# Symplectic Manifolds

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## Sasakian and Kähler Manifolds 1

I have been caught up in end of semester preparations, but further posts in the series on connections on contact foliated manifolds are forthcoming.  In the meantime, I thought I would post the following notes from a lecture on the relationship between Sasakian and Kahler manifolds that I gave to the UConn Complex Geometry seminar on Friday, 23 March, 2018.  I will be giving a second talk on Friday, 20 April, and the notes will subsequently appear here.  Much of the following is adapted from Banyaga and Houenou, A Brief Introduction to Symplectic and Contact manifolds.

## Introduction

In this post, I am interested in discussing the relationship between the following types of manifolds:

• Symplectic
• Contact
• Kahler
• Sasakian

Essentially by definition, Kahler manifolds are always symplectic (even dimensional), and Sasakian manifolds are always contact (odd dimensional.) We will see that there is a strong relationship between these two sets of structures.

Then, in order to investigate the conditions under which a compact contact manifold is Sasakian, we will introduce the Boothby-Wang fibration.

## Symplectic and Contact manifolds

### Symplectic manifolds

Definition:
A symplectic manifold $$(\mathbb{M},\omega)$$ is a $$2n$$-dimensional smooth manifold with a closed, nondegenerate differential 2-form $$\omega$$ called a symplectic form.

Since the symplectic form is a differential 2-form, it must be skew-symmetric, that is $$\omega(X,Y) = – \omega(Y,X)$$. Since $$\omega$$ is nondegenerate, $$\omega^n$$ is a volume form and thus $$\mathbb{M}$$ is oriented.

Definition:
An almost complex structure $$J$$ is an endomorphism of $$T\mathbb{M}$$ such that $$J^2 = -Id$$.

Definition:
We say that an almost complex structure $$J$$ is compatible with the symplectic manifold $$(\mathbb{M},\omega)$$ if

• $$\omega(X,Y) = \omega(JX,JY)$$ for all $$X,Y \in T\mathbb{M}$$, and
• The bilinear form $$g(X,Y) = \omega(X,JY)$$ is symmetric and positive-definite (and thus a Riemannian metric.)

Claim:
Let $$(\mathbb{M},\omega)$$ be a symplectic manifold. Then there exists a compatible almost complex structure, and moreover the set of all compatible almost complex structures is infinite and contractible.

Proof sketch: (Banyaga, Houenou)
Let $$g$$ be any Riemannian metric on $$\mathbb{M}$$ (which can always be done using the explicit construction on the basis elements) and consider the operator $$A = \tilde{g}^{-1} \circ \tilde{\omega}$$ where $$\tilde{g}(X)(Y) = g(X,Y)$$ and similarly $$\tilde{\omega}(X)(Y) = \omega(X,Y)$$. Then
$g(AX,Y) = \omega(X,Y).$
Set $$A^t$$ to be the adjoint of $$A$$ by $$g$$, that is
$g(A^tX,Y) = g(X,AY).$
We see that $$A$$ is skew-symmetric
$\begin{split} g(A^tX,Y) &= g(X,AY) \\ &= g(AY,X) \\ &= \omega(Y,X) \\ &= – \omega(X,Y) \\ &= – g(AX,Y) \\ \end{split}$
and also that $$A^tA$$ is positive-definite
$g(A^tAX,X) = g(AX,AX) > 0, \quad X \neq 0$
so $$A^tA$$ is diagonalizable with positive eigenvalues $$\{\lambda_1, \dots, \lambda_{2n}\}$$. Thus
$A^tA = B \cdot diag(\lambda_1,\dots, \lambda_{2n}) \cdot B^{-1}$
for some matrix $$B$$. Define $$R = \sqrt{A^tA} = B \cdot diag(\sqrt{\lambda_1},\dots, \sqrt{\lambda_{2n}}) \cdot B^{-1}$$ and also $$J = R^{-1}A$$. Then

• $$g(JX,JY) = g(X,Y)$$,
• $$JR = RJ$$, and
• $$J^t = -J$$ so that $$J^2 = -Id$$

It follows that
$\omega(JX,JY) = g(AJX,JY) = g(AX,Y) = \omega(X,Y)$
and
$\omega(X,JX) = g(AX,JX) = g(-JAX,X) = g(RX,X) > 0$
for all $$X \neq 0$$.
We define a new Riemannian metric
$g_J(X,Y) = \omega(X,JX) = \cdots = g(RX,Y)$
which depends on the original choice of $$g$$, of which there are infinitely many. We can construct an explicit homotopy between $$J_1 = J_{g_1}$$ and $$J_2 = J_{g_2}$$ by
$J_t = J_{(tg_1 + (1-t)g_2)}$

Example:

• $$\mathbb{R}^{2n}$$ with coordinates $$(x_1,\dots,x_n,y_1,\dots,y_n)$$ and 2-form
$\omega = dx_1 \wedge dy_1 + \cdots dx_n \wedge dy_n$
is symplectic, since clearly $$d\omega = 0$$ and $$\omega^n \neq 0$$.
• An even dimensional torus $$T^{2n} = \mathbb{R}^{2n}/\mathbb{Z}^{2n}$$ will be a symplectic manifold with $$\omega$$ descending to the quotient from the first example.

Theorem (Darboux)
Let $$(\mathbb{M},\omega)$$ be a symplectic manifold. Each point $$p \in \mathbb{M}$$ has an open neighborhood $$U$$ and a chart $$\phi \colon U \rightarrow \mathbb{R}^{2n}$$ such that $$\phi(p) = 0$$ and
$\phi^*(\omega’) = \omega\vert_U$
where $$\omega’$$ is as in example 1 above.

In other words, all symplectic manifolds look the same, locally.

### Contact manifolds

Definition:
A contact manifold $$(\mathbb{M}, \eta)$$ is a $$2n+1$$-dimensional smooth manifold with a differential 1-form $$\eta$$ such that $$\eta \wedge (d\eta)^n$$ is a volume form. $$\eta$$ is called a contact form.

Remark:
Recall, $$\eta \wedge (d\eta)^n$$ is a volume form if it is a nonvanishing $$2n+1$$-form. A contact form gives an orientation on $$\mathbb{M}$$. Observe that for any smooth, nonvanishing function $$\rho$$ on $$\mathbb{M}$$ the 1-form $$\eta’ = \rho\eta$$ will also be a contact form on $$\mathbb{M}$$.

We have the following
Claim:
Let $$(\mathbb{M},\eta)$$ be a contact manifold. There exists a unique vector field $$\xi$$ called the Reeb vector field such that $$\eta(\xi) = 1$$ and $$\iota_\xi d\eta = 0$$.

Proof:
Since $$\eta \wedge (d\eta)^n$$ is nonvanishing, $$d\eta$$ must have rank $$2n$$. Let $$\xi_p \in \ker d\eta$$, and find $$v_1, \dots, v_{2n}$$ so that $$\{\xi_p, v_1, \dots, v_{2n}\}$$ complete a basis of $$T_p\mathbb{M}$$. Then
$\begin{split} 0 &\neq (\eta \wedge (d\eta)^n)(\xi_p,v_1,\dots,v_{2n}) \\ &= \eta(\xi_p) \wedge (d\eta)^n(v_1,\dots,v_{2n}) + \sum_{i=1}^{2n} (-1)^i \eta(v_i) (d\eta)^n(v_1,\dots,v_{i-1},\xi_p,v_{i+1},\dots,v_{2n}) \\ &= \eta(\xi_p) \wedge (d\eta)^n(v_1,\dots,v_{2n}) \end{split}$
since $$\xi_p \in \ker d\eta$$. But then $$\eta(\xi_p) \neq 0$$ for all $$x$$. Normalizing and denoting the result again by $$\xi$$ we get
$\begin{split} \eta(\xi) &= 1 \\ \iota_\xi d\eta &= 0 \end{split}$
as desired.

Claim:
It is always possible to find a Riemannian metric $$g$$ on $$\mathbb{M}$$ such that $$g(X,\xi) = \eta(X)$$. Such a metric is called compatible with the contact structure.

We can sometimes construct a contact manifold from a symplectic one.

Claim: (Contactization of a symplectic manifold)
Let $$(\mathbb{M}, \omega)$$ be a symplectic manifold such that $$\omega$$ is an exact form, that is there exists a 1-form $$\lambda$$ with $$\omega = d\lambda$$. Then $$\mathbb{M}’ = \mathbb{M} \times \mathbb{R}$$ is a contact manifold with contact form $$\eta = \pi^*\lambda + dt$$ where $$t \colon \mathbb{M} \times \mathbb{R} \rightarrow \mathbb{R}$$ and $$\pi \colon \mathbb{M} \times \mathbb{R} \rightarrow \mathbb{M}$$ are the canonical projections.

Proof:
Notice that $$d\eta = d\pi^*\lambda + d^2t = \pi^*d\lambda = \pi^*\omega$$. Thus $$\eta \wedge (d\eta)^n = \eta \wedge (\pi^*\omega)^n$$ has rank $$2n+1$$ and therefore must be a volume form on $$\mathbb{M}’$$.

From a contact manifold we can also construct a symplectic manifold on its cone $$\mathbb{R}^+ \times \mathbb{M}$$. This process is referred to the symplectization of $$\mathbb{M}$$ (see Boyer-Galicki, pg 203.) We will discuss this in further detail.

Claim: (Symplectization of a contact manifold)
Let $$\eta$$ be a 1-form on a $$2n+1$$-dimensional manifold $$\mathbb{M}$$. Then $$\eta$$ is a contact form on $$\mathbb{M}$$ if and only if the 2-form $$\omega = d(r^2\eta) = 2rdr\wedge\eta + r^2d\eta$$ is a symplectic form over the cone $$C(\mathbb{M})$$.

Proof:
If $$(\mathbb{M}, \eta)$$ is a contact manifold, then taking $$\omega = d(r^2\eta)$$ gives a closed, nondegenerate 2-form on $$C(\mathbb{M})$$.

If $$\omega = d(r^2\eta)$$ is a symplectic form on $$C(\mathbb{M})$$, then since $$\omega$$ is closed we see that $$\tilde\eta = r^2\eta$$ is a 1-form on $$C(\mathbb{M}) = \mathbb{M} \times \mathbb{R}^+$$. Then, restricting $$\tilde\eta \vert_{M \times \{1\}} = \eta$$ we see that $$\eta$$ must be a nondegenerate 1-form on $$\mathbb{M}$$. Since $$\omega^{n+1} = (d(r^2\eta))^{n+1} \neq 0$$, it must be that $$(d\eta)^n \neq 0$$ on $$\mathbb{M}$$, and we can conclude that $$\eta$$ is a contact form on $$\mathbb{M}$$.

We will be interested in this example later.

Example:

• $$\mathbb{R}^{2n+1}$$ with coordinates $$(x_1,\dots,x_n,y_1,\dots,y_n,z)$$ and 1-form
$\eta = \sum_{i=1}^{2n}x_idy_i + dz$
is a contact manifold, and has Reeb field
$\xi = \frac{\partial}{\partial z}$
• $$T^3$$ with 1-form
$\eta = \cos(z) dx + \sin(z) dy$
is a contact manifold with Reeb field
$\xi = \cos(z) \frac{\partial}{\partial x} + \sin(z) \frac{\partial}{\partial y}$
• $$S^{2n+1} \subset \mathbb{R}^{2n+2}$$ with 1-form
$\eta = \frac{1}{2}\left(\sum_{i=1}^{n+1} x_idy_x – y_i dx_i\right)$
is a contact manifold, and has Reeb field
$\xi = \sum_{i=1}^n x_i\frac{\partial}{\partial y_i} – y_i \frac{\partial}{\partial x_i}$

Theorem: (Martinet)
Every orientable 3-manifold admits a contact structure.

There is a well-known theorem describing locally the behavior of all contact forms.
Theorem: (Darboux)
Let $$\eta$$ be a contact form on a $$2n+1$$-dimensional manifold $$\mathbb{M}$$. For each point $$p \in \mathbb{M}$$ there exists an open neighborhood $$U$$ of $$p$$ and a chart $$\phi \colon U \rightarrow \mathbb{R}^{2n+1}$$ with $$\phi(p) = 0$$ and
$\phi^*(\eta’) = \eta\vert_U$
where $$\eta’$$ is the standard contact form
$\eta’ = \sum_{i=1}^n x_idy_i + dz$

## Kahler and Sasakian manifolds

### Kahler manifolds

From the complex point of view, a Kahler manifold is defined as follows.
Definition:
An almost Kahler manifold $$(\mathbb{M},J,h)$$ is a smooth manifold with almost complex structure
$J \in End(T\mathbb{M})$
(that is, $$J^2 = -Id$$) and hermitian scalar product
$h \colon T\mathbb{M} \times T\mathbb{M} \rightarrow \mathbb{C}$
(that is, $$h(X,\bar{Y}) = \overline{h(\bar{X},Y)}$$ and $$h(X,\bar{X}) > 0$$ for all $$X \neq 0$$) such that the associated differential 2-form
$\omega(X,Y) = Re\ h(JX,Y)$
is closed.

This can be strengthened as follows.

Definition:
An almost Kahler manifold $$(\mathbb{M},J,h)$$ such that the almost complex structure $$J$$ is integrable is called a Kahler manifold.

It is easy to show that K\”ahler manifolds are always even dimensional (this is a consequence of the existence of an almost complex structure,) and so
Proposition:
A Kahler manifold $$(\mathbb{M},J,h)$$ is a symplectic manifold $$(\mathbb{M},\omega)$$ when equipped with the 2-form
$\omega(X,Y) = Re\ h(JX,Y)$

In fact, there is an equivalent definition of K\”ahler manifolds from the symplectic perspective.
Definition:
A Kahler manifold $$(\mathbb{M},\omega,J)$$ is a symplectic manifold with symplectic form $$\omega$$ and an integrable almost complex structure $$J \in End(T\mathbb{M})$$ such that $$g(X,Y) = \omega(X,JY)$$ is symmetric and positive definite, and thus a Riemannian metric on $$\mathbb{M}$$.

From this definition, we will recover the hermitian scalar product as $$h = g – i\omega$$.

### Sasakian manifolds

Definition:
A contact metric structure on a contact manifold $$(\mathbb{M},\eta)$$ is a triple $$(\xi, J,g)$$ where $$\xi$$ is the Reeb field associated to $$\eta$$, $$g$$ is a Riemannian metric on $$\mathbb{M}$$ and $$J$$ is a $$(1,1)$$-tensor field satisfying

• $$J(\xi) = 0$$,
• $$J^2(X) = -X + \eta(X)\xi$$,
• $$d\eta(X,Y) = g(X,JY)$$, and
• $$g(X,Y) = g(JX,JY) + \eta(X)\eta(Y)$$.

Notice that $$g$$ is then compatible with the contact structure.

Remark: A triple $$(\xi,J,g)$$ that meet conditions 1 and 2 are referred to as an almost contact structure on a contact manifold $$(\mathbb{M},\eta)$$.

Remark: Notice that if $$(\mathbb{M},\eta)$$ is the contactization of a symplectic manifold $$(\mathbb{B},\omega)$$ then $$J$$ restricted to $$\mathbb{B}$$ is an almost contact structure. In fact, by choosing an almost complex structure $$J$$ on $$(\mathbb{B},d\eta,g)$$ (with compatible Riemannian metric $$g$$) and extending $$J$$ it to $$\mathbb{M}$$ by setting $$J(\xi) = 0$$ and extending the $$g$$ by $$g(X,Y) = g(JX,JY) + \eta(X)\eta(Y)$$ we will recover a contact metric structure on $$\mathbb{M}$$.

Example:
$$\mathbb{R}^3$$ with the form
$\eta = dz – ydx$
is contact, by the above. The Reeb field is
$V_3 = \xi = \frac{\partial}{\partial z}$
and the contact distribution $$\mathbb{B} = \ker \eta$$ is spanned by
$V_1 = \frac{\partial}{\partial y} \text{ and } V_2 = y\frac{\partial}{\partial z} + \frac{\partial}{\partial x}$
the compatible metric $$g$$ must satisfy
$g(V_i,V_j) = \delta_{ij}$
so a computation gives us that
$g = \left(\begin{array}{ccc} 1+y^2 & 0 & -y \\ 0 & 1 & 0 \\ -y & 0 & 1 \end{array}\right)$
and we define the almost contact structure by
$J(V_1) = -V_2, \quad J(V_2) = V_1, \quad J(V_3) = J(\xi) = 0$

Theorem:
Every contact manifold admits infinitely many contact metric structures, all of which are homotopic.

Definition:
Let $$(\mathbb{M},g)$$ be a Riemannian manifold. Its Riemannian cone is the Riemannian manifold $$C(\mathbb{M}) = \mathbb{R}^+ \times \mathbb{M}$$ with cone metric
$g_{C(\mathbb{M})} = dr^2 + r^2g$
where $$r \in \mathbb{R}^+$$.

It is clear that there is a one-to-one correspondence between Riemannian metrics on $$\mathbb{M}$$ and cone metrics on $$C(\mathbb{M})$$. Henceforth, denote $$\Psi = r\frac{\partial}{\partial r}$$. We have the following

Claim:
Let $$(\mathbb{M},\xi,\eta,J)$$ be an almost contact manifold. Then we can define a section $$I$$ of the endomorphism bundle of $$TC(\mathbb{M})$$ by
$IY = JY + \eta(Y)\Psi, \quad I\Psi = -\xi$
for $$Y \in T\mathbb{M}$$ (where we abuse notation by identifying $$T(\mathbb{M})$$ with $$T(\mathbb{M}) \times \{0\} \subset TC(\mathbb{M})$$.) Then $$I$$ is an almost complex structure on $$C(\mathbb{M})$$.

Proof:
We verify directly. First, for $$X = \rho\Psi$$,
$I^2 X = I(-\rho\xi) = -\rho J\xi – \rho\eta(\xi)\Psi = -\rho\Psi = -X$
and for $$Y \in T\mathbb{M}$$,
$I^2Y = I(JY + \eta(Y)\Psi) = J^2Y + \eta(JY)\Psi – \eta(Y)\xi = -Y$

Since for any $$X \in TC(\mathbb{M})$$ it holds that $$X = \rho\Psi + Y$$ with $$\rho$$ a smooth function and $$Y \in T(\mathbb{M})$$, we are done.

Recalling the symplectization of a contact manifold, we have the following.

Corollary:
There is a one-to-one correspondence between the contact metric structures $$(\xi,\eta,J,g)$$ on $$\mathbb{M}$$ and almost K\”ahler structures $$(dr^2 + r^2g, d(r^2\eta),I)$$ on $$C(\mathbb{M})$$.

Definition:
An almost contact structure $$(\xi, \eta, J)$$ is said to be normal if the corresponding almost complex structure $$I$$ on $$C(\mathbb{M})$$ is integrable, or equivalently if $$(C(M), dr^2 + r^2g, d(r^2\eta),I)$$ is Kahler.

Definition:
A manifold $$\mathbb{M}$$ with a normal almost contact metric structure $$(\xi,\eta,J,g)$$ is called a Sasakian manifold.

In some sense, then, Sasakian manifolds are an odd-dimensional counterpart to Kahler manifolds.

Example:
$$S^{2n+1} \rightarrow S^{2n+1} \times \mathbb{R} = \mathbb{C}^{n+1}$$.

## Boothby-Wang Fibration

We want to understand the necessary conditions for a contact manifold to be Sasakian. To this end, we strengthen the notion of a contact structure. We hereafter assume our manifolds to be compact.

Definition:
Let $$(\mathbb{M},\eta)$$ be a compact contact manifold. The Reeb field $$\xi$$ generates a dynamical system on $$\mathbb{M}$$; if the orbits of $$\xi$$ are periodic with period 1 we call $$(\mathbb{M},\eta)$$ a regular contact manifold.

Remark: If the orbits are periodic with period $$\lambda(p)$$ (which will be a nonvanishing constant on each orbit of $$\xi$$) then we can define $$\eta’ = \frac{1}{\lambda(p)}\eta$$ which will then make $$(\mathbb{M},\eta’)$$ a regular contact manifold. It is necessary to show that $$\lambda(p)$$ is smooth.

Example:
Any Reeb field on the torus $$T^3$$ generates a noncompact integral curve diffeomorphic to $$\mathbb{R}$$, and thus is not a regular contact form. This holds generally for tori, and is a theorem of Blair.

This then gives rise to the following characterization of regular contact manifolds.
Theorem: (Boothby-Wang)
If $$(\mathbb{M},\eta)$$ is a compact, regular contact manifold then

• $$\mathbb{M}$$ is a principal fiber bundle over the set of orbits $$\mathbb{B}$$ with group and fiber $$S^1$$,
• $$\eta$$ is a connection form in this bundle, and
• the base space $$\mathbb{B}$$ is a symplectic manifold whose symplectic form $$\omega$$ given by $$\pi^*\omega = d\eta$$ determines an integral cocycle on $$\mathbb{B}$$, that is $$\omega$$ is a representative of $$H^2(\mathbb{M},\mathbb{Z})$$.

Proof sketch:

• Since $$\xi$$ is never $$0$$, the integral curves must be closed, compact submanifolds of dimension 1, and thus homeomorphic to $$S^1$$. Then $$\xi$$ generates a periodic global one parameter group of transformations on $$\mathbb{M}$$, i.e. an $$S^1$$-action, that leaves no point fixed. We can conclude that $$\pi \colon \mathbb{M} \rightarrow \mathbb{B}$$ is a principal fiber bundle with group and fiber $$S^1$$.
• Notice that $$\mathcal{L}_\xi\eta = 0$$ and $$\mathcal{L}_\xi d\eta = 0$$. Let $$A = \frac{d}{dt}$$ be a basis for the Lie algebra $$\mathfrak{S}^1$$ of $$S^1$$, and set $$\tilde\eta = \eta A$$. We need to show that for $$B \in \mathfrak{S}^1$$, $$\tilde\eta(B^*) = B$$ (where $$B^*$$ is the vector on $$\mathbb{M}$$ induced by $$B$$) and that $$R^*_t\eta(X) = ad(t^{-1})X$$. The first follows since $$A = \xi$$, and the second follows from the fact that $$R^*_t\eta = \eta$$ and the fact that $$S^1$$ is abelian.
• This is essentially a reversal of the contactization of a symplectic manifold. Since $$d\eta$$ has rank $$2n$$ and $$\iota_\xi d\eta = 0$$, it is clear that $$\omega$$ on $$\mathbb{B}$$ given by $$\pi^*\omega = d\eta$$ will be a volume form on $$\mathbb{B}$$, making it a symplectic manifold. Moreover, $$\omega$$ is necessarily exact, and so determines an element of $$H^2(\mathbb{M},\mathbb{R})$$, what remains to be shown is that it is, in fact, integral. This follows from a theorem of Kobayashi.

Moreover, the converse holds as well
Theorem: (Boothby-Wang, converse)
If $$(\mathbb{B},\omega)$$ is a symplectic manifold such that $$\omega$$ is an integral cocycle, there is a principal $$S^1$$ bundle $$\mathbb{M}$$ over $$\mathbb{B}$$ and a 1-form $$\eta$$ on $$\mathbb{M}$$ such that $$(\mathbb{M},\eta)$$ is a contact manifold and the Reeb field of $$(\mathbb{M},\eta)$$ generates the action of $$S^1$$ on the bundle.

Proof sketch:
The same theorem of Kobayashi gives the existence of a circle bundle $$\pi \colon \mathbb{M} \rightarrow \mathbb{B}$$ with connection $$\tilde\eta$$ and structure equation $$d\tilde\eta = \pi^*\omega$$. It holds that $$(d\tilde\eta)^n = \pi^*\omega^n \neq 0$$, so that $$\tilde\eta \wedge (d\tilde\eta)^n \neq 0$$ is a volume form. Letting $$A$$ be a basis for $$\mathfrak{S}^1$$ and defining $$\eta$$ by $$\tilde\eta = \eta A$$ we have that $$\eta(A) = 1$$ and if $$\iota_Xd\omega = 0$$ then $$\iota_{\pi(X)}\omega = 0$$ so $$\pi(X) = 0$$ which implies that $$X$$ is vertical. Thus $$A = \xi$$, the associated vector field to $$\eta$$.

Recall that a Hodge manifold is a Kahler manifold $$(\mathbb{M},g,\omega,J)$$ such that the symplectic form is an integral cocycle.

Corollary:
If $$\mathbb{B}$$ is a compact Hodge manifold , then it has over it a canonically associated circle bundle which is a regular contact manifold.

Example:
The Hopf Fibration: $$S^1 \rightarrow S^3 \rightarrow S^2$$.

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

Theorem: (Hatakeyama)
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. Moreover, if the almost complex structure on $$\mathbb{B}$$ is integrable and the curvature form on $$\mathbb{B}$$ associated to the contact form of $$\mathbb{M}$$ is of type $$(1,1)$$ then the almost contact structure on $$\mathbb{M}$$ is normal.

The proof of the theorem is roughly along these lines: The construction of the contact metric structure is similar to the construction in the converse of Boothby-Wang, and the normality condition then follows from consideration of the Nijenhius tensor on B. The Newlander-Nirenberg theorem implies that J is integrable if and only if N = 0, which here can be shown to imply that the almost contact structure is normal.

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.

## References

• Banyaga, A.; Houenou, D. F. A Brief Introduction to Symplectic and Contact Manifolds; Nankai Tracts in Mathematics, Vol. 15; World Scientific: New Jersey, 2017.
• W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math., 68(1958), 721-734.
• Foreman, Brendan. Complex contact manifolds and hyperkähler geometry. Kodai Math. J. 23 (2000), no. 1, 12–26. doi:10.2996/kmj/1138044153. https://projecteuclid.org/euclid.kmj/1138044153
• Hatakeyama, Yoji. Some notes on differentiable manifolds with almost contact structures. Tohoku Math. J. (2) 15 (1963), no. 2, 176–181. doi:10.2748/tmj/1178243844. https://projecteuclid.org/euclid.tmj/1178243844
• Kobayashi, Shoshichi. Principal fibre bundles with the 1-dimensional toroidal group. Tohoku Math. J. (2) 8 (1956), no. 1, 29–45. doi:10.2748/tmj/1178245006. https://projecteuclid.org/euclid.tmj/1178245006
• Morimoto, Akihiko. On normal almost contact structures. J. Math. Soc. Japan 15 (1963), no. 4, 420–436. doi:10.2969/jmsj/01540420. https://projecteuclid.org/euclid.jmsj/1260976537