ON SASAKIAN–EINSTEIN GEOMETRY

Abstract

We introduce a multiplication ⋆ (we call it a join) on the space of all compact Sasakian-Einstein orbifolds [Formula: see text] and show that [Formula: see text] has the structure of a commutative associative topological monoid. The set [Formula: see text] of all compact regular Sasakian–Einstein manifolds is then a submonoid. The set of smooth manifolds in [Formula: see text] is not closed under this multiplication; however, the join [Formula: see text] of two Sasakian–Einstein manifolds is smooth under some additional conditions which we specify. We use this construction to obtain many old and new examples of Sasakain–Einstein manifolds. In particular, in every odd dimension greater that five we obtain spaces with arbitrary second Betti number.