3-Sasakian manifold

Sasakian and $3$-Sasakian spaces are odd-dimensional companions of Kähler and hyper-Kähler manifolds, respectively. A Riemannian manifold $( \mathcal{S} , g )$ of dimension $m$ is called Sasakian if the holonomy group of the metric cone $( C ( \mathcal{S} ) , \overline { g } ) = ( \mathbf{R} _ { + } \times \mathcal{S} , d r ^ { 2 } + r ^ { 2 } g )$ reduces to a subgroup of $U ( ( m + 1 ) / 2 )$. In particular, $m = 2l + 1$, $l \geq 1$, and such a cone is a Kähler manifold. Let $I$ be a complex structure on $C ( \mathcal S )$. Then $\xi = I ( \partial _ { r } )$ restricted to $\mathcal{S}$ is a unit Killing vector field (cf. also Killing vector) with the property that the sectional curvature of every section containing $\xi $ equals one. Such a $\xi $ is called the characteristic vector field on $\mathcal{S}$ and its properties can be used as an alternative characterization of a Sasakian manifold.

Similarly, one says that $( \mathcal{S} , g )$ is a $3$-Sasakian manifold if the holonomy group of the metric cone $( C ( \mathcal{S} ) , \overline { g } )$ reduces to a subgroup of $\operatorname { Sp } ( ( m + 1 ) / 4 )$. In particular, $m = 4 n + 3$, $n \geq 1$, and the cone is a hyper-Kähler manifold. When $\mathcal{S}$ is $3$-Sasakian, the hyper-Kähler structure on the associated cone $C ( \mathcal S )$ can be used to define three vector fields $\xi ^ {a } = I ^ { a} ( \partial _ { r } )$, $a = 1,2,3$, where $\{ I ^ { 1 } , I ^ { 2 } , I ^ { 3 } \}$ is a hypercomplex structure on $C ( \mathcal S )$. It follows that, when restricted to $\mathcal{S}$, $\{ \xi ^ { 1 } , \xi ^ { 2 } , \xi ^ { 3 } \}$ are Killing vector fields such that $g ( \xi ^ { a } , \xi ^ { b } ) = \delta _ { a b }$ and $[ \xi ^ {a } , \xi ^ { b } ] = 2 \epsilon _ { a b c } \xi ^ { c }$. Hence, they are orthonormal and locally define an isometric $ \operatorname {SO} ( 3 )$ (or $\operatorname{SU} ( 2 )$) action on $\mathcal{S}$. In turn, the triple $\{ \xi ^ { 1 } , \xi ^ { 2 } , \xi ^ { 3 } \}$ yields $\eta ^ { a } ( Y ) = g ( \xi ^ { a } , Y )$ and $\Phi ^ { a } ( Y ) = \nabla _ { Y } \xi ^ { a }$ for each $a = 1,2,3$. The collection of tensors $\{ \xi ^ { a } , \eta ^ { a } , \Phi ^ { a } \}_{a = 1,2,3}$ is traditionally called the $3$-Sasakian structure on $( \mathcal{S} , g )$. 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 $3$-Sasakian manifold is an Einstein manifold with positive Einstein constant $\lambda = \operatorname { dim } ( \mathcal{S} ) - 1$. If $\mathcal{S}$ is complete, it is compact with finite fundamental group. If $( \mathcal{S} , g )$ is compact, the characteristic vector fields $\{ \xi ^ { 1 } , \xi ^ { 2 } , \xi ^ { 3 } \}$ are complete and define a $3$-dimensional foliation $\mathcal{F} _ { 3 }$ on $\mathcal{S}$. The leaves of this foliation are necessarily compact, since $\{ \xi ^ { 1 } , \xi ^ { 2 } , \xi ^ { 3 } \}$ defines a locally free $\operatorname{SU} ( 2 )$ action on $\mathcal{S}$. Hence, the foliation $\mathcal{F} _ { 3 }$ is automatically almost-regular and the space of leaves is a compact orbifold, denoted by $\mathcal{O}$. The leaves of $\mathcal{F} _ { 3 }$ are totally geodesic submanifolds of constant sectional curvature equal one (cf. also Totally-geodesic manifold). They are all $3$-dimensional homogeneous spherical space forms $S ^ { 3 } / \Gamma$, where $\Gamma \subset \operatorname{SU} ( 2 )$ is a finite subgroup (cf. also Space forms). In particular, the leaves are $3$-Sasakian manifolds themselves. The space of leaves $\mathcal{O}$ is a compact positive quaternion Kähler orbifold. The principal leaves are always diffeomorphic to either $S ^ { 3 }$ or $ \operatorname {SO} ( 3 )$. A compact $3$-Sasakian manifold is said to be regular if $\mathcal{F} _ { 3 }$ is regular, i.e., if all the leaves are diffeomorphic. In this case $\mathcal{O}$ is a smooth manifold (cf. Differentiable manifold). For any $\tau = ( \tau _ { 1 } , \tau _ { 2 } , \tau _ { 3 } ) \in \mathbf{R} ^ { 3 }$ such that $\tau _ { 1 } ^ { 2 } + \tau _ { 3 } ^ { 2 } + \tau _ { 3 } ^ { 2 } = 1$, the vector field $\xi ( \tau ) = \tau _ { 1 } \xi ^ { 1 } + \tau _ { 2 } \xi ^ { 2 } + \tau _ { 3 } \xi ^ { 3 }$ has the Sasakian property. Hence, a $3$-Sasakian manifold has a $2$-sphere worth of Sasakian structures (just as hyper-Kähler manifold has an $S ^ { 2 }$-worth of complex structures). When $\mathcal{S}$ is compact, the vector field $\xi ( \tau )$ defines a $1$-dimensional foliation $\mathcal{F} _ { \tau } \subset \mathcal{F} _ { 3 } \subset \mathcal{S}$ with compact leaves. Such a foliation gives $\mathcal{S}$ an isometric locally free circle action $U ( 1 ) _ { \tau } \subset \operatorname { SU } ( 2 )$. The space of leaves $\mathcal{Z} = \mathcal{S} / \mathcal{F} _ { \tau }$ is a compact Kähler–Einstein orbifold of positive scalar curvature. It is a simply-connected normal projective algebraic variety (cf. Projective algebraic set). $\mathcal Z$ has a complex contact structure and it is a $\mathbf{Q}$-factorial Fano variety (cf. also Fano variety). It is an orbifold twistor space of $\mathcal{O}$. All the foliations associated to $\mathcal{S}$ can be described in the the following diagram of orbifold fibrations:

All four geometries in the above diagram are Einstein. Both $\mathcal{S}$ and $\mathcal Z$ admit second, non-isometric Einstein metrics of positive scalar curvature. Every $3$-Sasakian manifold is a spin manifold (cf. Spinor structure). When $\mathcal{S}$ is complete, simply-connected and not of constant curvature, it admits $n + 2$ Killing spinors, where $\operatorname { dim } ( {\cal S} ) = 4 n + 3$. The holonomy group of $\mathcal{S}$ never reduces to a proper subgroup of $\operatorname{SO} ( 4 n + 3 )$ and the metric admits no infinitesimal deformations.

For every compact semi-simple Lie group $G$ one has a corresponding diagram $^{ \bigtriangleup } _ { \bigtriangledown } ( G / K )$ with ${\cal O} = G / \operatorname { Sp } ( 1 ) . K$ being a symmetric positive quaternion Kähler manifold (a Wolf space) and $\mathcal{Z} = G / U ( 1 ) . K$. In particular, every every $3$-Sasakian $G$-homogeneous space is regular and it is one of the spaces

\begin{equation*} \operatorname { Sp } ( n + 1 ) / \operatorname { Sp } ( n ) , \quad \operatorname { Sp } ( n + 1 ) / \operatorname { Sp } ( n ) \times \mathbf{Z} _ { 2 }, \end{equation*}

\begin{equation*} \operatorname {SU} ( m ) / S ( U ( m - 2 ) \times U ( 1 ) ) , \operatorname {SO} ( k ) / \operatorname {SO} ( k - 4 ) \times \operatorname {Sp} ( 1 ), \end{equation*}

\begin{equation*} G _ { 2 } / \operatorname { Sp } ( 1 ) , \quad F _ { 4 } / \operatorname { Sp } ( 3 ) , E _ { 6 } / \operatorname{SU} ( 6 ) , \quad E _ { 7 } / \operatorname { Spin } ( 12 ) , \quad E _ { 8 } / E _ { 7 }. \end{equation*}

Here, $n \geq 0$, $\operatorname{Sp} ( 0 )$ denotes the trivial group, $m > 3$, and $k \geq 7$. Hence, there is one-to-one correspondence between the simple Lie algebras and the simply-connected $3$-Sasakian homogeneous manifolds.

There is a conjecture that all complete regular $3$-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 $\operatorname { dim } ( {\cal S} ) = 7$ or $11$ ($\operatorname { dim } ( \mathcal{O} ) = 4$ or $8$). More generally, it is know that in each dimension $4 n + 3$, there are only finitely many complete regular $3$-Sasakian manifolds, all of them having $b _ { 2 } ( \mathcal{S} ) \leq 1$ with equality holding only when $\mathcal{S} = \text{SU} ( m ) / S ( U ( m - 2 ) \times U ( 1 ) )$. Furthermore, it was shown by K. Galicki and S. Salamon [a4] that each Betti number of such an $\mathcal{S}$ must satisfy the linear relation

\begin{equation*} \sum _ { k = 1 } ^ { n } k ( n + 1 - k ) ( n + 1 - 2 k ) b _ { 2 k } = 0 \end{equation*}

with odd Betti numbers $b _ { 2i + 1} ( \mathcal{S} ) = 0$ for $ { i } \leq n$. In fact the vanishing of odd Betti numbers holds true in the irregular case as well. There are, however, examples of an $11$-dimensional irregular $3$-Sasakian manifold for which $b _ { 2 } \neq b _ { 4 }$ and of $15$-dimensional manifolds with $b _ { 2 } \neq b _ { 6 }$. 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 $3$-Sasakian reduction. The examples are bi-quotients of unitary groups of the form $\mathcal{S} ( p ) = U ( 1 ) _ { p } \backslash U ( n + 2 ) / U ( n )$. The $( 4 n + 3 )$-dimensional family ${\cal S} ( \operatorname{p} )$ depends on positive integral "weights" $\operatorname{p} = ( p _ { 1 } , \dots , p _ { n + 2} )$ which are pairwise relatively prime. The integral cohomology ring of ${\cal S} ( \operatorname{p} )$ depends on the weight vector $\text{p}$ and one gets infinitely many homotopy types of compact simply-connected $3$-Sasakian manifolds in each allowable dimension $\geq 7$. Other irregular examples were constructed later in dimension $7$, $11$, $15$ by Boyer, Galicki, Mann, and E. Rees [a6]. The same method of $3$-Sasakian reduction was used to obtain families of compact simply-connected $3$-Sasakian $7$-manifolds with an arbitrary second Betti number. All these examples are toric, i.e., having $T ^ { 2 } \times \operatorname { Sp } ( 1 )$ or $T ^ { 2 } \times SO ( 3 )$ as the group of isometries with the $2$-torus action preserving the $3$-Sasakian structure. R. Bielawski [a7] showed that, in any dimension $4 n + 3$, a toric $3$-Sasakian manifold is necessarily diffeomorphic to one of the quotients obtained in [a6]. Examples of compact $3$-Sasakian manifold which are not toric can also be constructed.

After their introduction in 1969, $3$-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 $3$-Sasakian manifolds returned in two different areas. One of them is the study of $7$-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 $3$-Sasakian manifolds is undertaken.

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

References

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[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
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