On positivity in Sasaki geometry

Abstract

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

Locally homogeneous aspherical Sasaki manifolds

“…We describe in detail a natural procedure which explicitly constructs a new Sasaki manifold from a pair of given regular Sasaki manifolds. This correponds to a variant of the join construction as is discussed in [11] for the compact case. In our context we apply the join in the construction of homogeneous Sasaki manifolds.…”