## Lecture 5. Riemannian foliations and horizontal Brownian motion

In many interesting cases, we do not actually have a globally defined Riemannian sumersion but a Riemannian foliation.

Definition:  Let $\mathbb M$ be a smooth and connected $n+m$ dimensional manifold. A $m$-dimensional foliation $\mathcal{F}$ on $\mathbb M$ is defined by a maximal collection of pairs $\{ (U_\alpha, \pi_\alpha), \alpha \in I \}$ of open subsets $U_\alpha$ of $\mathbb M$ and submersions $\pi_\alpha: U_\alpha \to U_\alpha^0$ onto open subsets of $\mathbb{R}^n$ satisfying:

•  $\cup_{\alpha \in I} U_\alpha =\mathbb M$;
•  If $U_\alpha \cap U_\beta \neq \emptyset$, there exists a local diffeomorphism $\Psi_{\alpha \beta}$ of $\mathbb{R}^n$ such that $\pi_\alpha=\Psi_{\alpha \beta} \pi_\beta$ on $U_\alpha \cap U_\beta$.

The maps $\pi_\alpha$ are called disintegrating maps of $\mathcal{F}$. The connected components of the sets $\pi_\alpha^{-1}(c)$, $c \in \mathbb{R}^n$, are called the plaques of the foliation. A foliation arises from an integrable sub-bundle of $T\mathbb M$, to be denoted by $\mathcal{V}$ and referred to as the vertical distribution. These are the vectors tangent to the leaves, the maximal integral sub-manifolds of $\mathcal{V}$.

Foliations have been extensively studied and numerous books are devoted to them. We refer in particular to the book by Tondeur.

In the sequel, we shall only be interested in Riemannian foliations with bundle like metric.

Definition: Let $\mathbb M$ be a smooth and connected $n+m$ dimensional Riemannian manifold. A $m$-dimensional foliation $\mathcal{F}$ on $\mathbb M$ is said to be Riemannian with a bundle like metric if the disintegrating maps $\pi_\alpha$ are Riemannian submersions onto $U_\alpha^0$ with its given Riemannian structure. If moreover the leaves are totally geodesic sub-manifolds of $\mathbb M$, then we say that the Riemannian foliation is totally geodesic with a bundle like metric.

Observe that if we have a Riemannian submersion $\pi : (\mathbb M,g) \to (\mathbb{B},j)$, then $\mathbb M$ is equipped with a Riemannian foliation with bundle like metric whose leaves are the fibers of the submersion. Of course, there are many Riemannian foliations with bundle like metric that do not come from a Riemannian submersion.

Example: (Contact manifolds) Let $(\mathbb M,\theta)$ be a $2n+1$-dimensional smooth contact manifold. On $\mathbb M$ there is a unique smooth vector field $T$, the so-called Reeb vector field, that satisfies

$\theta(T)=1,\quad \mathcal{L}_T(\theta)=0,$
where $\mathcal{L}_T$ denotes the Lie derivative with respect to $T$. On $\mathbb M$ there is a foliation, the Reeb foliation, whose leaves are the orbits of the vector field $T$. As it is well-known, it is always possible to find a Riemannian metric $g$ and a $(1,1)$-tensor field $J$ on $\mathbb M$ so that for every vector fields $X, Y$

$g(X,T)=\theta(X),\quad J^2(X)=-X+\theta (X) T, \quad g(X,JY)=(d\theta)(X,Y).$
The triple $(\mathbb M, \theta,g)$ is called a contact Riemannian manifold. We see then that the Reeb foliation is totally geodesic with bundle like metric if and only if the Reeb vector field $T$ is a Killing field, that is,

$\mathcal{L}_T g=0.$
In that case $(\mathbb M, \theta,g)$ is called a K-contact Riemannian manifold.

Since Riemannian foliations with a bundle like metric can locally be desribed by a Riemannian submersion, we can define a horizontal Laplacian $\Delta_{\mathcal H}$. This allows to define horizontal Brownian motions.
Let $\mathbb M$ be a smooth and connected manifold with dimension $n+m$. In the sequel, we assume that $\mathbb M$ is equipped with a Riemannian foliation $\mathcal{F}$ with bundle-like metric $g$ and totally geodesic $m$-dimensional leaves.
The sub-bundle $\mathcal{V}$ formed by vectors tangent to the leaves is referred to as the set of  vertical directions. The sub-bundle $\mathcal{H}$ which is normal to $\mathcal{V}$ is referred to as the set of horizontal directions. The metric $g$ can be split as

$g=g_\mathcal{H} \oplus g_{\mathcal{V}}.$
On the Riemannian manifold $(\mathbb M,g)$ there is the Levi-Civita connection that we denote by $D$, but this connection is not adapted to the study of follations because the horizontal and the vertical bundle may not be parallel. More adapted to the geometry of the foliation is the Bott’s connection that we now define. It is an easy exercise to check that, since the foliation is totally geodesic, there exists a unique affine connection $\nabla$ such that:

• $\nabla$ is metric, that is, $\nabla g =0$;
• For $X,Y \in \Gamma^\infty(\mathcal H)$, $\nabla_X Y \in \Gamma^\infty(\mathcal H)$;
• For $U,V \in \Gamma^\infty(\mathcal V)$, $\nabla_U V \in \Gamma^\infty(\mathcal V)$
•  For $X,Y \in \Gamma^\infty(\mathcal H)$, $T(X,Y) \in \Gamma^\infty(\mathcal V)$ and for $U,V \in \Gamma^\infty(\mathcal V)$, $T(U,V) \in \Gamma^\infty(\mathcal H)$, where $T$ denotes the torsion tensor of $\nabla$
• For $X \in \Gamma^\infty(\mathcal H), U \in \Gamma^\infty(\mathcal V)$, $T(X,U)=0$.

In terms of the Levi-Civita connection, the Bott connection writes

$\nabla_X Y = \begin{cases} ( D_X Y)_{\mathcal{H}} , \quad X,Y \in \Gamma^\infty(\mathcal{H}) \\ [X,Y]_{\mathcal{H}}, \quad X \in \Gamma^\infty(\mathcal{V}), Y \in \Gamma^\infty(\mathcal{H}) \\ [X,Y]_{\mathcal{V}}, \quad X \in \Gamma^\infty(\mathcal{H}), Y \in \Gamma^\infty(\mathcal{V}) \\ ( D_X Y)_{\mathcal{V}}, \quad X,Y \in \Gamma^\infty(\mathcal{V}) \end{cases}$
Observe that for horizontal vector fields $X,Y$ the torsion $T(X,Y)$ is given by

$T(X,Y)=-[X,Y]_\mathcal{V}.$
Also observe that for $X,Y \in \Gamma^\infty(\mathcal{V})$ we actually have $( D_X Y)_{\mathcal{V}}= D_X Y$ because the leaves are assumed to be totally geodesic.

For local computations, it is convenient to work in normal frames.

Lemma: [B., Kim, Wang 2016]  Let $x \in \mathbb M$. Around $x$, there exist a local orthonormal horizontal frame $\{X_1,\cdots,X_n \}$ and a local orthonormal vertical frame $\{Z_1,\cdots,Z_m \}$ such that the following structure relations hold

$[X_i,X_j]=\sum_{k=1}^n \omega_{ij}^k X_k +\sum_{k=1}^m \gamma_{ij}^k Z_k$
$[X_i,Z_k]=\sum_{j=1}^m \beta_{ik}^j Z_j,$
where $\omega_{ij}^k, \gamma_{ij}^k, \beta_{ik}^j$ are smooth functions such that:

$\beta_{ik}^j=- \beta_{ij}^k.$
Moreover, at $x$, we have
$\omega_{ij}^k=0, \beta_{ij}^k=0.$

For later use, we record the fact that in this frame the Christofell symbols of the Bott connection are given by
$\begin{cases} \nabla_{X_i} X_j =\frac{1}{2} \sum_{k=1}^n \left( \omega_{ij}^k +\omega_{ki}^j+\omega_{kj}^i\right)X_k \\ \nabla_{Z_j} X_i =0 \\ \nabla_{X_i} Z_j=\sum_{k=1}^m \beta_{ij}^{k} Z_k \end{cases}$

Also observe that, since the foliation is totally geodesic, the horizontal Laplacian is locally given by

$\Delta_\mathcal{H}=\sum_{i=1}^n X_i^2 -\nabla_{X_i} X_i$

A horizontal orthonormal map at $x \in \mathbb M$ is an isometry $u: \mathbb{R}^n \to \mathcal{H}_x$. The horizontal orthonormal map bundle will be denoted by $\mathcal{O}_\mathcal{H} (\mathbb M)$.
The Bott connection allows to lift vector fields on $\mathbb M$ into vector fields on $\mathcal{O}_\mathcal{H} (\mathbb M)$. Let $e_1,\cdots,e_n$ be the canonical basis of $\mathbb{R}^n$. We denote by $A_i$ the vector field on $\mathcal{O}_\mathcal{H} (\mathbb M)$ such that $A_i (x,u)$ is the lift of $u(e_i) \in \mathcal{H}_x$.

We can locally write the vector fields $A_i$‘s in terms of the normal frames constructed in the previous subsection. We consider $x \in \mathbb M$ and a normal horizontal orthonormal frame $X_1,\cdots,X_n$ around $x$ as in the previous section.

If $u: \mathbb{R}^n \to \mathcal{H}_y$ is an isometry, we can find an orthogonal matrix $e_i^j$ such that $u(e_i)=\sum_{j=1}^n e_i^j X_j$. It is then easy to prove that

$A_i=\sum_{j=1}^n e_i^j \bar{X}_j-\sum_{j,k,l,m=1}^n e_i^j e_m^l \Gamma_{jl}^k \frac{\partial}{\partial e_m^k}$
where the $\Gamma_{jl}^k$‘s are the Christoffels symbols of the Bott connection and $\bar{X}_j$ is the vector field on $\mathcal{O}_\mathcal{H} (\mathbb M)$ defined by

$\bar{X}_j f (y,u)=\lim_{t \to 0} \frac{ f( e^{tX_j}(y), u)-f(y,u)}{t}$
In particular, at the center $x$ of the frame we have,

$A_i=\sum_{j=1}^n e_i^j \bar{X}_j.$

The main result is the following.

Proposition:  Let $\pi: \mathcal{O}_\mathcal{H} (\mathbb M) \to \mathbb M$ be the bundle projection map. For a smooth $f:\mathbb M \to \mathbb{R}$,
$\left( \sum_{i=1}^n A_i^2 \right)(f\circ \pi)=\Delta_\mathcal{H} f \circ \pi.$

Proof: It is enough to prove this identity at the center of the frame $x$. Using the fact that $\Gamma_{jl}^k(x)=0$, we see that, at $x$,

$\sum_{i=1}^n A_i^2 =\sum_{j=1}^n \bar{X}_j^2.$
The conclusion follows then easily $\square$

As a corollary, we obtain:

Corollary: Let $(B_t)_{t \ge 0}$ be a $n$-dimensional Brownian motion and let $(U_t)_{t \ge 0}$ be a solution of the stochastic differential equation

$dU_t =\sum_{i=1}^n A_i (U_t) \circ dB^i_t, \quad t < \mathbf{e}$
then $X_t=\pi(U_t)$ is a horizontal Brownian motion on $\mathbb M$, that is a diffusion process with generator $\frac{1}{2}\Delta_\mathcal{H}$.

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