On the Blair's conjecture for contact metric three-manifolds

Abstract

We prove that a bicontact metric three-manifold either is flat or admits some positive sectional curvature at any point. In particular, the Blair's conjecture is true for a bicontact metric three-manifold. Moreover, we prove that a contact metric three-manifold for which the contact distribution cannot be decomposed as a sum of two one-dimensional distributions, admits some positive sectional curvature; this extends the main result of [4].