An Application of Convex Integration to Contact Geometry
Hansjörg Geiges,Jesús Gonzalo +1 more
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In this paper, Gromov's convex integration technique and the h-principle were used to show that every closed, orientable 3-manifold admits a parallelization by the Reeb vector fields of a triple of contact forms with equal volume form.Abstract:
We prove that every closed, orientable 3-manifold M admits a parallelization by the Reeb vector fields of a triple of contact forms with equal volume form. Our proof is based on Gromov's convex integration technique and the h-principle. Similar methods can be used to show that M admits a parallelization by contact forms with everywhere linearly independent Reeb vector fields. We also prove a generalization of this latter result to higher dimensions. If M is a closed (2n + l)-manifold with contact form w whose contact distribution ker w admits k everywhere linearly independent sections, then M admits k + 1 linearly independent contact forms with linearly independent Reeb vector fields.read more
Citations
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Journal ArticleDOI
Moduli of contact circles
Hansjörg Geiges,Jesús Gonzalo +1 more
TL;DR: In this paper, the authors introduced Teichmuller and moduli spaces for so-called taut contact circles and showed that these geometrically defined deformation spaces are equivalent to certain spaces of representations of the fundamental group in appropriate Lie groups.
Journal ArticleDOI
A Survey of Riemannian Contact Geometry
TL;DR: A survey of the Riemannian contact geometry can be found in this paper, where the authors present a presentation of the five lectures on contact geometry that the author gave at the RIEMain in Contact Conference 2018 in Cagliari, Italy.
Journal ArticleDOI
Fifteen years of contact circles and contact spheres
TL;DR: In this article, the main results about contact circles and contact spheres are reviewed, and two as yet unpublished proofs are presented for the existence of contact spheres on 3-manifolds.
Book ChapterDOI
Contact Forms in Geometry and Topology
TL;DR: In this paper, the authors give an introduction to existence problems of contact structures and some historical considerations pointing important steps in the development of contact geometry are presented, and the existence of contact forms is studied in the next Section.
References
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Book
Partial Differential Relations
TL;DR: In this article, a survey of basic problems and results is presented, and methods to prove the h-Principle are presented, as well as methods to validate the h -Principle.
Journal ArticleDOI
Branched covers and contact structures
TL;DR: Every closed, orientable three-manifold has a parallelization by three contact forms.
Journal ArticleDOI
Closed oriented $3$-manifolds as $3$-fold branched coverings of $S^{3}$ of special type.
TL;DR: In this article, it was shown that the curve in a 3-manifold M which covers the branch set in S3 bounds a disc in M. The first author [Amer. J. Math. No. 98 (1976), no. 4, 989-992] and the second author [Quart. Math No. 27 (1976, no. 105, 85-94] have shown that any closed orientable 3-MANIFold M is a 3fold cover of S3 branched over a knot.
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