## Algebraic & Geometric Topology

### The curvature of contact structures on $3$–manifolds

#### Abstract

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

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

Dates
Revised: 24 July 2008
Accepted: 27 July 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796897

Digital Object Identifier
doi:10.2140/agt.2008.8.1567

Mathematical Reviews number (MathSciNet)
MR2443254

Zentralblatt MATH identifier
1146.53025

#### Citation

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. https://projecteuclid.org/euclid.agt/1513796897

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