Remarks on contact structures and vector fields on isolated complete intersection singularities

Let(M,g)be a closed, connected, orientedC∞Riemannian 3-manifold with tangentially oriented flowF. Suppose thatFadmits a basic transverse volume formμand mean curvature one-formκwhich is horizontally closed. Let{X,Y}be any pair of basic vector fields, soμ(X,Y)=1. Suppose further that the globally defined vector𝒱[X,Y]tangent to the flow satisfies[Z.𝒱[X,Y]]=fZ𝒱[X,Y]for any basic vector fieldZand for some functionfZdepending onZ. Then,𝒱[X,Y]is either always zero andH, the distribution orthogonal to the flow inT(M), is integrable with minimal leaves, or𝒱[X,Y]never vanishes andHis a contact structure. If additionally,Mhas a finite-fundamental group, then𝒱[X,Y]never vanishes onM, by the above together with a theorem of Sullivan (1979). In this caseHis always a contact structure. We conclude with some simple examples.

Download Full-text

Vector fields on a complete intersection

Functional Analysis and Its Applications ◽

10.1007/bf01080081 ◽

1992 ◽

Vol 25 (4) ◽

pp. 283-284 ◽

Cited By ~ 1

Author(s):

A. G. Aleksandrov

Keyword(s):

Complete Intersection ◽

Vector Fields

Download Full-text

Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics

Compositio Mathematica ◽

10.1112/s0010437x1100529x ◽

2011 ◽

Vol 147 (5) ◽

pp. 1613-1634 ◽

Cited By ~ 13

Author(s):

Eveline Legendre

Keyword(s):

Finite Number ◽

Scalar Curvature ◽

Vector Fields ◽

Existence Result ◽

Constant Scalar Curvature ◽

Contact Structures ◽

Toric Geometry ◽

Contact Manifolds ◽

Reeb Vector ◽

Toric Contact Manifolds

AbstractWe study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least five. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using an existence result of E. Legendre [Toric geometry of convex quadrilaterals, J. Symplectic Geom. 9 (2011), 343–385], we show that a co-oriented compact toric contact 5-manifold whose moment cone has four facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on S2×S3 admitting two non-isometric and non-transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.

Download Full-text