I’ve been looking at codimension 3 sub-Riemannian manifolds, so I’m posting out of order. The planned sequence is
- The Bott connection on foliated manifolds,
- Tanno’s connection on contact manifolds,
- The equivalence of Bott and Tanno’s connections on \(K\)-contact manifolds with the Reeb foliation,
- 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.
