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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.

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.

Reviewer: Jürgen Berndt (Hull)

### MSC:

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

53C30 | Differential geometry of homogeneous manifolds |