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.

MINIMALITY, HARMONICITY AND CR GEOMETRY FOR REEB VECTOR FIELDS

2010 ◽  
Vol 21 (09) ◽  
pp. 1189-1218 ◽  
Author(s):  
DOMENICO PERRONE
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Harmonic Map ◽  
Contact Manifold ◽  
Contact Metric Manifold ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Contact Metric Manifolds ◽  
Cr Map ◽  
Kaluza Klein

Let (M, g) be a Riemannian manifold and T1 M its unit tangent sphere bundle. Minimality and harmonicity of unit vector fields have been extensively studied by considering on T1M the Sasaki metric [Formula: see text]. This metric, and other well-known Riemannian metrics on T1 M, are particular examples of Riemannian natural metrics. In this paper we equip T1 M with a Riemannian natural metric [Formula: see text] and in particular with a natural contact metric structure. Then, we study the minimality for Reeb vector fields of contact metric manifolds and of quasi-umbilical hypersurfaces of a Kähler manifold. Several explicit examples are given. In particular, the Reeb vector field ξ of a K-contact manifold is minimal for any [Formula: see text] that belongs to a family depending on two parameters of metrics of the Kaluza–Klein type. Next, we show that the Reeb vector field ξ of a K-contact manifold defines a harmonic map [Formula: see text] for any Riemannian natural metric [Formula: see text]. Besides this, if the Reeb vector ξ of an almost contact metric manifold is a CR map then the induced almost CR structure on M is strictly pseudoconvex and ξ is a pseudo-Hermitian map; if in addition ξ is geodesic then [Formula: see text] is a harmonic map. Moreover, the Reeb vector field ξ of a contact metric manifold is a CR map iff ξ is Killing and [Formula: see text] is a special metric of the Kaluza–Klein type. Finally, in the final section, we obtain that there is a family of strictly pseudoconvex CR structures on T1S2n+1 depending on one parameter, for which a Hopf vector field ξ determines a pseudo-harmonic map (in the sense of Barletta–Dragomir–Urakawa [8]) from S2n+1 to T1S2n+1.

Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

2019 ◽  
Vol 16 (03) ◽  
pp. 1950039 ◽  
Author(s):  
V. Venkatesha ◽  
Devaraja Mallesha Naik
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Sasakian Manifold ◽  
Contact Metric Manifold ◽  
Reeb Vector Field ◽  
3 Dimensional ◽  
Flow Vector ◽  
Contact Metric Manifolds ◽  
Yamabe Soliton

If [Formula: see text] is a 3-dimensional contact metric manifold such that [Formula: see text] which admits a Yamabe soliton [Formula: see text] with the flow vector field [Formula: see text] pointwise collinear with the Reeb vector field [Formula: see text], then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if [Formula: see text] is endowed with a Yamabe soliton [Formula: see text], then either [Formula: see text] is flat or it has constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, we show that if [Formula: see text] is non-flat, then either [Formula: see text] is a Sasakian manifold of constant curvature [Formula: see text] or [Formula: see text] is an infinitesimal automorphism of the contact metric structure on [Formula: see text].

scholarly journals Stability of contact metric manifolds and unit vector fields of minimum energy

2007 ◽  
Vol 76 (2) ◽  
pp. 269-283 ◽  
Author(s):  
D. Perrone ◽  
L. Vergori
Keyword(s):  
Vector Field ◽  
Sectional Curvature ◽  
Minimum Energy ◽  
Energy Functional ◽  
Holomorphic Sectional Curvature ◽  
Sasakian Manifolds ◽  
Contact Metric Manifold ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Contact Metric Manifolds

In this paper we obtain criteria of stability for ηEinstein k-contact manifolds, for Sasakian manifolds of constant ϕ-sectional curvature and for 3-dimensional Sasakian manifolds. Moreover, we show that a stable compact Einstein contact metric manifold M is Sasakian if and only if the Reeb vector field ξ minimises the energy functional. In particular, the Reeb vector field of a Sasakian manifold M of constant ϕ-holomorphic sectional curvature +1 minimises the energy functional if and only if M is not simply connected.

Ricci almost solitons and contact geometry

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Amalendu Ghosh
Keyword(s):  
Vector Field ◽  
Contact Manifold ◽  
Ricci Soliton ◽  
Jacobi Field ◽  
Reeb Vector ◽  
Potential Vector ◽  
Reeb Vector Field ◽  
Rigidity Results ◽  
Parallel Ricci Tensor ◽  
Potential Vector Field

Abstract We prove that on a K-contact manifold, a Ricci almost soliton is a Ricci soliton if and only if the potential vector field V is a Jacobi field along the Reeb vector field ξ. Then we study contact metric as a Ricci almost soliton with parallel Ricci tensor. To this end, we consider Ricci almost solitons whose potential vector field is a contact vector field and prove some rigidity results.

scholarly journals Reeb periodic orbits after a bypass attachment

2013 ◽  
Vol 35 (2) ◽  
pp. 615-672
Author(s):  
ANNE VAUGON
Keyword(s):  
Periodic Orbits ◽  
Contact Structure ◽  
Convex Surface ◽  
Three Dimensional ◽  
Contact Manifold ◽  
Contact Structures ◽  
Contact Homology ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Manifold With Boundary

AbstractOn a three-dimensional contact manifold with boundary, a bypass attachment is an elementary change of the contact structure consisting in the attachment of a thickened half-disc with a prescribed contact structure along an arc on the boundary. We give a model bypass attachment in which we describe the periodic orbits of the Reeb vector field created by the bypass attachment in terms of Reeb chords of the attachment arc. As an application, we compute the contact homology of a product neighbourhood of a convex surface after a bypass attachment, and the contact homology of some contact structures on solid tori.

CERTAIN RESULTS ON THREE-DIMENSIONAL CONTACT METRIC MANIFOLDS

2010 ◽  
Vol 03 (04) ◽  
pp. 577-591 ◽  
Author(s):  
Amalendu Ghosh
Keyword(s):  
Ricci Tensor ◽  
Three Dimensional ◽  
Tensor Fields ◽  
Contact Metric Manifold ◽  
Ricci Operator ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
Parallel Ricci Tensor

In this paper we study 3-dimensional contact metric manifolds satisfying certain conditions on the tensor fields *-Ricci tensorS*, h(= ½Lξφ), τ(= Lξg = 2hφ) and the Ricci operator Q. First, we prove that a 3-dimensional non-Sasakian contact metric manifold satisfies. [Formula: see text] (where ⊕X,Y,Z denotes the cyclic sum over X,Y,Z) if and only if M is a generalized (κ, μ)-space. Next, we prove that a 3-dimensional contact metric manifold with vanishing *-Ricci tensor is a generalized (κ, μ)-space. Finally, some results on 3-dimensional contact metric manifold with cyclic η-parallel h or cyclic η-parallel τ or η-parallel Ricci tensor are presented.

On ruled surface in 3-dimensional almost contact metric manifold

2017 ◽  
Vol 14 (05) ◽  
pp. 1750076 ◽  
Author(s):  
Murat Kemal Karacan ◽  
Nural Yuksel ◽  
Hasibe Ikiz
Keyword(s):  
Cross Product ◽  
Ruled Surface ◽  
Contact Metric Manifold ◽  
Almost Contact Metric Manifold ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
Special Cases ◽  
Distribution Parameters ◽  
Almost Contact Metric ◽  
Phd Thesis

In this paper, we study ruled surface in 3-dimensional almost contact metric manifolds by using surface theory defined by Gök [Surfaces theory in contact geometry, PhD thesis (2010)]. We also studied the theory of curves using cross product defined by Camcı. In this study, we obtain the distribution parameters of the ruled surface and then some results and theorems are presented with special cases. Moreover, some relationships among asymptotic curve and striction line of the base curve of the ruled surface have been found.

scholarly journals Some curvature tensors in N(k)-contact metric manifold

BIBECHANA ◽  
2018 ◽  
Vol 16 ◽  
pp. 55-63
Author(s):  
Riddhi Jung Shah
Keyword(s):  
Ricci Tensor ◽  
Einstein Manifold ◽  
Contact Metric Manifold ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
Curvature Tensors

The purpose of this paper is to study W7and W9-curvature tensors on N(k)-contact metric manifolds. We prove that a N(k)-contact metric manifold satisfying the condition W7( xi,X).W9=0 is eta-Einstein manifold. We also obtain the Ricci tensor S of type (0, 2) for phi-W9flat and divW9=0 conditions on N(k)-contact metric manifolds. Finally, we give an example of 3-dimensional N(k)-contact metric manifold.BIBECHANA 16 (2019) 55-63

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.

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