Biquard and Hladky Connections (Connections 4)

I’ve been looking at codimension 3 sub-Riemannian manifolds, so I’m posting out of order. 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. Connections on codimension 3 sub-Riemannian manifolds.

Definition 4.1

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 4.2 (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 4.3

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 4.4

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 4.5 (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 4.6

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 \(Z \in \mathcal{H}\),

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

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 \((\mathfrak{sp}(n)\oplus \mathfrak{sp}(1))^\perp \subset \mathfrak{gl}(4n)\), which doesn’t seem to be the case.

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 \(\mathbb{M}\).

Copyright 2018 Gianmarco Molino All Rights Reserved