Reducibility in Sasakian Geometry

TL;DR

This paper explores the structure of Sasakian manifolds, establishing a decomposition theorem, classifying certain contact structures, and demonstrating splitting of extremal metrics in toric cases, advancing understanding of geometric reducibility.

Contribution

It provides the Sasaki version of the de Rham Decomposition Theorem, introduces cone reducibility, and classifies contact structures on $S^3$ bundles over Riemann surfaces, including splitting results for extremal metrics.

Findings

Established Sasaki de Rham Decomposition Theorem under mild assumptions.

Classified contact structures on $S^3$ bundles over Riemann surfaces with torus actions.

Proved that extremal Sasaki metrics on Sasaki joins split in the toric case.

Abstract

The purpose of this paper is to study reducibility properties in Sasakian geometry. First we give the Sasaki version of the de Rham Decomposition Theorem; however, we need a mild technical assumption on the Sasaki automorphism group which includes the toric case. Next we introduce the concept of {\it cone reducible} and consider $S^3$ bundles over a smooth projective algebraic variety where we give a classification result concerning contact structures admitting the action of a 2-torus of Reeb type. In particular, we can classify all such Sasakian structures up to contact isotopy on $S^3$ bundles over a Riemann surface of genus greater than zero. Finally, we show that in the toric case an extremal Sasaki metric on a Sasaki join always splits.