Biquard and Hladky Connections (Connections 6)

I’m skipping ahead in the sequence of posts about connections a bit, since I’ve been looking at codimension 3 sub-Riemannian manifolds. The planned sequence is

  1. The Bott connection on foliated manifolds,
  2. Tanno’s connection on contact manifolds,
  3. The equivalence of Bott and Tanno’s connections on \(K\)-contact manifolds with the Reeb foliation,
  4. The Tanaka-Webster connection on Sasakian manifolds,
  5. Connections on 3-Sasakian manifolds,
  6. Connections on codimension 3 sub-Riemannian manifolds.

 

Definition. Let \((\mathbb{M},g,\mathcal{H})\) be a \(4n+3\)-dimensional sub-Riemannian manifold with codimension \(3\) distribution \(\mathcal{H}\) such that

  • \(\mathcal{H}\) has a \(Sp(n)Sp(1)\)-structure, that is there exists a rank 3 bundle \(\mathcal{Q}\) consisting of \((1,1)\)-tensors on \(\mathcal{H}\) locally generated by three almost-complex structures \(I_1,I_2,I_3\) on \(\mathcal{H}\) satisfying the quaternion relations \(I_1I_2I_3 = -id\) which are hermitian compatible with the metric, that is
    \[g(I_j \cdot, I_j \cdot) = g(\cdot, \cdot)\]
    for \(j \in \{1,2,3\}\).
  • \(\mathcal{H}\) is locally given as the kernel of a \(1\)-form \(\eta = (\eta_1,\eta_2,\eta_3)\) with values in \(\mathbb{R}^3\) such that
    \[2g(I_jX,Y) = d\eta_j(X,Y)\]
    for \(j \in \{1,2,3\}\).

We then call \((\mathbb{M},g,\mathcal{H},\mathcal{Q})\) a quaternionic contact manifold or qc manifold.

Remark. These are interesting because they are an example of sub-Riemannian manifolds where
\[(\mathcal{L}_{X_\mathcal{V}}g)(Y_\mathcal{H},Z_\mathcal{H}) \neq 0\]

In this setting, we have two reference connections, the Hladky connection and Biquard connection.

Theorem. (Biquard) Let \((\mathbb{M},g,\mathcal{H},\mathcal{Q})\) be a quaternionic contact manifold of dimension \(4n+3 > 7\). Then there exists a unique connection \(\nabla^{Bi}\) with torsion \(T^{Bi}\) on \(\mathbb{M}\) and a unique supplementary distibution \(\mathcal{V}\) to \(\mathcal{H}\) such that

  • \(\mathcal{H}, \mathcal{V},\) and \(g\) are parallel for \(\nabla^{Bi}\);
  • \(T^{Bi}(\mathcal{H},\mathcal{H}) \subseteq \mathcal{V}, T^{Bi}(\mathcal{H},\mathcal{V}) \subseteq \mathcal{H}\);
  • for \(X \in \mathcal{V},\) the operator \(T^{Bi}(X, \cdot) \colon \mathcal{H} \rightarrow \mathcal{H}\) is in \((\mathfrak{sp}(n)\oplus \mathfrak{sp}(1))^\perp \subset \mathfrak{gl}(4n)\).

The connection \(\nabla^{Bi}\) is called the Biquard connection on \((\mathbb{M},g,\mathcal{H},\mathcal{Q})\).

Biquard also described the vertical space \(\mathcal{V}\) as being locally generated by vector fields \(\{\xi_1,\xi_2,\xi_3\}\) such that
\[\begin{split}
\eta_j(\xi_k) &= \delta_{jk}, \\
(\iota_{\xi_j}d\eta_j)_\mathcal{H} &= 0, \\
(\iota_{\xi_j}d\eta_k)_\mathcal{H} &= -(\iota_{\xi_k}d\eta_j)_\mathcal{H}
\end{split}\]
The fields \(\{\xi_1,\xi_2,\xi_3\}\) are called Reeb vector fields, in keeping with the nomenclature for contact manifolds.

Remark. The condition \(T^{Bi}(\mathcal{H},\mathcal{H}) \subseteq \mathcal{V}\) is equivalent to \(T^{Bi}(X,Y) = -[X,Y]_\mathcal{V}\) for all \(X,Y \in \mathcal{H}\).

Remark. Biquard showed moreover that for a qc manifold \((\mathbb{M},g,\mathcal{H},\mathcal{Q})\) of dimension \(7\) there may not be any such fields. Duchemin has shown that the Biquard connection exists for a \(7\) dimensional qc manifold if we assume the existence of the Reeb vector fields.

We now introduce the concept of an \(r\)-graded sub-Riemannian manifold in order to define Hladky’s connection.

Definition. We call a sub-Riemannian manifold \((\mathbb{M},g,\mathcal{H})\) equipped with a choice of supplementary distribution \(\mathcal{V}\) (that is \(T\mathbb{M} = \mathcal{H} \oplus \mathcal{V}\)) a sub-Riemannian manifold with complement or sRC manifold.

We say that a sRC manifold \((\mathbb{M},g,\mathcal{H},\mathcal{V})\) is r-graded if there are smooth constant rank bundles \(\mathcal{V}^{(j)}, 0 < j \leq r\), such that
\[\mathcal{V} = \mathcal{V}^{(1)} \oplus \cdots \oplus \mathcal{V}^{(r)}\]
and
\[\mathcal{H} \oplus \mathcal{V}^{(j)} \oplus [\mathcal{H},\mathcal{V}^{(j)}] \subseteq \mathcal{H} \oplus \mathcal{V}^{(j)} \oplus \mathcal{V}^{(j+1)}\]
for all \(0 \leq j \leq r\) with the convention that \(\mathcal{V}^{(0)} = \mathcal{H}\) and \(\mathcal{V}^{(j)} = 0\) for \(j > r\).

A metric extension for an r-graded sRC manifold \((\mathbb{M},g,\mathcal{V},\mathcal{H})\) is a Riemannian metric \(\tilde g\) that agrees with \(g\) on \(\mathcal{H}\) and makes the split
\[T\mathbb{M} = \mathcal{H} \oplus_{1 \leq g \leq r} \mathcal{V}^{(j)}\]
orthogonal.

For convenience, we shall denote by \(X^{(j)}\) a section of \(\mathcal{V}^{(j)}\) and set
\[\hat{\mathcal{V}}^{(j)} = \bigoplus_{k \neq j} \mathcal{V}^{(k)}\]

Lie derivatives are not tensorial in general, but we can define on an sRC manifold with metric extension the symmetric tensor \(B^{(j)}\) by
\[B^{(j)}(X,Y,Z) = (\mathcal{L}_Zg)(X,Y)\]
for \(X,Y \in \mathcal{V}^{(j)}, Z \in \hat{\mathcal{V}}^{(j)}\) and setting \(B^{(j)} = 0\) on the orthogonal complement of \(\mathcal{V}^{(j)} \times \mathcal{V}^{(j)} \times \hat{\mathcal{V}}^{(j)}\).

We contract these to tensors \(C^{(j)} \colon T\mathbb{M} \times T\mathbb{M} \rightarrow \mathcal{V}^{(j)}\) defined by
\[g(C^{(j)} (X,Y), Z^{(j)}) = B^{(j)}(X,Z^{(j)},Y)\]

Remark. The tensors \(B^{(0)}\) and \(C^{(0)}\) rely only on the sRC structure, and are independent of the grading and metric extension

Definition. If \(g\) is a metric extension of a r-graded sRC manifold then there exists a unique connection \({\nabla^{Hl}}^{(r)}\) with torsion \({T^{Hl}}^{(r)}\) such that

  • \({\nabla^{Hl}}^{(r)}\) is metric, that is \({\nabla^{Hl}}^{(r)} g = 0\);
  • \(\mathcal{V}^{(j)}\) is parallel for all \(j\);
  • \({T^{Hl}}^{(r)}(\mathcal{V}^{(j)},\mathcal{V}^{(j)}) \subseteq \hat{\mathcal{V}}^{(j)}\) for all \(j\);
  • \(g({T^{Hl}}^{(r)}(X^{(j)}, Y^{(k)}),Z^{(j)}) = g({T^{Hl}}^{(r)}(Z^{(j)}, Y^{(k)}),X^{(j)})\) for all \(j,k\).

Furthermore, if \(X,Y \in \mathcal{H}\) then \({\nabla^{Hl}}^{(r)}(X)\) and \({T^{Hl}}^{(r)}(X,Y)\) are independent of the choice of grading and metric extension.

The Hladky connection can be expressed explicitly for vector fields \(X,Y,Z \in V^{(j)}, T \in \hat{\mathcal{V}}^{(j)}\) by
\[\begin{split}
g({\nabla^{Hl}}^{(r)}_XY, T) &= 0 \\
g({\nabla^{Hl}}^{(r)}_XY, Z) &= g(\nabla^g_XY,Z) \\
{\nabla^{Hl}}^{(r)}_TY &= [T,Y]_j + \frac{1}{2}C^{(j)}(Y,T) \\
\end{split}\]
where \(\nabla^g\) is the Levi-Civita connection.

Remark. An r-graded sRC manifold also admits a k-grading (for all \(1 \leq k < r\)) given by
\[ \tilde{\mathcal{V}}^{(j)} = \mathcal{V}^{(j)}, 0 \leq j < k, \qquad \tilde{\mathcal{V}}^{(k)} = \bigoplus_{j \geq k} \mathcal{V}^{(j)} \]
and then associated to each k-grading there is a connection \({\nabla^{Hl}}^{(k)}\). For this entire family of connections, \({\nabla^{Hl}}^{(j)} X^{(k)} = {\nabla^{Hl}}^{(r)}X^{(k)}\) whenever \(0 \leq k < j\), so in particular for a horizontal vector field \(X\) it holds that
\[{\nabla^{Hl}}^{(1)}X = {\nabla^{Hl}}^{(2)}X = \cdots {\nabla^{Hl}}^{(r)}X\]
and so the differences between the connections \({\nabla^{Hl}}^{(k)}X\) can be viewed as a choice of how to differentiate vertical vector fields.

All we need is the trivial 1-grading, but I wonder if the connections associated to higher gradings may be interesting.

Theorem. (Hladky) Let \((\mathbb{M},g,\mathcal{H},\mathcal{V})\) be an r-graded sRC manifold with extended metric. We will call \({\nabla^{Hl}} = {\nabla^{Hl}}^{(1)}\) the Hladky connection on \(\mathbb{M}\).

Corollary. The Hladky connection is uniquely determined on a 1-graded sRC manifold with metric extension by the properties

  • \(\mathcal{H}, \mathcal{V},\) and \(g\) are parallel for \({\nabla^{Hl}}\);
  • \({T^{Hl}}(\mathcal{H},\mathcal{H}) \subseteq \mathcal{V}, {T^{Hl}}(\mathcal{V},\mathcal{V}) \subseteq \mathcal{H},\);
  • \(g({T^{Hl}}(X,Z),Y) = g({T^{Hl}}(Y,Z),X)\) for \(X,Y \in \mathcal{V}, Z \in \mathcal{H}\) or \(X,Y \in \mathcal{H}, Z \in \mathcal{V}\).

Remark. If \((\mathcal{L}_{X_\mathcal{V}}g)(Y_\mathcal{H}, Z_\mathcal{H}) = 0\) then \(B^{(j)} = C^{(j)} = 0\) and the Hladky connection is equivalent to the Bott and Tanno connections. This occurs, for example, in the K-contact case.

Let \((\mathbb{M},g,\mathcal{H},\mathcal{Q})\) be a qc manfold (assuming the existence of the Reeb fields in dimension 7.) By the defining theorem for Biquard’s connection there is a unique distribution \(\mathcal{V}\) such that the Biquard connection is well defined. Then given an orthogonal extension \(\tilde{g}\) of the metric to \(\mathcal{V}\), \((\mathbb{M},\tilde{g},\mathcal{H},\mathcal{V})\) will be a 1-graded sRC manifold with metric extension and thus have an Hladky connection by defining theorem for Hladky’s connection.

If we extend \(g\) to \(\mathcal{V} = span(\xi_1,\xi_2,\xi_3)\) by requiring \(g(\xi_j, \xi_k) = \delta_{jk}\), it is known that \(\nabla^{Bi} g = 0\), in agreement with \({\nabla^{Hl}}\).

QUESTION: Do the Hladky and Biquard connections agree for this extension? Do they even agree on \(\mathcal{H}\)?

Using the explicit expression for the Hladky connection, we see that for \(X \in \mathcal{V},Y \in \mathcal{H}\),

\[\begin{split}
{T^{Hl}}(X,Y) &= {\nabla^{Hl}}_XY – {\nabla^{Hl}}_YX – [X,Y] \\
&= [X,Y]_\mathcal{H} + \frac{1}{2} C^\mathcal{H}(Y,X) – [Y,X]_\mathcal{V} – \frac{1}{2}C^\mathcal{V}(X,Y) – [X,Y] \\
&= \frac{1}{2} \left( C^\mathcal{H}(Y,X) – C^\mathcal{V}(X,Y) \right) \\
\end{split}\]

using this, we can get expressions for the horizontal and vertical components. For \(Z \in \mathcal{V}\),
\[\begin{split}
g({T^{Hl}}(X,Y), Z) &= \frac{1}{2} g \left( C^\mathcal{H}(Y,X) – C^\mathcal{V}(X,Y), Z \right) \\
&= -\frac{1}{2} B^\mathcal{V}(X,Z,Y) \\
&= -\frac{1}{2} (\mathcal{L}_Y g)(X,Z) \\
&= 0 \\
\end{split}\]
as desired, so for \(X \in \mathcal{V}\), we have that \({T^{Hl}}(X, \cdot) \colon \mathcal{H} \rightarrow \mathcal{H}\) in agreement with \(T^{Bi}\). Moreover for \(Z \in \mathcal{H}\),

\[\begin{split}
g({T^{Hl}}(X,Y), Z) &= \frac{1}{2} g \left( C^\mathcal{H}(Y,X) – C^\mathcal{V}(X,Y), Z \right) \\
&= \frac{1}{2} B^\mathcal{H}(Y,Z,X) \\
&= \frac{1}{2} (\mathcal{L}_X g)(Y,Z) \\
\end{split}\]

I think this is sufficient to show that if \(\frac{1}{2} (\mathcal{L}_X g)(Y,Z) = 0\) then \({\nabla^{Hl}} = \nabla^{Bi}\). Otherwise, we need to determine if \({T^{Hl}}(X, \cdot) \colon \mathcal{H} \rightarrow \mathcal{H}\) is in \((\mathfrak{sp}(n)\oplus \mathfrak{sp}(1))^\perp \subset \mathfrak{gl}(4n)\).

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 \(\mathbb{M}\).

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
\[\begin{split}
0 &\neq (\eta \wedge (d\eta)^n)(\xi_p,v_1,\dots,v_{2n}) \\
&= \eta(\xi_p) \wedge (d\eta)^n(v_1,\dots,v_{2n}) \\
&\quad + \sum_{i=1}^{2n} (-1)^i \eta(v_i) (d\eta)^n(v_1,\dots,v_{i-1},\xi_p,v_{i+1},\dots,v_{2n} \\
&= \eta(\xi_p) \wedge (d\eta)^n(v_1,\dots,v_{2n})
\end{split}\]
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:
\(S^{2n+1} \hookrightarrow S^{2n+1} \times \mathbb{R} = \mathbb{C}^{n+1}\).

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: \(S^1 \hookrightarrow S^3 \rightarrow S^2\).

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

Tanno’s Connection on Contact Manifolds (Connections 2)

This post is the second of a series on connections on foliated manifolds.

  1. The Bott connection on foliated manifolds,
  2. Tanno’s connection on contact manifolds,
  3. The equivalence of Bott and Tanno’s connections on \(K\)-contact manifolds with the Reeb foliation,
  4. The Tanaka-Webster connection on Sasakian manifolds,
  5. Connections on 3-Sasakian manifolds,
  6. Connections on codimension 3 sub-Riemannian manifolds.

We’ll be considering Tanno’s connection, which is well adapted to contact structures and thus appropriate for studying the Reeb foliation. Here I assume the reader is familiar with contact manifolds, (Koszul) connections, and quite a few other things.

Throughout this post, all manifolds will be smooth.

2. Tanno’s Connection on Contact Manifolds

We call \((\mathbb{M}, \theta)\) a contact manifold if \(\mathbb{M}\) is a \(2n+1\) dimensional manifold and \(\theta\) is a 1-form such that \(\theta \wedge (d\theta)^n\) is a volume form on \(\mathbb{M}\).

Proposition 2.1

Let \((\mathbb{M}, \theta)\) be a contact manifold. There exist on \(\mathbb{M}\) a unique vector field \(\xi\), a Riemannian metric \(g\), and a \((1,1)\)-tensor field \(J\) such that

  1. \(\theta(\xi) = 1\), \(\iota_\xi d\theta = 0\),
  2. \(g(X,\xi ) = \theta(X)\) for all vector fields \(X\),
  3. \(2g(X,JY) = d\theta(X,Y)\), \(J^2X = -X + \theta(X)\xi\) for all vector fields \(X,Y\).

\(\xi\) is called the Reeb vector field, and such a metric is said to be compatible with the contact structure.

A contact manifold \((\mathbb{M}, \theta)\) can be canonically equipped with a codimension 1 foliation \(\mathcal{F}_\xi\) by choosing the horizontal distribution to be \(\mathcal{H} = \ker \theta\) and the vertical distribution \(\mathcal{V}\) to be generated by the Reeb vector field \(\xi\) . This is known as the Reeb foliation.

Proof of some of the above (well-known) claims will be forthcoming in post 6, see also [bh17] for an introduction to contact manifolds.

Theorem 2.2 (Tanno’s Connection)

Let \((\mathbb{M}, \theta, \xi, g, J, \mathcal{F}_\xi)\) as above. There exists a unique connection \(\nabla^T\) on \(T\mathbb{M}\) satisfying

  1. \(\nabla^T\theta = 0\),
  2. \(\nabla^T\xi = 0\),
  3. \(\nabla^T\) is metric, i.e. \(\nabla^Tg = 0\),
  4. \(T^T(X,Y) = d\theta(X,Y)\xi\) for any \(X,Y \in \Gamma^\infty(\mathcal{H})\),
  5. \(T^T(\xi,JY) = -JT^T(\xi,Y)\) for any \(Y \in \Gamma^\infty(T\mathbb{M})\),
  6. \((\nabla^T_XJ)(Y) = Q(Y,X)\) for any \(X,Y \in \Gamma^\infty(T\mathbb{M})\),

where the Tanno tensor \(Q\) is the \((1,2)\)-tensor field determined by

\[Q^i_{jk} = \nabla^g_kJ^i_j + \xi^iJ^r_j\nabla^g_k\theta_r + J^i_r\nabla^g_k\xi^r\theta_j\]

or equivalently

\[Q(X,Y) = (\nabla^g_YJ)X + [(\nabla^g_Y\theta)JX]\xi + \theta(X)J(\nabla^g_Y\xi).\]

This connection is known as Tanno’s connection, or sometimes as the generalized Tanaka connection. Just as with Bott’s connection, the proof proceeds in two parts.

Proof.
Part 1. (Uniqueness)

We have the usual metric relations

\[\begin{align}
g(\nabla^T_XY,Z) + g(Y, \nabla^T_XZ) &= X \cdot g(Y,Z) \\
g(\nabla^T_YZ,X) + g(Z, \nabla^T_YX) &= Y \cdot g(Z,X) \\
g(\nabla^T_ZX,Y) + g(X, \nabla^T_ZY) &= Z \cdot g(X,Y)
\end{align}\]

which can be summed to show that

\[ 2g(\nabla^T_XY, Z) = g(\nabla^T_XY – \nabla^T_YX, Z) + g(\nabla^T_ZX – \nabla^T_XZ , Y) + g(\nabla^T_ZY – \nabla^T_YZ, X) \\
+ X \cdot(Y,Z) + Y \cdot g(Z,X) – Z \cdot g(X,Y).\]

By definition,

\[\nabla^T_XY – \nabla^T_YX = [X,Y] + T^T(X,Y) \]

so it remains to find an expression for \(T^T\) independent of the connection.

For vertical vector fields \(X,Y\),

\[\begin{aligned}
T^T(X,Y) &= \nabla^T_XY – \nabla^T_YX – [X,Y] \\
&= \theta(Y)\nabla^T_X\xi + X \cdot \theta(Y) – \theta(X)\nabla^T_Y\xi – Y \cdot \theta(X) – [X,Y] \\
&= X \cdot \theta(Y) – Y \cdot \theta(X) – [X,Y] \\
\end{aligned}\]

using the the fact that the Reeb vector field is parallel.

For horizontal fields \(X,Y\)

\[T^T(X,Y) = d\theta(X,Y)\xi\]

is given as condition 4.

Finally, for \(X\) vertical and \(Y\) horizontal we have

\[\begin{aligned}
T^T(X,Y) &= -\theta(X)T^T(\xi,J^2Y) \\
&= \theta(X)JT^T(\xi,JY) \\
&= -\theta(X)J^2T^T(\xi,Y) \\
&= -J^2T^T(X,Y) \\
&=T^T(X,Y) – \theta(T^T(X,Y))\xi \\
\theta(T^T(X,Y))\xi &= 0 \\
\end{aligned}\]

from which we conclude that \(T^T(X,Y)\) is horizontal, and also

\[\begin{aligned}
\nabla^T_XY &= -\nabla^T_X(J^2Y) \\
&= -(\nabla^T_XJ)(JY) – J(\nabla^T_X(JY)) \\
&= -Q(JY,X) – J((\nabla^T_XJ)Y – J(\nabla^T_XY)) \\
&= -Q(JY,X) – JQ(Y,X) – J^2(\nabla^T_XY) \\
&= -Q(JY,X) – JQ(Y,X) – \nabla^T_XY + \theta(\nabla^T_XY)\xi \\
2\nabla^T_XY &= -Q(JY,X) – JQ(Y,X) + \theta(\nabla^T_XY)\xi \\
\end{aligned}\]

which we can apply to the expression for the torsion giving us that

\[\begin{aligned}
2T^T(X,Y) &= 2\nabla^T_XY – 2\nabla^T_YX – 2[X,Y] \\
&= -Q(JY,X) – JQ(Y,X) + \theta(\nabla^T_XY)\xi \\
&\qquad – (-Q(JX,Y) – JQ(X,Y) + \theta(\nabla^T_YX)\xi ) – 2[X,Y] \\
&= -Q(JY,X) – JQ(Y,X) + \theta(\nabla^T_XY – \nabla^T_YX)\xi + JQ(X,Y) – 2[X,Y] \\
&= -Q(JY,X) – JQ(Y,X) + JQ(X,Y) + \theta(T^T(X,Y) + [X,Y])\xi – 2[X,Y] \\
&= -Q(JY,X) – JQ(Y,X) + JQ(X,Y) – \theta([X,Y])\xi – 2[X,Y]. \\
\end{aligned}\]

From this, we can write an expression for \(g(\nabla^T_XY,Z)\) independent of \(\nabla^T\), so it must be unique.
Remark. Notice that we did not need to use condition 1 (that \(\nabla^T\theta = 0\)) to prove uniqueness.

Part 2. (Existence)

Following Tanno’s original paper [tan89], we define a connection \(\nabla\) by its Christoffel symbols

\[\overline{\Gamma^i_{jk}} = \Gamma^i_{jk} + \theta_jJ^i_k – \nabla^g_j\xi^i\theta_k + \xi^i\nabla^g_j\theta_k\]

or equivalently in coordinate-free notation,

\[\nabla_XY = \nabla^g_XY + \theta(X)JY – \theta(Y)\nabla^g_X\xi + [(\nabla^g_X\theta)Y]\xi \]

where the \(\Gamma^i_{jk}\) denote the Christoffel symbols of the Levi-Civita connection \(\nabla^g\). We claim that \(\nabla\) is in fact Tanno’s connection.

To prove this, we will verify the conditions explicitly.

Condition 1

We have that

\[\begin{aligned}
(\nabla \theta) (X, Y) &= (\nabla_X\theta)(Y) \\
&= X \cdot \theta(Y) – \theta(\nabla_XY) \\
&= X \cdot \theta(Y) – \theta(\nabla^g_XY + \theta(X)JY – \theta(Y)\nabla^g_X\xi + [(\nabla^g_X\theta)Y]\xi) \\
&= X \cdot \theta(Y) – \theta(\nabla^g_XY) – \theta(X)\theta(JY) + \theta(Y)\theta(\nabla^g_X\xi) – [(\nabla^g_X\theta)Y]\theta(\xi) \\
&= X \cdot \theta(Y) – \theta(\nabla^g_XY) – X \cdot \theta(Y) + \theta(\nabla^g_XY) \\
&= 0
\end{aligned}\]

using, in particular, that \(\theta(J(Y)) = 0\) since \(J \colon T\mathbb{M} \rightarrow \mathcal{H} = \ker \theta\), and also that \(\theta(\nabla^g_X\xi) = 0\) since \(\nabla^g_X\xi \in \mathcal{H}\). Thus \(\nabla\) satisfies condition 1.

Condition 2

Similarly,

\[\begin{aligned}
(\nabla \xi)(X) &= \nabla_X\xi \\
&= \nabla^g_X\xi + \theta(X)J\xi – \theta(\xi)\nabla^g_X\xi + [(\nabla^g_X\theta)\xi]\xi \\
&= \nabla^g_X\xi – \nabla^g_X\xi + [X \cdot \theta(\xi) – \theta(\nabla^g_X\xi)]\xi \\
&= 0
\end{aligned}\]

which proves that \(\nabla\) satisfies condition 2.

Condition 3

Again, we show condition 3 directly,

\[\begin{aligned}
(\nabla g) (X,Y,Z) &= (\nabla_Xg)(Y,Z) \\
&= X \cdot g(Y,Z) – g(\nabla_XY, Z) – g(Y, \nabla_XZ) \\
&= X \cdot g(Y,Z) – g(\nabla^g_XY, Z) – g(Y, \nabla^g_XZ) \\
&\qquad – g(\theta(X)JY – \theta(Y)\nabla^g_X\xi + [(\nabla^g_X\theta)Y]\xi, Z) \\
&\qquad – g(Y, \theta(X)JZ – \theta(Z)\nabla^g_X\xi + [(\nabla^g_X\theta)Z]\xi) \\
&= (\nabla^gg)(X,Y,Z) \\
&\qquad – g(\theta(X)JY – \theta(Y)\nabla^g_X\xi + [(\nabla^g_X\theta)Y]\xi, Z) \\
&\qquad – g(Y, \theta(X)JZ – \theta(Z)\nabla^g_X\xi + [(\nabla^g_X\theta)Z]\xi) \\
&= – g([(\nabla^g_X\theta)Y]\xi – \theta(Y)\nabla^g_X\xi, Z) \\
&\qquad – g(Y, [(\nabla^g_X\theta)Z]\xi – \theta(Z)\nabla^g_X\xi) \\
&\qquad – g(\theta(X)JY, Z) – g(Y, \theta(X)JZ) \\
&= – \theta(Z)([(\nabla^g_{X_\mathcal{H}}\theta)Y] – g(Y,\nabla^g_{X_\mathcal{H}}\xi)) \\
&\qquad – \theta(Y)([(\nabla^g_{X_\mathcal{H}}\theta)Z] – g(Z,\nabla^g_{X_\mathcal{H}}\xi)) \\
&\qquad – \theta(X)[d\theta(Z,Y) + d\theta(Y,Z)] \\
&= – \theta(Z)(X_\mathcal{H}\cdot g(Y,\xi) – g(\nabla^g_{X_\mathcal{H}}Y,\xi) – g(Y,\nabla^g_{X_\mathcal{H}}\xi)) \\
&\qquad – \theta(Y)(X_\mathcal{H}\cdot g(Z,\xi) – g(\nabla^g_{X_\mathcal{H}}Z,\xi) – g(Z,\nabla^g_{X_\mathcal{H}}\xi)) \\
&= – \theta(Z)(\nabla^gg)(X_\mathcal{H},Y,\xi) – \theta(Y)(\nabla^gg)(X_\mathcal{H},Z,\xi) \\
&= 0
\end{aligned}\]

using, in particular, that \(d\theta(Y,Z) + d\theta(Z,Y) = 0\) and \(g(X,\zeta) = \theta(X)\).

Condition 4

To prove that conditions 4 and 5 hold, we will want an explicit expression for the torsion, which we write as

\[\begin{aligned}
T(X,Y) &= \nabla_XY – \nabla_YX – [X,Y] \\
&= \nabla^g_XY + \theta(X)JY – \theta(Y)\nabla^g_X\xi + [(\nabla^g_X\theta)Y]\xi \\
&\qquad – \nabla^g_YX – \theta(Y)JX + \theta(X)\nabla^g_Y\xi – [(\nabla^g_Y\theta)X]\xi \\
&\qquad – [X,Y] \\
&= \theta(X)(JY + \nabla^g_Y\xi) – \theta(Y)(JX + \nabla^g_X\xi) + ([(\nabla^g_X\theta)Y] – [(\nabla^g_Y\theta)X])\xi \\
&= \theta(X)(JY + \nabla^g_Y\xi) – \theta(Y)(JX + \nabla^g_X\xi) + d\theta(X,Y)\xi \\
\end{aligned}\]

Then to check condition 4, we assume \(X,Y \in \mathcal{H} = \ker \theta\) so that

\[\begin{aligned}
T(X,Y) &= \theta(X)(JY + \nabla^g_Y\xi) – \theta(Y)(JX + \nabla^g_X\xi) + d\theta(X,Y)\xi \\
&= d\theta(X,Y)\xi
\end{aligned}\]

using the expansion of the exterior derivative on 1-forms given by a torsion free connection.

Condition 5

For condition 5, again let \(Y\) be any vector field, so that

\[\begin{aligned}
T(\xi,Y) &= \theta(\xi)(JY + \nabla^g_Y\xi) – \theta(Y)(J\xi + \nabla^g_\xi\xi) + d\theta(\xi,Y)\xi \\
&= JY + \nabla^g_Y\xi \\
\end{aligned}\]

Now, if \(Y\) is a vertical field the conclusion is clear. For \(Y\) a horizontal field we claim that \(\nabla^g_{JY}\xi + J\nabla^g_Y\xi = 2Y\) (which will be shown subsequently) and it holds that

\[\begin{aligned}
-JT(\xi, Y) &= -J^2Y – J\nabla^g_Y \xi \\
&= -J^2Y – (2Y – \nabla^g_{JY}\xi) \\
&= J^2Y + \nabla^g_{JY}\xi \\
&= T(\xi, JY) \\
\end{aligned}\]

and condition 5 follows from the linearity of \(T\). We complete the case with the following due to F. Baudoin.

Lemma. For horizontal \(X,Y\) it holds that \(\theta((\nabla^g_XJ)Y) = \theta((\nabla^g_YJ)X)\).

Proof. Recall that \(\theta(\nabla^g_YJ)X) = g((\nabla^g_YJ)X,\xi)\). Differentiating \(g(JX,\xi) = 0\) with respect to \(Y\) we see that

\[g((\nabla^g_YJ)X,\xi) + g(JX,\nabla^g_Y\xi) = 0\]

so it is enough to prove that

\[g(JX, \nabla^g_Y\xi) = g(JY,\nabla^g_X\xi)\]

or equivalently

\[d\theta(X,\nabla^g_Y\xi) = d\theta(Y,\nabla^g_X\xi).\]

We have that

\[d\theta(X, \nabla^g_Y\xi) = d\theta(X,\nabla^g_\xi Y + [Y,\xi]) = d\theta(X,\nabla^g_\xi Y) + d\theta(X,[Y,\xi]).\]

Using \(\nabla^g_\xi d\theta = 0\),

\[d\theta(X,\nabla^g_\xi Y) = \xi \cdot d\theta(X,Y) – d\theta(\nabla^g_\xi X,Y)\]

and similarly using \(\mathcal{L}_\xi d\theta = d\mathcal{L}_\xi \theta = 0\),

\[-d\theta(X,[Y,\xi]) = \xi \cdot d\theta(X,Y) – d\theta([\xi,X],Y).\]

From which we see that

\[d\theta(X,\nabla^g_Y\xi) = -d\theta(\nabla^g_\xi X,Y) + d\theta([\xi,X],Y) = -d\theta(\nabla^g_X\xi,Y).\]

proving the lemma.

Claim. For horizontal \(X\) it holds that \(\nabla^g_{JX}\xi + J\nabla^g_X\xi = 2X\).

Proof. Let \(Y\) be horizonal. It holds that

\[\begin{aligned}
g(\nabla^g_{JX}\xi, Y) &= – g(\xi, \nabla^g_{JX}Y) \\
&= -\theta(\nabla^g_{JX}Y) \\
&= -\theta(\nabla^g_Y(JX)) – \theta([JX,Y]) \\
&= d\theta(JX,Y) – \theta(\nabla^g_Y(JX)) \\
&= 2g(X,Y) – \theta(\nabla^g_Y(JX)).
\end{aligned}\]

On the other hand,

\[\begin{aligned}
g(J\nabla_X\xi,Y) &= – g(\nabla^g_X\xi, JY) \\
&= g(\xi, \nabla^g_X(JY)) \\
&= \theta(\nabla^g_X(JY))
\end{aligned}\]

thus applying the last lemma, the conclusion follows.

Condition 6

For the final condition,

\[\begin{aligned}
(\nabla_XJ)Y &= \nabla_X(JY) – J(\nabla_XY) \\
&= \nabla^g_X(JY) + \theta(X)J(JY) – \theta(JY)\nabla^g_X\xi + [(\nabla^g_X\theta)(JY)]\xi \\
&\qquad – J(\nabla^g_XY + \theta(X)JY – \theta(Y)\nabla^g_X\xi + [(\nabla^g_X\theta)Y]\xi) \\
&= \nabla^g_X(JY) + \theta(X)J^2Y + [(\nabla^g_X\theta)JY]\xi – J(\nabla^g_XY) – \theta(X)J^2Y + \theta(Y)J(\nabla^g_X\xi) \\
&= \nabla^g_X(JY) – J(\nabla^g_XY) + [(\nabla^g_X\theta)JY]\xi + \theta(Y)J(\nabla^g_X\xi) \\
&= (\nabla^g_XJ)Y + [(\nabla^g_X\theta)JY]\xi + \theta(Y)J(\nabla^g_X\xi) \\
&= Q(Y,X) \\
\end{aligned}\]

completing the proof.

We finish by remarking that the case of interest to us is when \(Q=0\); this condition is equivalent to \((M,\theta,J)\) being a strongly pseudoconvex CR manifold. Moreover, \(\xi\) will be a Killing field, and the foliation will be totally geodesic with bundle-like metric.

References

[bh17] A. Banyaga, and D. Houenou. A Brief Introduction to Symplectic and Contact Manifolds. Vol. 15, World Scientific, 2017.

[tan89] S. Tanno. Variational problems on contact Riemannian manifolds. Trans. Amer. Math. Soc., 314(1):349–379, 1989.

Copyright 2018 Gianmarco Molino All Rights Reserved