View source for 3-Sasakian manifold

Sasakian and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t1200102.png" />-Sasakian spaces are odd-dimensional companions of Kähler and hyper-Kähler manifolds, respectively. A [[Riemannian manifold|Riemannian manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t1200103.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t1200104.png" /> is called Sasakian if the [[Holonomy group|holonomy group]] of the metric cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t1200105.png" /> reduces to a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t1200106.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t1200107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t1200108.png" />, and such a cone is a [[Kähler manifold|Kähler manifold]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t1200109.png" /> be a [[Complex structure|complex structure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001010.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001011.png" /> restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001012.png" /> is a unit Killing vector field (cf. also [[Killing vector|Killing vector]]) with the property that the [[Sectional curvature|sectional curvature]] of every section containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001013.png" /> equals one. Such a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001014.png" /> is called the characteristic vector field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001015.png" /> and its properties can be used as an alternative characterization of a Sasakian manifold.

Similarly, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001016.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001018.png" />-Sasakian manifold if the holonomy group of the metric cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001019.png" /> reduces to a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001020.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001022.png" />, and the cone is a hyper-Kähler manifold. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001023.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001024.png" />-Sasakian, the hyper-Kähler structure on the associated cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001025.png" /> can be used to define three vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001028.png" /> is a hypercomplex structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001029.png" />. It follows that, when restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001031.png" /> are Killing vector fields such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001033.png" />. Hence, they are orthonormal and locally define an isometric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001034.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001035.png" />) action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001036.png" />. In turn, the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001037.png" /> yields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001039.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001040.png" />. The collection of tensors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001041.png" /> is traditionally called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001043.png" />-Sasakian structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001044.png" />. This is the way such structures were first introduced in the work of C. Udrişte [[#References|[a1]]] and Y. Kuo [[#References|[a2]]] in 1969 and 1970.

Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001045.png" />-Sasakian manifold is an Einstein manifold with positive Einstein constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001046.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001047.png" /> is complete, it is compact with finite [[Fundamental group|fundamental group]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001048.png" /> is compact, the characteristic vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001049.png" /> are complete and define a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001050.png" />-dimensional [[Foliation|foliation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001051.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001052.png" />. The leaves of this foliation are necessarily compact, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001053.png" /> defines a locally free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001054.png" /> action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001055.png" />. Hence, the foliation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001056.png" /> is automatically almost-regular and the space of leaves is a compact orbifold, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001057.png" />. The leaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001058.png" /> are totally geodesic submanifolds of constant sectional curvature equal one (cf. also [[Totally-geodesic manifold|Totally-geodesic manifold]]). They are all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001059.png" />-dimensional homogeneous spherical space forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001060.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001061.png" /> is a finite subgroup (cf. also [[Space forms|Space forms]]). In particular, the leaves are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001062.png" />-Sasakian manifolds themselves. The space of leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001063.png" /> is a compact positive quaternion Kähler orbifold. The principal leaves are always diffeomorphic to either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001064.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001065.png" />. A compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001066.png" />-Sasakian manifold is said to be regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001068.png" /> is regular, i.e., if all the leaves are diffeomorphic. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001069.png" /> is a smooth manifold (cf. [[Differentiable manifold|Differentiable manifold]]). For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001070.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001071.png" />, the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001072.png" /> has the Sasakian property. Hence, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001073.png" />-Sasakian manifold has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001074.png" />-sphere worth of Sasakian structures (just as hyper-Kähler manifold has an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001075.png" />-worth of complex structures). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001076.png" /> is compact, the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001077.png" /> defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001078.png" />-dimensional foliation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001079.png" /> with compact leaves. Such a foliation gives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001080.png" /> an isometric locally free circle action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001081.png" />. The space of leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001082.png" /> is a compact Kähler–Einstein orbifold of positive scalar curvature. It is a simply-connected normal projective algebraic variety (cf. [[Projective algebraic set|Projective algebraic set]]). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001083.png" /> has a complex [[Contact structure|contact structure]] and it is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001085.png" />-factorial Fano variety (cf. also [[Fano variety|Fano variety]]). It is an orbifold twistor space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001086.png" />. All the foliations associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001087.png" /> can be described in the the following diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001088.png" /> of orbifold fibrations:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001089.png" /></td> </tr></table>

All four geometries in the above diagram are Einstein. Both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001091.png" /> admit second, non-isometric Einstein metrics of positive scalar curvature. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001092.png" />-Sasakian manifold is a spin manifold (cf. [[Spinor structure|Spinor structure]]). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001093.png" /> is complete, simply-connected and not of constant curvature, it admits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001094.png" /> Killing spinors, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001095.png" />. The holonomy group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001096.png" /> never reduces to a proper subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001097.png" /> and the metric admits no infinitesimal deformations.

For every compact semi-simple [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001098.png" /> one has a corresponding diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t12001099.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010100.png" /> being a symmetric positive quaternion Kähler manifold (a Wolf space) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010101.png" />. In particular, every every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010102.png" />-Sasakian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010103.png" />-homogeneous space is regular and it is one of the spaces

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010104.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010105.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010106.png" /></td> </tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010108.png" /> denotes the trivial group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010109.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010110.png" />. Hence, there is one-to-one correspondence between the simple Lie algebras and the simply-connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010111.png" />-Sasakian homogeneous manifolds.

There is a conjecture that all complete regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010113.png" />-Sasakian manifolds are homogeneous. It is a simple translation of the corresponding conjecture due to C. LeBrun and S. Salamon [[#References|[a3]]] that all positive quaternion Kähler manifolds are symmetric. This is known to be true when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010114.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010115.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010116.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010117.png" />). More generally, it is know that in each dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010118.png" />, there are only finitely many complete regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010119.png" />-Sasakian manifolds, all of them having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010120.png" /> with equality holding only when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010121.png" />. Furthermore, it was shown by K. Galicki and S. Salamon [[#References|[a4]]] that each [[Betti number|Betti number]] of such an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010122.png" /> must satisfy the linear relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010123.png" /></td> </tr></table>

with odd Betti numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010124.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010125.png" />. In fact the vanishing of odd Betti numbers holds true in the irregular case as well. There are, however, examples of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010126.png" />-dimensional irregular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010127.png" />-Sasakian manifold for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010128.png" /> and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010129.png" />-dimensional manifolds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010130.png" />. These were constructed explicitly by C. Boyer, K. Galicki and B. Mann [[#References|[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 [[#References|[a10]]], [[#References|[a11]]], [[#References|[a12]]], [[#References|[a13]]], [[#References|[a14]]], [[#References|[a15]]], using a method called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010132.png" />-Sasakian reduction. The examples are bi-quotients of unitary groups of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010133.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010134.png" />-dimensional family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010135.png" /> depends on positive integral "weights" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010136.png" /> which are pairwise relatively prime. The integral cohomology ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010137.png" /> depends on the weight vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010138.png" /> and one gets infinitely many homotopy types of compact simply-connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010139.png" />-Sasakian manifolds in each allowable dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010140.png" />. Other irregular examples were constructed later in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010141.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010142.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010143.png" /> by Boyer, Galicki, Mann, and E. Rees [[#References|[a6]]]. The same method of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010144.png" />-Sasakian reduction was used to obtain families of compact simply-connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010145.png" />-Sasakian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010146.png" />-manifolds with an arbitrary second Betti number. All these examples are toric, i.e., having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010147.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010148.png" /> as the group of isometries with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010149.png" />-torus action preserving the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010150.png" />-Sasakian structure. R. Bielawski [[#References|[a7]]] showed that, in any dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010151.png" />, a toric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010152.png" />-Sasakian manifold is necessarily diffeomorphic to one of the quotients obtained in [[#References|[a6]]]. Examples of compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010153.png" />-Sasakian manifold which are not toric can also be constructed.

After their introduction in 1969, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010154.png" />-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 [[#References|[a16]]], [[#References|[a17]]], [[#References|[a18]]], [[#References|[a19]]], [[#References|[a20]]], [[#References|[a21]]], [[#References|[a22]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010155.png" />-Sasakian manifolds returned in two different areas. One of them is the study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010156.png" />-manifolds admitting Killing spinors, in the work of T. Friedrich and I. Kath [[#References|[a8]]]. The other is the work [[#References|[a10]]], [[#References|[a11]]], [[#References|[a12]]], [[#References|[a13]]], [[#References|[a14]]], [[#References|[a15]]], of Boyer–Galicki–Mann, in which the first irregular examples are constructed and a systematic study of geometry and topology of compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010157.png" />-Sasakian manifolds is undertaken.

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Udrişte, "Structures presque coquaternioniennes" ''Bull. Math. Soc. Sci. Math. Roum.'' , '''12''' (1969) pp. 487–507 {{MR|0296849}} {{ZBL|0213.48205}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Y.-Y. Kuo, "On almost contact 3-structure" ''Tôhoku Math. J.'' , '''22''' (1970) pp. 325–332 {{MR|0278225}} {{ZBL|0205.25801}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> K. Galicki, S. Salamon, "On Betti numbers of 3-Sasakian manifolds" ''Geom. Dedicata'' , '''63''' (1996) pp. 45–68 {{MR|1413621}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> C. LeBrun, S.M. Salamon, "Strong rigidity of positive quaternion–Kähler manifolds" ''Invent. Math.'' , '''118''' (1994) pp. 109–132</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> C.P. Boyer, K. Galicki, B.M. Mann, "A note on smooth toral reductions of spheres" ''Manuscripta Math.'' , '''95''' (1998) pp. 149–158 {{MR|1603301}} {{ZBL|0913.53020}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> 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 {{MR|1608567}} {{ZBL|0901.53033}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> R. Bielawski, "Complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010158.png" />-invariant hyperkähler <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120010/t120010159.png" />-manifolds" ''MPI preprint'' , '''65''' (1998) (www.mpim-bonn.mpg.de/html/preprints/preprints.html)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> T. Friedrich, I. Kath, "Compact seven-dimensional manifolds with Killing spinors" ''Comm. Math. Phys.'' , '''133''' (1990) pp. 543–561</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> C.P. Boyer, K. Galicki, "3-Sasakian Manifolds" C. LeBrun (ed.) M. Wang (ed.) , ''Essays on Einstein Manifolds'' , Internat. Press (to appear) {{MR|1798609}} {{MR|1645769}} {{MR|1433200}} {{MR|1293878}} {{MR|1249451}} {{ZBL|1008.53047}} {{ZBL|0942.53030}} {{ZBL|0901.53033}} {{ZBL|0889.53029}} {{ZBL|0814.53037}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> C.P. Boyer, K. Galicki, B.M. Mann, "Quaternionic reduction and Einstein manifolds" ''Commun. Anal. Geom.'' , '''1''' (1993) pp. 1–51 {{MR|1243524}} {{ZBL|0856.53038}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> 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 {{MR|1293878}} {{ZBL|0889.53029}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> 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 {{MR|1258497}} {{ZBL|0842.53033}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> C.P. Boyer, K. Galicki, B.M. Mann, "3-Sasakian manifolds" ''Proc. Japan Acad. Ser. A'' , '''69''' (1993) pp. 335–340 {{MR|1249451}} {{ZBL|0814.53037}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> C.P. Boyer, K. Galicki, B.M. Mann, "Hypercomplex structures on Stiefel manifolds" ''Ann. Global Anal. Geom.'' , '''14''' (1996) pp. 81–105 {{MR|1375068}} {{ZBL|0843.53030}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> C.P. Boyer, K. Galicki, B.M. Mann, "New examples of inhomogeneous Einstein manifolds of positive scalar curvature" ''Bull. London Math. Soc.'' , '''28''' (1996) pp. 401–408 {{MR|1258497}} {{ZBL|0842.53033}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> S. Ishihara, M. Konishi, "Fibered Riemannian spaces with Sasakian 3-structure" , ''Differential Geometry, in Honor of K. Yano'' , Kinokuniya (1972) pp. 179–194</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> S. Ishihara, "Quaternion Kählerian manifolds and fibered Riemannian spaces with Sasakian 3-structure" ''Kodai Math. Sem. Rep.'' , '''25''' (1973) pp. 321–329</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> T. Kashiwada, "A note on a Riemannian space with Sasakian 3-structure" ''Nat. Sci. Rep. Ochanomizu Univ.'' , '''22''' (1971) pp. 1–2 {{MR|0303449}} {{ZBL|0228.53033}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> M. Konishi, "On manifolds with Sasakian 3-structure over quaternion Kählerian manifolds" ''Kodai Math. Sem. Rep.'' , '''26''' (1975) pp. 194–200</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> Y.-Y. Kuo, S. Tachibana, "On the distribution appeared in contact 3-structure" ''Taita J. Math.'' , '''2''' (1970) pp. 17–24 {{MR|0309004}} {{ZBL|0231.53053}} </TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> S. Tachibana, W.N. Yu, "On a Riemannian space admitting more than one Sasakian structure" ''Tôhoku Math. J.'' , '''22''' (1970) pp. 536–540 {{MR|0275329}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> S. Tanno, "Killing vectors on contact Riemannian manifolds and fiberings related to the Hopf fibrations" ''Tôhoku Math. J.'' , '''23''' (1971) pp. 313–333 {{MR|0287477}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top"> S. Tanno, "On the isometry of Sasakian manifolds" ''J. Math. Soc. Japan'' , '''22''' (1970) pp. 579–590 {{MR|271874}} {{ZBL|0197.48004}} </TD></TR></table>