The Sasaki Join, Hamiltonian 2-forms, and Sasaki-Einstein Metrics

Abstract

By combining the join construction from Sasakian geometry with the Hamiltonian 2-form construction from Kähler geometry, we recover Sasaki-Einstein metrics discovered by physicists. Our geometrical approach allows us to give an algorithm for computing the topology of these Sasaki-Einstein manifolds. In particular, we explicitly compute the cohomology rings for several cases of interest and give a formula for homotopy equivalence in one particular 7-dimensional case. We also show that our construction gives at least a two dimensional cone of both Sasaki-Ricci solitons and extremal Sasaki metrics.

38 pages, paragraph added to introduction and Proposition 4.1 added, Proposition 4.15 corrected, Remark 5.5 added, and explanation for irregular Sasaki-Einstein structures expanded. Reference added

References in corpus (2)

• On the Equivalence Problem for Toric Contact Structures on S^3-bundles over S^2$
• The Sasaki Join, Hamiltonian 2-forms, and Constant Scalar Curvature

Cited by in corpus (4)

• Simply Connected Manifolds with Infinitely Many Toric Contact Structures and Constant Scalar Curvature Sasaki Metrics
• The Sasaki Join, Hamiltonian 2-forms, and Constant Scalar Curvature
• The Sasaki Join and Admissible Kähler Constructions
• On the Topology of some Sasaki-Einstein Manifolds