On Positivity in Sasaki Geometry

Abstract

It is well known that if the dimension of the Sasaki cone is greater than one, then all Sasakian structures are either positive or indefinite. We discuss the phenomenon of type changing within a fixed Sasaki cone. Assuming henceforth that the dimension of the Sasaki cone is greater than one, there are three possibilities, either all elements are positive, all are indefinite, or both positive and indefinite Sasakian structures occur. We illustrate by examples how the type can change as we move in the Sasaki cone. If there exists a Sasakian structure in the cone whose total transverse scalar curvature is non-positive, then all elements of the Sasaki cone are indefinite. Furthermore, we prove that if the first Chern class is a torsion class or represented by a positive definite (1,1) form, then all elements of the Sasaki cone are positive.