On positivity in Sasaki geometry

Abstract

It is well known that if the dimension of the Sasaki cone $${\mathfrak {t}}^+$$ is greater than one, then all Sasakian structures in $${\mathfrak {t}}^+$$ are either positive or indefinite. We discuss the phenomenon of type changing within a fixed Sasaki cone. Assuming henceforth that $$\dim {\mathfrak {t}}^+>1$$ there are three possibilities, either all elements of $${\mathfrak {t}}^+$$ are positive, all are indefinite, or both positive and indefinite Sasakian structures occur in $${\mathfrak {t}}^+$$. We illustrate by examples how the type can change as we move in $${\mathfrak {t}}^+$$. If there exists a Sasakian structure in $${\mathfrak {t}}^+$$ whose total transverse scalar curvature is non-positive, then all elements of $${\mathfrak {t}}^+$$ 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 $${\mathfrak {t}}^+$$ are positive.