Algebraic & Geometric Topology

The curvature of contact structures on $3$–manifolds

Vladimir Krouglov

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We study the sectional curvature of plane distributions on 3–manifolds. We show that if a distribution is a contact structure it is easy to manipulate its curvature. As a corollary we obtain that for every transversally oriented contact structure on a closed 3–dimensional manifold, there is a metric such that the sectional curvature of the contact distribution is equal to 1. We also introduce the notion of Gaussian curvature of the plane distribution. For this notion of curvature we get similar results.

Article information

Algebr. Geom. Topol., Volume 8, Number 3 (2008), 1567-1579.

Received: 4 February 2008
Revised: 24 July 2008
Accepted: 27 July 2008
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx]
Secondary: 53B21: Methods of Riemannian geometry

contact structure uniformization curvature


Krouglov, Vladimir. The curvature of contact structures on $3$–manifolds. Algebr. Geom. Topol. 8 (2008), no. 3, 1567--1579. doi:10.2140/agt.2008.8.1567.

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