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On Sasakian-Einstein geometry. (English) Zbl 1022.53038

In the work under review the construction of many new Sasakian-Einstein manifolds is given. By making use of this construction we can also obtain some well-known examples of such manifolds. We recall that a Sasakian manifold, which is an Einstein manifold, is called a Sasakian-Einstein manifold. Let \({\mathcal{SE}}\) denote the space of all compact Sasakian-Einstein orbifolds and let \(\star \) be a multiplication (called a join) defined on the space \({\mathcal{SE}}\). It was shown that \(({\mathcal{SE}}, \star)\) has a structure of a commutative associative topological monoid and that the set \({\mathcal{R}} \subset {\mathcal{SE}}\) of all compact regular Sasakian-Einstein manifolds is a submonoid. The set of smooth manifolds in \({\mathcal{SE}}\) is not closed under this multiplication. However, under some additional conditions, the join \({\mathcal{S_{1}}} \star {\mathcal{S}_{2}}\) of two Sasakian-Einstein manifolds, is a smooth manifold. By making use of this construction, examples of Sasakian-Einstein manifolds of dimension greater than 5 can be obtained.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C12 Foliations (differential geometric aspects)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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