I’d like to begin this blog by discussing some ideas relevant to my current research; to that end, this will be the first in a series of posts about connections on foliated 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. Connections on codimension 3 sub-Riemannian manifolds.

For this post, I assume that the reader is familiar with Riemannian manifolds, (Koszul) connections, the Levi-Civita connection, foliated manifolds, basic vector fields, and quite a few other things.

Throughout this post, all manifolds will be smooth, oriented, connected, Riemannian, and complete with respect to their metric.

1. The Bott Connection on Foliated Manifolds

Let \( (\mathbb{M}, g, \mathcal{F})\) be a Riemannian manifold of dimension \( n+m\), equipped with a foliation \( \mathcal{F}\) which has totally geodesic, \( m\)-dimensional leaves and a bundle-like metric \( g\). The sub-bundle \( \mathcal{V}\) of \( T\mathbb{M}\) formed by vectors tangent to the leaves is referred to as the vertical distribution, and the sub-bundle \( \mathcal{H}\) of \( T\mathbb{M}\) which is normal (under \( g\)) to \( \mathcal{V}\) is referred to as the horizontal distribution.

Our first task will be to define the Bott connection on foliated manifolds. Heuristically, this connection is interesting because it is well adapted to the foliation, making both the vertical and horizontal distributions parallel while also being metric.

Theorem 1.1 (Bott Connection).

For \( (\mathbb{M}, g, \mathcal{F})\) as before, there exists a unique connection \( \nabla^B\) over \( T\mathbb{M}\) satisfying the following:

1. \( \nabla^B\) is metric. That is, \( \nabla^B g = 0\).
2. If \( Y\) is an horizontal vector field, \( \nabla^B_XY\) is horizontal for all vector fields \( X\).
3. If \( Z\) is a vertical vector field, \( \nabla^B_XZ\) is vertical for all vector fields \( X\).
4. For horizontal vector fields \( X_1,X_2\) and vertical vector fields \( Z_1,Z_2\), it holds that \( T^B(X_1,X_2)\) is vertical and that \( T^B(X_1,Z_1) = T^B(Z_1,Z_2) = 0\), where \( T^B(X,Y) = \nabla^B_XY – \nabla^B_YX – [X,Y]\) is the torsion tensor associated to \( \nabla^B\).

This connection is referred to as the Bott connection on \( (\mathbb{M}, g, \mathcal{F})\). The proof will proceed in two parts.

Proof.

Part 1. (Uniqueness)

We begin by showing that the Bott connection is necessarily unique. Let \( X,Y,Z\) be vector fields. Because \( \nabla^B\) is metric, we have the relations

\[\begin{align} g(\nabla^B_XY,Z) + g(Y, \nabla^B_XZ) &= X \cdot g(Y,Z) \\
g(\nabla^B_YZ,X) + g(Z, \nabla^B_YX) &= Y \cdot g(Z,X) \\
g(\nabla^B_ZX,Y) + g(X, \nabla^B_ZY) &= Z \cdot g(X,Y) \end{align}\]

as well as the torsion relations

\[\begin{align} \nabla^B_XY – \nabla^B_YX &= [X,Y] – \pi_\mathcal{V}[\pi_\mathcal{H}X, \pi_\mathcal{H}Y] \\
\nabla^B_ZX – \nabla^B_XZ &= [Z,X] – \pi_\mathcal{V}[\pi_\mathcal{H}Z, \pi_\mathcal{H}X] \\
\nabla^B_ZY – \nabla^B_YZ &= [Z,Y] – \pi_\mathcal{V}[\pi_\mathcal{H}Z, \pi_\mathcal{H}Y] \end{align}\]

which follow from

\[\begin{align} T^B(X,Y) &= T^B(\pi_\mathcal{V}X + \pi_\mathcal{H}X, \pi_\mathcal{V}Y + \pi_\mathcal{H}Y) \\
&= T^B(\pi_\mathcal{V}X, \pi_\mathcal{V}Y) + T^B(\pi_\mathcal{H}X, \pi_\mathcal{V}Y) + T^B(\pi_\mathcal{V}X, \pi_\mathcal{H}Y) + T^B(\pi_\mathcal{H}X, \pi_\mathcal{H}Y) \\
&= T^B(\pi_\mathcal{H}X, \pi_\mathcal{H}Y) \\
&= \pi_\mathcal{V}\left(\nabla^B_{\pi_\mathcal{H}X}\pi_\mathcal{H}Y – \nabla^B_{\pi_\mathcal{H}Y}\pi_\mathcal{H}X – [\pi_\mathcal{H}X,\pi_\mathcal{H}Y]\right) \\
&= -\pi_\mathcal{V}[\pi_\mathcal{H}X, \pi_\mathcal{H}Y] \end{align}\]

Alternately summing the metric relations, we find

\[ 2g(\nabla^B_XY, Z) = g(\nabla^B_XY – \nabla^B_YX, Z) + g(\nabla^B_ZX – \nabla^B_XZ , Y) + g(\nabla^B_ZY – \nabla^B_YZ, X) \\
+ X \cdot(Y,Z) + Y \cdot g(Z,X) – Z \cdot g(X,Y)\]

which, applying the torsion relations, reduces to

\[\begin{align} 2g(\nabla^B_XY, Z) &= g([X,Y] – \pi_\mathcal{V}[\pi_\mathcal{H}X, \pi_\mathcal{H}Y], Z) \\
&\quad + g([Z,X] – \pi_\mathcal{V}[\pi_\mathcal{H}Z, \pi_\mathcal{H}X], Y) \\
&\quad + g([Z,Y] – \pi_\mathcal{V}[\pi_\mathcal{H}Z, \pi_\mathcal{H}Y], X) \\
&\quad + X \cdot(Y,Z) + Y \cdot g(Z,X) – Z \cdot g(X,Y)\end{align}\]

The right side of this expression, while a bit messy, is independant of the connection and thus determines the Bott connection uniquely. (Notice, this is the same proceedure that is carried out for the Levi-Civita connection, but isn’t quite as clean thanks to the torsion.)

Part 2. (Existence)

To see that the Bott connection exists, we construct it explicitly in terms of \( \nabla^g\), the Levi-Cevita connection on \( \mathbb{M}\) associated to the metric \( g\). Recall that \( \nabla^g\) is the unique connection on \( \mathbb{M}\) that is both metric and torsion free (i.e. \( T^g(X,Y) = 0\).) We define a connection \( \nabla\) on \(T\mathbb{M}\) by

\[ \nabla_XY = \begin{cases}
\pi_\mathcal{H}\nabla^g_XY & X,Y \in \Gamma^\infty(\mathcal{H}) \\
\pi_\mathcal{H}[X,Y] & X \in \Gamma^\infty(\mathcal{V}), Y \in \Gamma^\infty(\mathcal{H}) \\
\pi_\mathcal{V}[X,Y] & X \in \Gamma^\infty(\mathcal{H}), Y \in \Gamma^\infty(\mathcal{V}) \\
\pi_\mathcal{V}\nabla^g_XY & X,Y \in \Gamma^\infty(\mathcal{V})
\end{cases}\]

That \( \nabla\) is a connection is clear, verifying the Leibniz property directly. We claim that \( \nabla\) satisfies the conditions of the Bott connection. Conditions 2 and 3 are immediate, by definition. The rest of the proof will follow by cases, decomposing vector fields as \( X = \pi_\mathcal{V}X + \pi_\mathcal{H}X\) and using the additive properties of connections.

To show that condition 4 holds, let \( X_i \in \Gamma^\infty(\mathcal{H})\) and \( Z_i \in \Gamma^\infty(\mathcal{V})\). Then

\[\begin{align} T(X_1,X_2) &= \nabla_{X_1}X_2 – \nabla_{X_2}X_1 – [X_1,X_2] \\
&= \pi_\mathcal{H}\nabla_{X_1}X_2 – \pi_\mathcal{H}\nabla^g_{X_2}X_1 – (\nabla^g_{X_1}X_2 – \nabla^g_{X_2}X_1) \\
&= -\pi_\mathcal{V}\nabla^g_{X_1}X_2 + \pi_\mathcal{V}\nabla^g_{X_2}X_1 \\
&= -\pi_\mathcal{V} [X_1,X_2]\end{align}\]

using the fact that the Levi-Civita connection is torsion free. Similarly,

\[\begin{align} T(Z_1,Z_2) &= \nabla_{Z_1}Z_2 – \nabla_{Z_2}Z_1 – [Z_1,Z_2] \\
&= \pi_\mathcal{V}\nabla_{Z_1}Z_2 – \pi_\mathcal{V}\nabla^g_{Z_2}Z_1 – (\nabla^g_{Z_1}Z_2 – \nabla^g_{Z_2}Z_1) \\
&= -\pi_\mathcal{H}\nabla^g_{Z_1}Z_2 + \pi_\mathcal{H}\nabla^g_{Z_2}Z_1 \\
&= 0 \end{align}\]

where the last step follows since the vertical distribution being totally geodesic implies that \( \nabla^g_{Z_i}Z_j\) is vertical whenever both \( Z_i\) and \( Z_j\) are both vertical. Finally,

\[\begin{align} T(X_1,Z_1) &= \nabla_{X_1}Z_1 – \nabla_{Z_1}X_1 – [X_1,Z_1] \\
&= \pi_\mathcal{V}[X_1,Z_1] – \pi_\mathcal{H}[Z_1,X_1] – [X_1,Z_1] \\
&= 0\end{align}\]

which shows that \( \nabla\) satisfies condition 4.

It remains to be shown that \( \nabla\) is metric. We have that \( \nabla g\) is given by

\[ (\nabla g)(X,Y,Z) = X \cdot (g(Y,Z)) – g(\nabla_XY,Z) – g(Y,\nabla_XZ)\]

for any vector fields \( X,Y,Z\).

First, if \( Y \in \Gamma^\infty(\mathcal{H}), Z \in \Gamma^\infty(\mathcal{V})\) we have by the definition of \( \nabla\) that \( \nabla_XY \in \Gamma^\infty(\mathcal{H}), \nabla_XZ \in \Gamma^\infty(\mathcal{V})\) and since the metric splits orthogonally as \( g = g_\mathcal{V} \oplus g_\mathcal{H}\) each of the terms on the right side vanish, and similarly for \( Y \in \Gamma^\infty(\mathcal{V}), Z \in \Gamma^\infty(\mathcal{H})\). Thus we only need to consider the cases where \( Y,Z\) are both vertical or both horizonal.

Now, if \( X,Y,Z \in \Gamma^\infty(\mathcal{H})\), we see that

\[\begin{align} (\nabla g)(X,Y,Z) &= X \cdot (g(Y,Z)) – g(\nabla_XY,Z) – g(Y,\nabla_XZ) \\
&= X \cdot (g(Y,Z)) – g(\pi_\mathcal{H}\nabla^g_XY,Z) – g(Y,\pi_\mathcal{H}\nabla^g_XZ) \\
&= X \cdot (g(Y,Z)) – g(\nabla^g_XY,Z) + g(\pi_\mathcal{V}\nabla^g_XY,Z) \\
&\quad – g(Y,\nabla^g_XZ) + g(Y,\pi_\mathcal{V}\nabla^g_XZ) \\
&= (\nabla^gg)(X,Y,Z) + g(\pi_\mathcal{V}\nabla^g_XY,Z) + g(Y,\pi_\mathcal{V}\nabla^g_XZ) \\
&=0\end{align}\]

using the fact that the Levi-Cevita connection is metric, and the orthogonality of the horizontal and vertical distributions. A similar computation holds for \( X,Y,Z \in \Gamma^\infty(\mathcal{V})\).

It is useful here to recall that since \( g\) is a bundle-like metric, \( (M, g, \mathcal{F})\) is given locally as a submersion \(\phi \colon (V_{T\mathbb{M}},g\vert_{V_{T\mathbb{M}}}) \rightarrow (U_\mathcal{H},g_\mathcal{H})\); moreover there exists a basis of the plaque \( U_\mathcal{H}\) given by basic vector fields, so by the additivity of connections we can always consider the horizontal component of vector fields to be basic.

Then, for \( X \in \Gamma^\infty(\mathcal{V}), Y,Z \in \Gamma^\infty(\mathcal{H})\),

\[\begin{align}(\nabla g)(X,Y,Z) &= X \cdot (g(Y,Z)) – g(\nabla_XY,Z) – g(Y,\nabla_XZ) \\
&= X \cdot (g(Y,Z)) – g(\pi_\mathcal{H}[X,Y],Z) – g(Y,\pi_\mathcal{H}[X,Z]) \\
&= 0\end{align}\]

since the Lie bracket \( [X,Y]\) of a vertical vector field and a basic vector field is always vertical.

Finally, for \( X \in \Gamma^\infty(\mathcal{H}), Y,Z \in \Gamma^\infty(\mathcal{V})\),

\[\begin{align} (\nabla g)(X,Y,Z) &= X \cdot (g(Y,Z)) – g(\nabla_XY,Z) – g(Y,\nabla_XZ) \\
&= X \cdot (g(Y,Z)) – g(\pi_\mathcal{V}[X,Y],Z) – g(Y,\pi_\mathcal{V}[X,Z]) \\
&= X \cdot (g(Y,Z)) – g([X,Y],Z) + g(\pi_\mathcal{H}[X,Y],Z) \\
&\quad – g(Y,[X,Z]) + g(Y,\pi_\mathcal{H}[X,Z]) \\
&= X \cdot (g(Y,Z)) – g([X,Y],Z) – g(Y,[X,Z]) \\
&= X \cdot (g(Y,Z)) – g(\nabla^g_XY,Z) + g(\nabla^g_YX,Z) \\
&\quad – g(Y,\nabla^g_XZ) + g(Y,\nabla^g_ZX) \\
&= (\nabla^gg)(X,Y,Z) + g(\nabla^g_YX,Z) + g(Y,\nabla^g_ZX) \\
&= \mathcal{L}_Xg(Y,Z) \\
&= 0\end{align}\]

since the vertical distribution is totally geodesic if and only if the flow generated by a basic field is an isometry. From the above, we have that \( \nabla\) satisfies the conditions, and thus \( \nabla = \nabla^B\) is the Bott connection, completing the proof.