On positivity in Sasaki geometry

  • Charles P. BoyerEmail author
  • Christina W. Tønnesen-Friedman
Original Paper


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.


Positive Sasakian structures Positivity of Ricci curvature Join construction 

Mathematics Subject Classification (2000)

Primary: 53C25 



The authors are grateful to Vestislav Apostolov for carefully reading our paper and suggesting some important clarifications. We also thank Hongnian Huang, and Eveline Legendre for their interest in and comments on our work.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of New MexicoAlbuquerqueUSA
  2. 2.Department of MathematicsUnion CollegeSchenectadyUSA

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