The curvature of contact structure on 3-manifolds

TL;DR

This paper investigates the sectional curvature of contact structures on 3-manifolds, demonstrating the ability to manipulate this curvature and establishing the existence of metrics with constant negative curvature for such structures.

Contribution

It introduces methods to control the curvature of contact structures and defines Gaussian curvature for plane distributions, extending curvature analysis in 3-manifolds.

Findings

Existence of metrics with sectional curvature -1 for contact structures

Introduction of Gaussian curvature for plane distributions

Methods to manipulate curvature of contact structures

Abstract

We study the sectional curvature of plane distributions on 3-manifolds. We show that if the distribution is a contact structure it is easy to manipulate this curvature. As a corollary we obtain that for every transversally oriented contact structure on a closed 3-dimensional manifold $M$ 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 the similar results.