Tightness in contact metric 3-manifolds

TL;DR

This paper explores the relationship between Riemannian curvature conditions and the tightness of contact structures on 3-manifolds, establishing new geometric criteria for tightness and universal tightness.

Contribution

It proves an analog of the sphere theorem linking positive curvature to tight contact structures and describes conditions for universal tightness in nonpositive curvature settings.

Findings

Positive 4/9-pinched curvature implies tightness of contact structures.

Universal cover of the manifold has the standard contact structure on S^3.

Geometric conditions for universal tightness in nonpositive curvature.

Abstract

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,\xi) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure \xi is tight; in particular, the contact structure pulled back to the universal cover is the standard contact structure on S^3. We also describe geometric conditions in dimension three for \xi to be universally tight in the nonpositive curvature setting.