Tightness in contact metric 3-manifolds

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,ξ) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure ξis 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 ξto be universally tight in the nonpositive curvature setting.

29 pages. Added the sphere theorem, removed high dimensional material and an alternate approach to the three dimensional tightness radius estimates

References in corpus (5)

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• On the existence of tight contact structures on Seifert fibered 3-manifolds
• Quantitative Darboux theorems in contact geometry
• Tight Beltrami fields with symmetry
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• Contact structures induced by skew fibrations of R^3
• Contact structures and Beltrami fields on the torus and the sphere
• Dynamics and Topology of Conformally Anosov Contact 3-Manifolds
• Fibrations of ${\mathbb{R}}^3$ by oriented lines
• Quantitative Darboux theorems in contact geometry
• Dividing sets as nodal sets of an eigenfunction of the Laplacian
• On the existence of critical compatible metrics on contact $3$-manifolds
• Singular Contact Geometry and Beltrami Fields in Cholesteric Liquid Crystals
• Ricci Curvature, Reeb Flows and Contact 3-Manifolds