3-Sasakian manifold

Sasakian and -Sasakian spaces are odd-dimensional companions of Kähler and hyper-Kähler manifolds, respectively. A Riemannian manifold of dimension is called Sasakian if the holonomy group of the metric cone reduces to a subgroup of . In particular, , , and such a cone is a Kähler manifold. Let be a complex structure on . Then restricted to is a unit Killing vector field (cf. also Killing vector) with the property that the sectional curvature of every section containing equals one. Such a is called the characteristic vector field on and its properties can be used as an alternative characterization of a Sasakian manifold.

Similarly, one says that is a -Sasakian manifold if the holonomy group of the metric cone reduces to a subgroup of . In particular, , , and the cone is a hyper-Kähler manifold. When is -Sasakian, the hyper-Kähler structure on the associated cone can be used to define three vector fields , , where is a hypercomplex structure on . It follows that, when restricted to , are Killing vector fields such that and . Hence, they are orthonormal and locally define an isometric (or ) action on . In turn, the triple yields and for each . The collection of tensors is traditionally called the -Sasakian structure on . This is the way such structures were first introduced in the work of C. Udrişte [a1] and Y. Kuo [a2] in 1969 and 1970.

Every -Sasakian manifold is an Einstein manifold with positive Einstein constant . If is complete, it is compact with finite fundamental group. If is compact, the characteristic vector fields are complete and define a -dimensional foliation on . The leaves of this foliation are necessarily compact, since defines a locally free action on . Hence, the foliation is automatically almost-regular and the space of leaves is a compact orbifold, denoted by . The leaves of are totally geodesic submanifolds of constant sectional curvature equal one (cf. also Totally-geodesic manifold). They are all -dimensional homogeneous spherical space forms , where is a finite subgroup (cf. also Space forms). In particular, the leaves are -Sasakian manifolds themselves. The space of leaves is a compact positive quaternion Kähler orbifold. The principal leaves are always diffeomorphic to either or . A compact -Sasakian manifold is said to be regular if is regular, i.e., if all the leaves are diffeomorphic. In this case is a smooth manifold (cf. Differentiable manifold). For any such that , the vector field has the Sasakian property. Hence, a -Sasakian manifold has a -sphere worth of Sasakian structures (just as hyper-Kähler manifold has an -worth of complex structures). When is compact, the vector field defines a -dimensional foliation with compact leaves. Such a foliation gives an isometric locally free circle action . The space of leaves is a compact Kähler–Einstein orbifold of positive scalar curvature. It is a simply-connected normal projective algebraic variety (cf. Projective algebraic set). has a complex contact structure and it is a -factorial Fano variety (cf. also Fano variety). It is an orbifold twistor space of . All the foliations associated to can be described in the the following diagram of orbifold fibrations:

All four geometries in the above diagram are Einstein. Both and admit second, non-isometric Einstein metrics of positive scalar curvature. Every -Sasakian manifold is a spin manifold (cf. Spinor structure). When is complete, simply-connected and not of constant curvature, it admits Killing spinors, where . The holonomy group of never reduces to a proper subgroup of and the metric admits no infinitesimal deformations.

For every compact semi-simple Lie group one has a corresponding diagram with being a symmetric positive quaternion Kähler manifold (a Wolf space) and . In particular, every every -Sasakian -homogeneous space is regular and it is one of the spaces

Here, , denotes the trivial group, , and . Hence, there is one-to-one correspondence between the simple Lie algebras and the simply-connected -Sasakian homogeneous manifolds.

There is a conjecture that all complete regular -Sasakian manifolds are homogeneous. It is a simple translation of the corresponding conjecture due to C. LeBrun and S. Salamon [a3] that all positive quaternion Kähler manifolds are symmetric. This is known to be true when or ( or ). More generally, it is know that in each dimension , there are only finitely many complete regular -Sasakian manifolds, all of them having with equality holding only when . Furthermore, it was shown by K. Galicki and S. Salamon [a4] that each Betti number of such an must satisfy the linear relation

with odd Betti numbers for . In fact the vanishing of odd Betti numbers holds true in the irregular case as well. There are, however, examples of an -dimensional irregular -Sasakian manifold for which and of -dimensional manifolds with . These were constructed explicitly by C. Boyer, K. Galicki and B. Mann [a5].

The first complete irregular examples that are not quotients of homogeneous spaces by a discrete group of isometries were obtained also by Boyer, Galicki and Mann [a10], [a11], [a12], [a13], [a14], [a15], using a method called -Sasakian reduction. The examples are bi-quotients of unitary groups of the form . The -dimensional family depends on positive integral "weights" which are pairwise relatively prime. The integral cohomology ring of depends on the weight vector and one gets infinitely many homotopy types of compact simply-connected -Sasakian manifolds in each allowable dimension . Other irregular examples were constructed later in dimension , , by Boyer, Galicki, Mann, and E. Rees [a6]. The same method of -Sasakian reduction was used to obtain families of compact simply-connected -Sasakian -manifolds with an arbitrary second Betti number. All these examples are toric, i.e., having or as the group of isometries with the -torus action preserving the -Sasakian structure. R. Bielawski [a7] showed that, in any dimension , a toric -Sasakian manifold is necessarily diffeomorphic to one of the quotients obtained in [a6]. Examples of compact -Sasakian manifold which are not toric can also be constructed.

After their introduction in 1969, -Sasakian manifolds were vigorously studied by a group of Japanese geometers, including S. Ishihara, T. Kashiwada, M. Konishi, Y. Kuo, S. Tachibana, S. Tanno, and W.N. Yu [a16], [a17], [a18], [a19], [a20], [a21], [a22], [a23]. This lasted until 1975, when the whole subject was relegated to an almost complete obscurity largely due to lack of any interesting examples. In the early 1990s -Sasakian manifolds returned in two different areas. One of them is the study of -manifolds admitting Killing spinors, in the work of T. Friedrich and I. Kath [a8]. The other is the work [a10], [a11], [a12], [a13], [a14], [a15], of Boyer–Galicki–Mann, in which the first irregular examples are constructed and a systematic study of geometry and topology of compact -Sasakian manifolds is undertaken.

For a detailed review of the subject and extensive bibliography see [a9].

References

[a1]	C. Udrişte, "Structures presque coquaternioniennes" Bull. Math. Soc. Sci. Math. Roum. , 12 (1969) pp. 487–507 MR0296849 Zbl 0213.48205
[a2]	Y.-Y. Kuo, "On almost contact 3-structure" Tôhoku Math. J. , 22 (1970) pp. 325–332 MR0278225 Zbl 0205.25801
[a3]	K. Galicki, S. Salamon, "On Betti numbers of 3-Sasakian manifolds" Geom. Dedicata , 63 (1996) pp. 45–68 MR1413621
[a4]	C. LeBrun, S.M. Salamon, "Strong rigidity of positive quaternion–Kähler manifolds" Invent. Math. , 118 (1994) pp. 109–132
[a5]	C.P. Boyer, K. Galicki, B.M. Mann, "A note on smooth toral reductions of spheres" Manuscripta Math. , 95 (1998) pp. 149–158 MR1603301 Zbl 0913.53020
[a6]	C.P. Boyer, K. Galicki, B.M. Mann, E. Rees, "Compact 3-Sasakian 7-manifolds with arbitrary second Betti number" Invent. Math. , 131 (1998) pp. 321–344 MR1608567 Zbl 0901.53033
[a7]	R. Bielawski, "Complete -invariant hyperkähler -manifolds" MPI preprint , 65 (1998) (www.mpim-bonn.mpg.de/html/preprints/preprints.html)
[a8]	T. Friedrich, I. Kath, "Compact seven-dimensional manifolds with Killing spinors" Comm. Math. Phys. , 133 (1990) pp. 543–561
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[a11]	C.P. Boyer, K. Galicki, B.M. Mann, "The geometry and topology of 3-Sasakian manifolds" J. Reine Angew. Math. , 455 (1994) pp. 183–220 MR1293878 Zbl 0889.53029
[a12]	C.P. Boyer, K. Galicki, B.M. Mann, "New examples of inhomogeneous Einstein manifolds of positive scalar curvature" Math. Res. Lett. , 1 (1994) pp. 115–121 MR1258497 Zbl 0842.53033
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