Tightness in contact metric 3--manifolds
Joint work with Rafal Komendarczyk and Patrick Massot,
Invent. Math. 188 (2012), 621--657.

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

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