can be shown that the existence of a Riemannian metric on M implies that M satisfies the countability axiom. For any Riemannian metric g on M, we can define a natural metric d on M by setting the distance d(x, y) between two points x and y to be the infinimum of the lengths of all piecewise differentiable curves joining x and y. The Riemannian metric g is complete if the metric space M with d is complete. It is known that this is the case if and only if every bounded subset of M (with respect to d) is relatively compact. We shall say that a Riemannian metric g is bounded if M is bounded with respect to the metric d. We shall prove

The set N of all null geodesics of a globally hyperbolic (d + 1)-dimensional spacetime (M, g) is naturally a smooth (2d − 1)-dimensional contact manifold. The sky of an event x in M is the subset X of N consisting of all null geodesics through x, and is an embedded Legendrian submanifold of N diffeomorphic to S(d − 1). It was conjectured by Low that for d = 2 two events x and y are causally related if and only if X and Y are linked (in an appropriate sense). We use the contact structure and knot polynomial calculations to prove this conjecture in certain particular cases, and suggest that for d = 3 smooth linking should be replaced with Legendrian linking.

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

We disprove the generalized Chern–Hamilton conjecture on the existence of critical compatible metrics on contact 3‐manifolds. More precisely, we show that a contact 3‐manifold admits a critical compatible metric for the Chern–Hamilton energy functional if and only if it is Sasakian or its associated Reeb flow is ‐conjugate to an algebraic Anosov flow modeled on . In particular, this yields a complete topological classification of compact 3‐manifolds that admit critical compatible metrics. As a corollary, we prove that no contact structure on admits a critical compatible metric and that critical compatible metrics can only occur when the contact structure is tight.

Shape spaces are fundamental in a variety of applications including image registration, morphing, matching, interpolation, and shape optimization. In this work, we consider two-dimensional shapes represented by triangular meshes of a given connectivity. We show that the collection of admissible configurations representable by such meshes forms a smooth manifold. For this manifold of planar triangular meshes we propose a geodesically complete Riemannian metric. It is a distinguishing feature of this metric that it preserves the mesh connectivity and prevents the mesh from degrading along geodesic curves. We detail a symplectic numerical integrator for the geodesic equation in its Hamiltonian formulation. Numerical experiments show that the proposed metric keeps the cell aspect ratios bounded away from zero and thus avoids mesh degradation along arbitrarily long geodesic curves.

Generalized mass metric in Riemannian manifold plays a central role in dynamics and control of multibody system (MBS). In this work, two profitable aspects of multibody system dynamics studies, generalized mass metric in Riemannian geometry and recursive momentum formulation, are described. Firstly, we will derive an Adjoint-based expression of Riemannian metric and operator factorization of generalized mass tensor from a general-topology rigid MBS which comprises of a special Euclidian group SE(3) set. The specific expression can help to clearly understand what reasons lead to these components (Riemannian metric) of the generalized mass tensor and how they measure the curves of generalized velocity space. Meanwhile, the power algorithm of MBS is presented based on the Adjoint map of generalized velocity and generalized force. Next, from the generalized momentum definition depending on such Riemannian mass metric, recursive momentum equations of MBS dynamics are developed for progressively more complex systems: open-chains, topological trees, and closed-loop systems. In terms of the relation principle of impulse and momentum, a new method is proposed for describing conservative MBS form a given initial orientation and location to desired final ones without needing to solve motion process.

This survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of “tight” vs. “overtwisted” or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture.

In the first part of this paper the five-dimensional contact SO(3)-manifolds are classified up to equivariant coorientation preserving contactomorphisms. The construction of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an application that all simply connected 5-manifolds with singular orbits are realized by a Brieskorn manifold with exponents (k,2,2,2). The standard contact structure on such a manifold gives right-handed Dehn twists, and a second contact structure defined in the article gives left-handed twists. In an appendix we also describe the classification of five-dimensional contact SU(2)-manifolds.

Let ξ ¯ st \overline \xi _\textrm {st} be the contact structure naturally induced on the lens space L ( p , q ) = S 3 / Z / p Z L(p,q)=S^3/\mathbb {Z}/p\mathbb {Z} by the standard contact structure ξ st \xi _\textrm {st} on the three–sphere S 3 S^3 . We obtain a complete classification of the symplectic fillings of ( L ( p , q ) , ξ ¯ st ) (L(p,q),\overline \xi _\textrm {st}) up to orientation–preserving diffeomorphisms. In view of our results, we formulate a conjecture on the diffeomorphism types of the smoothings of complex two–dimensional cyclic quotient singularities.

The standard contact structure on the three-sphere is invariant under the action of the cyclic group of order p yielding the lens space L(p,q). Therefore, every lens space carries a natural quotient contact structure Q. A theorem of Eliashberg and McDuff classifies the symplectic fillings of (L(p,1), Q) up to diffeomorphism. We announce a generalization of that result to every lens space. In particular, we give an explicit handlebody decomposition of every symplectic filling of (L(p,q), Q) for every p and q. Our results imply that: (a) there exist infinitely many lens spaces L(p,q) with q>1 such that (L(p,q), Q) admits only one symplectic filling up to blowup and diffeomorphism; (b) for any natural number N, there exist infinitely many lens spaces L(p,q) such that (L(p,q), Q) admits more than N symplectic fillings up to blowup and diffeomorphism.

Conformal quaternionic contact curvature and the local sphere theorem

We prove several finiteness results for the class $M_{a,b,G,n}$ of $n$-manifolds that have fundamental groups isomorphic to $G$ and that can be given complete Riemannian metrics of sectional curvatures within $[a,b]$ where $a\le b<0$. In particular, if $M$ is a closed negatively curved manifold of dimension at least three, then only finitely many manifolds in the class $M_{a,b,\pi_1(M), n}$ are total spaces of vector bundles over $M$. Furthermore, given a word-hyperbolic group $G$ and an integer $n$ there exists a positive $\epsilon=\epsilon(n,G)$ such that the tangent bundle of any manifold in the class $M_{-1-\epsilon, -1, G, n}$ has zero rational Pontrjagin classes.

We prove some contact analogs of smooth embedding theorems for closed $$\pi $$ -manifolds. We show that a closed, k-connected, $$\pi $$ -manifold of dimension $$(2n+1)$$ that bounds a $$\pi $$ -manifold, contact embeds in the $$(4n-2k+3)$$ -dimensional Euclidean space with the standard contact structure. We also prove some isocontact embedding results for $$\pi $$ -manifolds and parallelizable manifolds.

In this paper, we propose a general procedure for establishing the geometric landscape connections of a Riemannian optimization problem under the embedded and quotient geometries. By applying the general procedure to the fixed-rank positive semidefinite (PSD) and general matrix optimization, we establish an exact Riemannian gradient connection under two geometries at every point on the manifold and sandwich inequalities between the spectra of Riemannian Hessians at Riemannian first-order stationary points (FOSPs). These results immediately imply an equivalence on the sets of Riemannian FOSPs, Riemannian second-order stationary points (SOSPs), and strict saddles of fixed-rank matrix optimization under the embedded and the quotient geometries. To the best of our knowledge, this is the first geometric landscape connection between the embedded and the quotient geometries for fixed-rank matrix optimization, and it provides a concrete example of how these two geometries are connected in Riemannian optimization. In addition, the effects of the Riemannian metric and quotient structure on the landscape connection are discussed. We also observe an algorithmic connection between two geometries with some specific Riemannian metrics in fixed-rank matrix optimization: there is an equivalence between gradient flows under two geometries with shared spectra of Riemannian Hessians. A number of novel ideas and technical ingredients—including a unified treatment for different Riemannian metrics, novel metrics for the Stiefel manifold, and new horizontal space representations under quotient geometries—are developed to obtain our results. The results in this paper deepen our understanding of geometric and algorithmic connections of Riemannian optimization under different Riemannian geometries and provide a few new theoretical insights to unanswered questions in the literature. Funding: X. Li was partially supported by National Key R&amp;D Program of China [Grants 2020YFA0711900, 2020YFA0711901], the National Natural Science Foundation of China [Grants 12271107, 62141407], the Young Elite Scientists Sponsorship Program by CAST [Grant 2019QNRC001], the “Chenguang Program” by the Shanghai Education Development Foundation and Shanghai Municipal Education Commission [Grant 19CG02], the Shanghai Science and Technology Program [21JC1400600]. Y. Luo and A. R. Zhang were partially supported by the National Science Foundation [CAREER-2203741].

In recent years, various generalizations of Einstein manifolds are actively studied, for example, manifolds with the trivial Schouten-Weyl tensor, and Ricci solitons, which were first considered by R. Hamilton. Ricci solitons on homogeneous (pseudo)Rieman-nian spaces and, in particular, on the Lie groups have been studied by many mathematicians. For example, there are no nontrivial homogeneous invariant Ricci solitons on three and four-dimensional Lie groups with a left-invariant Riemannian metric. A similar result was proved for the unimodular Lie groups with a left-invariant Riemannian metric in any dimension. However, this question is still an open problem for nonunimodular Lie groups of dimension more than 4. Another important example of Ricci solitons is the case of algebraic Ricci solitons on Lie groups, first considered by J. Lauret. Later, it was proved that every algebraic Ricci soliton on a Lie group with left-invariant (pseudo)Riemannian metric is a homogeneous Ricci soliton. This paper shows the existence of non-trivial algebraic and homogeneous invariant Ricci solitons on conformally flat Lie groups in the case of nondiagonalizable Ricci operator. Also, a non-diagonalizable Ricci operator on Lie groups with harmonic Weyl tensor is demonstrated.