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