# Sasakian and Kähler Manifolds 2

The following is essentially the content of the second talk I gave on Sasakian and Kähler manifolds.

## Sasakian Boothby-Wang Fibrations

Theorem (Newlander-Nirenberg)
An almost complex structure $$J$$ is integrable if and only if the Nijenhuis tensor
$N_J(X,Y) = -J^2[X,Y] + J([JX,Y] + [X,JY]) – [JX,JY]$
vanishes.

The Boothby-Wang fibration gives a canonical circle bundle over a symplectic manifold. Recall, of course, that Kähler manifolds are symplectic; there is the following interesting result:

Theorem (Hatakeyama)
Suppose that on a principle fiber bundle $$\pi \colon \mathbb{M} \rightarrow \mathbb{B}$$ over an almost complex manifold $$\mathbb{B}$$ with group $$S^1$$ we can define an almost contact structure $$(\xi,\eta,\Phi)$$. Then if the almost complex structure $$J = \Phi \vert_\mathbb{B}$$ is integrable and the curvature form $$\omega$$ given by $$\pi^*\omega = d\eta$$ on $$\mathbb{B}$$ associated to the contact form $$\eta$$ of $$\mathbb{M}$$ is of type $$(1,1)$$ with respect to the almost complex structure, then the almost contact structure on $$\mathbb{M}$$ is normal.

Proof:
By a paper of Sasaki and Hatakeyama normality of the contact metric structure is equivalent to the vanishing of the tensor
$N(X,Y) = [X,Y] + \Phi [\Phi X,Y] + \Phi [X,\Phi Y] – [\Phi X,\Phi Y] – \eta([X,Y])\xi – d\eta(X,Y)\xi$
which follows from considering the Nijenhuis tensor on the Riemannian cone over $$\mathbb{M}$$. Projecting onto the horizontal and vertical spaces, it is clear that $$N(X,Y) = 0$$ if and only if $$\pi(N(X,Y)) = 0$$ and $$\eta(N(X,Y)) = 0$$. Directly, it can be seen that
$\pi(N_p(X,Y)) = \bar{N}_{\pi(p)}(\pi X,\pi Y)$
where $$\bar{N}$$ is the Nijenhuis tensor associated to the almost complex structure $$J$$ on $$\mathbb{B}$$. Then the Newlander-Nirenberg theorem implies that $$N$$ will vanish only if $$J$$ is integrable.
Moreover,
$\eta(N(X,Y)) = -\eta([\Phi X,\Phi Y]) – d\eta(X,Y)$
and
$-\eta([\Phi X,\Phi Y]) = d\eta(\Phi X,\Phi Y)$
so that $$\eta(N(X,Y))$$ vanishes if any only if
$d\eta(\Phi X,\Phi Y) = d\eta(X,Y)$
which is equivalent to
$\omega(J(\pi X), J(\pi Y)) = \omega( \pi X, \pi Y)$
and so we are done.

Taking the last two results together gives

Theorem (Hatakeyama)
A necessary and sufficient condition for a compact manifold with a regular contact structure to admit an associated normal contact metric structure (and thus be Sasakian) is that the base manifold of the Boothby-Wang fibration of $$\mathbb{M}$$ is Hodge.

Proof:
One direction is the content of the Boothby-Wang theorem. If $$\mathbb{M}$$ is Hodge, it is Kähler, and thus the almost complex structure is integrable. From the first talk, the almost complex structure is compatible with the symplectic form $$\omega$$, which is precisely the statement
$\omega(JX,JY) = \omega(X,Y)$
and so we are done.

## 3-Sasakian manifolds

We want to introduce a generalization of the Kähler-Sasakian correspondence by allowing for the existence of triples of structures obeying a quaternionic relation. We begin with the following

Definition:
Let $$\mathbb{M}$$ be a $$4n$$-dimensional manifold with 3 integrable almost complex structures $$I_1,I_2,I_3$$ such that
$I_iI_j = -\delta_{ij}Id + 2 \epsilon_{ijk}I_k$
Then we call $$(\mathbb{M}, I,J,K)$$ a hyperkähler manifold.

To develop the corresponding Sasakian’ notion, we begin with extending the definition of a contact’ manifold.

Definition:
Let $$\mathbb{M}$$ be a $$4n+3$$-dimensional manifold such that there exists a family of contact structures $$\mathcal{S} = \{\eta(\tau),\xi(\tau),\Phi(\tau)\}$$ parameterized by $$\tau \in S^2$$ satisfying the relations

• $$\Phi(\tau) \circ \Phi(\tau’) – \eta(\tau) \otimes \eta(\tau’) = – \Phi(\tau \times \tau’) – (\tau \cdot \tau’)Id$$
• $$\Phi(\tau)\xi(\tau’) = – \xi(\tau \times \tau’)$$, and
• $$\eta(\tau) \circ \Phi(\tau’) = – \eta(\tau \times \tau’)$$

for all $$\tau, \tau’ \in S^2$$. We then call $$(\mathbb{M}, \{\eta(\tau),\xi(\tau),\Phi(\tau)\})$$ an almost hypercontact manifold. If moreover there exists a Riemannian metric $$g$$ on $$M$$ such that
$g(\Phi(\tau)X, \Phi(\tau)Y) = g(X,Y) – \eta(\tau)(X) \eta(\tau)(Y)$
for all $$\tau \in S^2$$ we call $$(\mathbb{M}, g, \{\eta(\tau),\xi(\tau),\Phi(\tau)\})$$ an almost hypercontact metric manifold.

Remark: Another standard definition comes from a choice of an orthonormal frame on $$\mathbb{R}^3$$, which we will refer to as an almost contact (metric) 3-structure.

Remark: Every compact, orientable 3-manifold admits an almost contact 3-structure.

Attempting to generalize the idea of Sasakian manifolds gives us the following

Proposition:
There exists a one-to-one correspondence between almost hypercontact structures on $$\mathbb{M}$$ and $$\Psi$$-invariant almost hypercomplex structures $$\mathcal{I}$$ on the cone $$C(\mathbb{M}) = \mathbb{M} \times \mathbb{R}^+$$.

and so we define

Definition:
Let $$(\mathbb{M},\mathcal{S}, g)$$ be an almost hypercontact metric manifold. Then if $$(C(\mathbb{M}), \mathcal{I}, g)$$ is hyperkähler, we call $$(\mathbb{M},\mathcal{S}, g)$$ 3-Sasakian.

Proposition:
If $$\mathcal{S} = \{\eta(\tau),\xi(\tau),\Phi(\tau)\}$$ is a 3-Sasakian structure on $$(\mathbb{M},g)$$ then

• $$g(\xi(\tau),\xi(\tau’)) = \tau \cdot \tau’$$
• $$[\xi(\tau),\xi(\tau’)] = 2\xi(\tau \times \tau’)$$
• $$\Phi(\tau) = -\nabla\xi(\tau)$$

Conversely, if $$\mathcal{S}_1,\mathcal{S}_2, \mathcal{S}_3$$ are Sasakian structures on $$(\mathbb{M},g)$$ with Reeb fields $$\xi_1, \xi_2, \xi_3$$ such that

• $$g(\xi_a,\xi_b) = \delta_{ab}$$
• $$[\xi_a,\xi_b] = 2\epsilon_{abc}\xi_c$$

then $$\mathcal{S} = \{\mathcal{S}_1,\mathcal{S}_2, \mathcal{S}_3\}$$ is a 3-Sasakian structure on $$\mathbb{M}$$.