On positive Sasakian geometry

Abstract

A Sasakian structure S = (ξ; η; φ; g) on a manifold M is called positive if its basic first Chern class c1(Fξ) can be represented by a positive (1,1)-form with respect to its transverse holomorphic CR-structure. We prove a theorem that says that every positive Sasakian structure can be deformed to a Sasakian structure whose metric has positive Ricci curvature. This provides us with a new technique for proving the existence of positive Ricci curvature metrics on certain odd dimensional manifolds. As an example we give a completely independent proof of a result of Sha and Yang that for every nonnegative integer k the 5-manifolds k#(S2 × S 3) admits metrics of positive Ricci curvature.