The curvature of contact structures on 3–manifolds

Krouglov, Vladimir

doi:10.2140/agt.2008.8.1567

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The curvature of contact structures on $3$–manifolds

Vladimir Krouglov

Algebraic & Geometric Topology 8 (2008) 1567–1579

DOI: 10.2140/agt.2008.8.1567

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.

Keywords

contact structure, uniformization, curvature

Mathematical Subject Classification 2000

Primary: 53D35

Secondary: 53B21

References

Publication

Received: 4 February 2008

Revised: 24 July 2008

Accepted: 27 July 2008

Published: 15 September 2008

Authors

Vladimir Krouglov
Department of Geometry Institute for Low Temperature Physics and Engineering 47 Lenin Ave Kharkov 61103 Ukraine

Volume 8, issue 3 (2008)