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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An Application of Convex Integration
to Contact Geometry


Authors: Hansjörg Geiges and Jesús Gonzalo
Journal: Trans. Amer. Math. Soc. 348 (1996), 2139-2149
MSC (1991): Primary 53C15, 53C23
MathSciNet review: 1361639
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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 $(2n+1)$-manifold with contact form $\omega $ whose contact distribution $\ker \omega$ admits $k$ everywhere linearly independent sections, then $M$ admits $k+1$ linearly independent contact forms with linearly independent Reeb vector fields.


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Additional Information

Hansjörg Geiges
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Email: geiges@math.ethz.ch

Jesús Gonzalo
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Email: jgonzalo@ccuam3.sdi.uam.es

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01678-9
PII: S 0002-9947(96)01678-9
Received by editor(s): December 8, 1992
Article copyright: © Copyright 1996 American Mathematical Society