Reducibility in Sasakian geometry

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 cone reducible and consider S 3 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 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.