Sasakian Geometry on Sphere Bundles II: Constant Scalar Curvature

TL;DR

This paper advances the construction of constant scalar curvature Sasaki metrics on sphere bundles by applying recent existence theorems, providing explicit examples in dimensions 5 and 7.

Contribution

It extends previous work by utilizing a new existence theorem to produce constant scalar curvature Sasaki metrics on sphere bundles, with explicit constructions in specific dimensions.

Findings

Existence of constant scalar curvature Sasaki metrics under certain conditions.

Explicit construction of such metrics on 5- and 7-dimensional sphere bundles.

Application of recent theorems to classical geometric structures.

Abstract

In a previous paper [BTF21] the authors employed the fiber join construction of Yamazaki [Yam99] together with the admissible construction of Apostolov, Calderbank, Gauduchon, and T{\o}nnesen-Friedman [ACGTF08a] to construct new extremal Sasaki metrics on odd dimensional sphere bundles over smooth projective algebraic varieties. In the present paper we continue this study by applying a recent existence theorem [BHLTF23] that shows that under certain conditions one can always obtain a constant scalar curvature Sasaki metric in the Sasaki cone. Moreover, we explicitly describe this construction for certain sphere bundles of dimension 5 and 7.