Sasakian Manifolds

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Sasakian and Kähler Manifolds 2

12 June, 2018 in Lecture Notes | No comments

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 \(\mathbb{M}\).

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Sasakian and Kähler Manifolds 1

11 April, 2018 in Lecture Notes | 1 comment

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

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