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arXiv · 0906.3487

Tightness in contact metric 3-manifolds

John B. Etnyre·Rafal Komendarczyk·Patrick Massot
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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.

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John B. Etnyre, Rafal Komendarczyk, Patrick Massot. 2011-09-03. Tightness in contact metric 3-manifolds. https://doi.org/10.1007/s00222-011-0355-2

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