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The geometry and topology of 3-Sasakian manifolds. (English) Zbl 0889.53029
Sasakian geometry is generally considered as the odd-dimensional analogue of Kähler geometry. A 3-Sasakian manifold is a Riemannian manifold equipped with three Sasakian structures which are related in a certain way. It was already known in the 1970s that 3-Sasakian geometry is closely related to quaternionic Kähler geometry. In the present paper the authors study in detail the geometry and topology of 3-Sasakian manifolds. Their results include the following ones.
Each 3-Sasakian manifold $$S$$ is an Einstein manifold with positive scalar curvature. The three structure vector fields of the Sasakian structures determine a three-dimensional foliation $$F$$ on $$S$$ whose leaves are three-dimensional homogeneous spherical space forms. The metric is bundle-like with respect to $$F$$ and the space of leaves $$S/F$$ is a quaternionic Kähler orbifold with positive scalar curvature. It follows that each complete 3-Sasakian manifold is compact with $$\pi_1(S)$$ finite. Given a complete 3-Sasakian manifold $$S$$, the product $$S\times \mathbb{R}^+$$ with the cone metric is hyperkähler with some additional properties which can be used to construct new examples of compact hypercomplex manifolds. The authors classify explicitly all 3-Sasakian manifolds with a transitive group of automorphisms of the 3-Sasakian structure. All of them fiber over Wolf spaces. They present a reduction procedure for obtaining new 3-Sasakian manifolds from a given one. There is a natural free circle action on the complex Stiefel manifolds of Hermitian orthonormal 2-frames in $$\mathbb{C}^n$$ depending on $$n$$-tuples $$p= (p_1, \dots, p_n)$$ for pairwise relatively prime positive integers. The corresponding quotient $$S(p)$$ is a compact, simply connected 3-Sasakian manifold. The authors compute explicitly the integral cohomology ring of $$S(p)$$. This implies that in each dimension $$4n+3 (n\geq 1)$$ there exist infinitely many distinct homotopy types of complete 3-Sasakian manifolds which are, with the exception of $$S(1,\dots,1)$$, inhomogeneous Einstein manifolds. Moreover, in dimension 7 some of these examples are even not homotopy equivalent to any compact homogeneous space.

##### MSC:
 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C30 Differential geometry of homogeneous manifolds
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