×

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.

MSC:

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

Citations:

Zbl 0205.258
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] A. L. Besse : Einstein Manifolds. Springer-Verlag, New York (1987). · Zbl 0613.53001
[2] C. P. Boyer, K. Galicki, and B. M. Mann: Quaternionic reduction and Einstein manifolds. Communications in Analysis and Geometry, vol. 1, no. 2, pp. 1-51 (1993). · Zbl 0856.53038
[3] C. P. Boyer, K. Galicki, and B. M. Mann: The geometry and topology of 3-Sasakian manifolds (submitted for publication). · Zbl 0889.53029
[4] C. P. Boyer, K. Galicki, and B. M. Mann: New examples of inhomogeneous Einstein manifolds of positive scalar curvature (submitted for publication). · Zbl 0842.53033
[5] C. P. Boyer, K. Galicki, and B. M. Mann: Hypercomplex structures on circle bundles (in preparation). · Zbl 0843.53030
[6] J. H. Eschenburg: New examples of manifolds with strictly positive curvature. Invent. Math., 66, 469-480 (1982). · Zbl 0484.53031
[7] T. Friedrich and I. Kath: Compact seven-dimensional manifolds with Killing spinors. Commun. Math. Phys., 133, 543-561 (1990). · Zbl 0722.53038
[8] N. J. Hitchin: Kahlerian twistor spaces. Proc. Lond. Math. Soc., 43, 133-150 (1981). · Zbl 0474.14024
[9] S. Ishihara: Quaternion Kahlerian manifolds. J. Diff. Geom., 9, 483-500 (1974). · Zbl 0297.53014
[10] S. Ishihara: Quaternion Kahlerian manifolds and fibered Riemannian spaces with Sasakian 3-structure. Kodai Math. Sem. Rep., 25, 321-329 (1973). · Zbl 0267.53023
[11] S. Ishihara and M. Konishi: Fibered Riemannian spaces with Sasakian 3-structure-. Differential Geometry, in honor of K. Yano. Kinokuniya, Tokyo, pp. 179-194 (1972). · Zbl 0252.53041
[12] T. Kashiwada: A note on a Riemannian space with Sasakian 3-structure. Nat. Sci. Reps. Ochanomizu Univ., 22, 1-2 (1971). · Zbl 0228.53033
[13] M. Konishi: On manifolds with Sasakian 3-structure over quaternion Kahlerian manifolds. Kodai Math. Sem. Reps., 26, 194-200 (1975). · Zbl 0308.53035
[14] Y.-Y. Kuo: On almost contact 3-structure. Tohoku Math. J., 22, 325-332 (1970). · Zbl 0205.25801
[15] Y.-Y. Kuo and S. Tachibana: On the distrubution appeared in contact 3-structure. Taita J. of Math., 2, 17-24 (1970). · Zbl 0231.53053
[16] Y. S. Poon and S. Salamon: Eight-dimensional quaternionic Kahler manifolds with positive scalar curvature. J. Diff. Geom., 33, 363-378 (1990). · Zbl 0733.53035
[17] S. Sasaki: On Differentiable manifolds with certain structures which are closely related to almost contact structure. Tahoku Math. J., 2, 459-476 (1960). · Zbl 0192.27903
[18] S. Sasaki: Spherical space forms with normal contact metric 3-structure. J. Diff. Geom., 6, 307-315 (1972). · Zbl 0245.53044
[19] I. Satake : The Gauss-Bonnet theorem for K-manifolds. J. Math. Soc. Japan, 9, no. 4, 464-476 (1957). · Zbl 0080.37403
[20] A. F. Swann : Hyperkahler and quaternionic Kahler geometry. Math. Ann., 289, 421-450 (1991). · Zbl 0711.53051
[21] S. Tanno: On the isometry of Sasakian manifolds. J. Math*. Soc. Japan, 22, 579-590 (1970). · Zbl 0197.48004
[22] S. Tachibana and W.N.Yu : On a Riemannian space admitting more than one Sasakian structure. Tohoku Math. J., 22, 536-540 (1970). · Zbl 0213.48301
[23] J. A. Wolf: Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. Math. Mech., 14, 1033-1047 (1965). · Zbl 0141.38202
[24] K. Yano and M. Kon: Structures on Manifolds. World Scientific, Singapore (1984). · Zbl 0557.53001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.