3-Sasakian manifolds
type: Item
instance of: scholarly article
title: 3-Sasakian manifolds
zbMATH Open document ID: 0814.53037
author: Charles P. Boyer
author: Krzysztof Galicki
author: Benjamin M. Mann
published in: Proceedings of the Japan Academy. Series A
publication date: 1995-05-29T00:00:00Z
review text: 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 \(X\), \(Y\). \textit{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.
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 549046
zbMATH Keywords: Sasakian structure
zbMATH Keywords: Sasakian 3-structure
zbMATH Keywords: Riemannian foliation
zbMATH Keywords: quaternionic Kähler orbifold
reviewed by: Joan Girbau
MaRDI profile type: Publication
full work available at URL: pjaa.69.335
OpenAlex ID: W1991564775
cites work: Q4720334
cites work: Quaternionic reduction and Einstein manifolds
cites work: Q4305108
cites work: New examples of inhomogeneous Einstein manifolds of positive scalar curvature
cites work: Hypercomplex structures on Stiefel manifolds
cites work: New examples of manifolds with strictly positive curvature
cites work: 7-dimensional compact Riemannian manifolds with Killing spinors
cites work: Kählerian Twistor Spaces
cites work: Quaternion Kählerian manifolds
cites work: Quaternion Kählerian manifolds and fibred Riemannian spaces with Sasakian $3$-structure
cites work: Q5665883
cites work: Q5635995
cites work: On manifolds with Sasakian $3$-structure over quaternion Kaehler manifolds
cites work: On almost contact 3-structure
cites work: Q5638960
cites work: Quaternionic Kähler 8-manifolds with positive scalar curvature
cites work: On differentiable manifolds with certain structures which are closely related to almost contact structure. I
cites work: Spherical space forms with normal contact metric 3-structure
cites work: The Gauss-Bonnet theorem for \(V\)-manifolds
cites work: HyperKähler and quaternionic Kähler geometry
cites work: On the isometry groups of Sasakian manifolds
cites work: On a Riemannian space admitting more than one Sasakian structures
cites work: Q5516627
cites work: Q3222791
DOI: 10.3792/PJAA.69.335
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description: scientific article; zbMATH DE number 549046