3-Sasakian manifolds. (English) Zbl 0814.53037

Let \((M,g)\) be a Riemannian manifold. If \(\xi\) is a Killing vector field of unit length, \(\eta\) is the 1-form dual to \(\xi\) with respect to \(g\) and \(\Phi\) is the tensor field of type \((1,1)\) defined by \(\Phi= \nabla\xi\), \(\xi\) is said to be a Sasakian structure if the two following conditions are fulfilled: \((\nabla_ X \Phi) (Y)= \eta(Y) X-g(X,Y)\xi\), \((\nabla_ X\eta) (Y)= g(X,\Phi Y)\) for any pair of vector fields \(X\), \(Y\). Y.-Y. Kuo introduced the notion of Sasakian 3-structure [Tôhoku Math. J., II. Ser. 22, 325-332 (1970; Zbl 0205.258)]. Three distinct Sasakian structures \(\{\xi^ a\}\) \((a=1,2,3)\) fulfilling \(g(\xi^ a, \xi^ b)= \delta_{ab}\) and \([\xi^ a, \xi^ b]= 2\varepsilon^{ abc} \xi^ c\) are said to be a Sasakian 3-structure. In this note the authors announce some of their results about the geometry of Sasakian 3-structures whose proofs are contained in some other papers (most of them submitted for publication). It is well known that every 3- Sasakian manifold has dimension \(4n+3\) and defines a Riemannian foliation of codimension \(4n\). The authors announce that the space of leaves of this foliation is a quaternionic Kähler orbifold of dimension \(4n\) with positive scalar curvature equal to \(16n (n+2)\). They classify homogeneous 3-Sasakian manifolds and give some non-homogeneous examples that are not covered by their classification theorem.


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


Zbl 0205.258
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