scholarly journals A Survey of Riemannian Contact Geometry

2019 ◽  
Vol 6 (1) ◽  
pp. 31-64 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Contact Structure ◽  
Positive Curvature ◽  
Contact Geometry ◽  
Contact Structures ◽  
Contact Manifolds ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
The Subject ◽  
Mere Existence ◽  
Curvature Functionals

AbstractThis 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.

On the non-regularity of tangent sphere bundles

1978 ◽  
Vol 82 (1-2) ◽  
pp. 13-17 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Riemannian Manifold ◽  
Large Class ◽  
Constant Curvature ◽  
Positive Constant ◽  
Contact Structure ◽  
Compact Riemannian Manifold ◽  
Contact Structures ◽  
Sphere Bundle ◽  
Contact Manifolds ◽  
Tangent Sphere

SynopsisClassically the tangent sphere bundles have formed a large class of contact manifolds; their contact structures are not in general regular, however. Specifically we prove that the natural contact structure on the tangent sphere bundle of a compact Riemannian manifold of non-positive constant curvature is not regular.

scholarly journals Legendrian contact homology and topological entropy

2019 ◽  
Vol 11 (01) ◽  
pp. 53-108 ◽  
Author(s):  
Marcelo R. R. Alves
Keyword(s):  
Growth Rate ◽  
Topological Entropy ◽  
Infinite Family ◽  
Contact Structures ◽  
Legendrian Knots ◽  
Contact Homology ◽  
Contact Manifolds ◽  
3 Dimensional ◽  
Reeb Flows ◽  
Legendrian Contact Homology

In this paper we study the growth rate of a version of Legendrian contact homology, which we call strip Legendrian contact homology, in 3-dimensional contact manifolds and its relation to the topological entropy of Reeb flows. We show that: if for a pair of Legendrian knots in a contact 3-manifold [Formula: see text] the strip Legendrian contact homology is defined and has exponential homotopical growth with respect to the action, then every Reeb flow on [Formula: see text] has positive topological entropy. This has the following dynamical consequence: for all Reeb flows (even degenerate ones) on [Formula: see text] the number of hyperbolic periodic orbits grows exponentially with respect to the period. We show that for an infinite family of 3-manifolds, infinitely many different contact structures exist that possess a pair of Legendrian knots for which the strip Legendrian contact homology has exponential growth rate.

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 Space time manifolds and contact structures

1990 ◽  
Vol 13 (3) ◽  
pp. 545-553 ◽  
Author(s):  
K. L. Duggal
Keyword(s):  
Contact Structure ◽  
Contact Manifold ◽  
Contact Structures ◽  
Timelike Vector ◽  
Contact Manifolds ◽  
New Class ◽  
Dimensional Spacetime ◽  
Globally Hyperbolic Spacetimes ◽  
Hyperbolic Spacetimes ◽  
Regular Contact

A new class of contact manifolds (carring a global non-vanishing timelike vector field) is introduced to establish a relation between spacetime manifolds and contact structures. We show that odd dimensional strongly causal (in particular, globally hyperbolic) spacetimes can carry a regular contact structure. As examples, we present a causal spacetime with a non regular contact structure and a physical model [Gödel Universe] of Homogeneous contact manifold. Finally, we construct a model of 4-dimensional spacetime of general relativity as a contact CR-submanifold.

scholarly journals On the Sign of the Curvature of a Contact Metric Manifold

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 892
Author(s):  
David E. Blair
Keyword(s):  
Positive Curvature ◽  
Contact Manifold ◽  
Contact Metric Manifold ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
Expository Article ◽  
Standard Contact ◽  
Negative Sectional Curvature

In this expository article, we discuss the author’s conjecture that an associated metric for a given contact form on a contact manifold of dimension ≥5 must have some positive curvature. In dimension 3, the standard contact structure on the 3-torus admits a flat associated metric; we also discuss a local example, due to Krouglov, where there exists a neighborhood of negative curvature on a particular 3-dimensional contact metric manifold. In the last section, we review some results on contact metric manifolds with negative sectional curvature for sections containing the Reeb vector field.

On curves in 3-dimensional normal almost contact metric manifolds

2020 ◽  
Vol 18 (01) ◽  
pp. 2150004
Author(s):  
Abdullah Yıldırım
Keyword(s):  
Contact Manifolds ◽  
3 Dimensional ◽  
And Topology ◽  
Contact Metric Manifolds ◽  
Almost Contact Metric ◽  
Almost Contact Metric Manifolds ◽  
Frenet Curves

The characterization of curves plays an important role in both geometry and topology of almost contact manifolds. Olszak found the equation [Formula: see text] on normal almost contact manifolds. The pair [Formula: see text] denotes the type of these manifolds. In this study, we obtained the curvatures of non-geodesic Frenet curves on [Formula: see text]-dimensional normal almost contact manifolds without neglecting [Formula: see text] and [Formula: see text], and provided the results of their characterization. We exemplified these results with examples.

scholarly journals Contact Structures of Sasaki Type and Their Associated Moduli

2019 ◽  
Vol 6 (1) ◽  
pp. 1-30
Author(s):  
Charles P. Boyer
Keyword(s):  
Scalar Curvature ◽  
Present Article ◽  
Einstein Metrics ◽  
Contact Structure ◽  
Constant Scalar Curvature ◽  
Contact Geometry ◽  
Contact Structures ◽  
Sasakian Geometry ◽  
Sasakian Structures

Abstract This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special type of Riemannian contact structure known as Sasakian geometry. An ultimate goal of this survey is to understand the moduli of classes of Sasakian structures as well as the moduli of extremal and constant scalar curvature Sasaki metrics, and in particular the moduli of Sasaki-Einstein metrics.

scholarly journals On Subcritically Stein Fillable 5-manifolds

2018 ◽  
Vol 61 (1) ◽  
pp. 85-96 ◽  
Author(s):  
Fan Ding ◽  
Hansjörg Geiges ◽  
Guangjian Zhang
Keyword(s):  
Fundamental Group ◽  
Contact Structure ◽  
Contact Structures ◽  
Homotopy Classification ◽  
Contact Manifolds ◽  
Almost Contact Structure ◽  
Diffeomorphism Type ◽  
Standard Contact ◽  
Simply Connected

AbstractWe make some elementary observations concerning subcritically Stein fillable contact structures on 5-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case where the fundamental group is finite cyclic, and we show that on the 5-sphere, the standard contact structure is the unique subcritically ?llable one. More generally, it is shown that subcritically fillable contact structures on simply connected 5-manifolds are determined by their underlying almost contact structure. Along the way, we discuss the homotopy classification of almost contact structures.

scholarly journals Certain Geometric Aspects of a Class of Almost Contact Structures on a Smooth Metric Manifold

2020 ◽  
pp. 24-38
Author(s):  
Ojiema M. Onyango ◽  
Mengich Beatrice ◽  
Wafula A. Wanambisi
Keyword(s):  
Riemannian Manifolds ◽  
Contact Structure ◽  
Contact Structures ◽  
Linear Combinations ◽  
Almost Contact Structure ◽  
Contact Metric Manifolds ◽  
Algebraic Properties ◽  
Almost Contact Metric ◽  
Codazzi Equations ◽  
Almost Contact Metric Manifolds

The classication of Smooth Geometrical Manifolds still remains an open problem. The concept of almost contact Riemannian manifolds provides neat descriptions and distinctions between classes of odd and even dimensional manifolds and their geometries. We construct an almost contact structure which is related to almost contact 3-structure carried on a smooth Riemannian manifold (M, gM) of dimension (5n + 4) such that gcd(2, n) = 1. Starting with the almost contact metric manifolds (N4n+3, gN) endowed with structure tensors (ϕi, ξj , ηk) such that 1 ≤ i, j, k ≤ 3 of types (1, 1), (1, 0), (0, 1) respectively, we establish that there exists a structure (ϕ4, ξ4, η4) on (N4n+3 ⊗ Rd) ≈ M; gcd(4, d) = 1, d|2n + 1, constructed as linear combinations of the three structures on (N4n+3, gN) . We study some algebraic properties of the tensors of the constructed almost contact structure and further explore the Geometry of the two manifolds (N4n+3⊗Rd) ≈ M and N4n+3 via a !submersion F : (N4n+3 ⊗Rd) ↩→ (N4n+3) and the metrics gM respective gN between them. This provides new forms of Gauss-Weigarten's equations, Gauss-Codazzi equations and the Ricci equations incorporating the submersion other than the First and second Fundamental coecients only. Fundamentally, this research has revealed that the structure (ϕ4, ξ4, η4) is constructible and it is carried on the hidden compartment of the manifold M∼=(N4n+3 ⊗ Rd) (d|2n + 1) which is related to the manifold (N4n+3).

scholarly journals Real open books and real contact structures

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
Ferit Öztürk ◽  
Nermin Salepci
Keyword(s):  
Contact Structure ◽  
Contact Manifold ◽  
Contact Structures ◽  
Open Book ◽  
Positive Real ◽  
The Real ◽  
3 Dimensional ◽  
Open Books ◽  
Real Contact ◽  
Open Book Decompositions

AbstractA real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. In this article we study open book decompositions on smooth real 3-manifolds that are compatible with the real structure.We call them real open book decompositions.We show that each real open book carries a real contact structure and two real contact structures supported by the same real open book decomposition are equivariantly isotopic. We also show that every real contact structure on a closed 3-dimensional real manifold is supported by a real open book. Finally, we conjecture that two real open books on a real contact manifold supporting the same real contact structure are related by positive real stabilizations and equivariant isotopy, and that the Giroux correspondence applies to real manifolds as well, namely that there is a one-to-one correspondence between the real contact structures on a real 3-manifold up to equivariant contact isotopy and the real open books up to positive real stabilization. Meanwhile, we study some examples of real open books and real Heegaard decompositions in lens spaces.

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