  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.
