In many interesting cases, we do not actually have a globally defined Riemannian sumersion but a Riemannian foliation.
Definition: Let be a smooth and connected dimensional manifold. A -dimensional foliation on is defined by a maximal collection of pairs of open subsets of and submersions onto open subsets of satisfying:
- If , there exists a local diffeomorphism of such that on .
The maps are called disintegrating maps of . The connected components of the sets , , are called the plaques of the foliation. A foliation arises from an integrable sub-bundle of , to be denoted by and referred to as the vertical distribution. These are the vectors tangent to the leaves, the maximal integral sub-manifolds of .
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 be a smooth and connected dimensional Riemannian manifold. A -dimensional foliation on is said to be Riemannian with a bundle like metric if the disintegrating maps are Riemannian submersions onto with its given Riemannian structure. If moreover the leaves are totally geodesic sub-manifolds of , then we say that the Riemannian foliation is totally geodesic with a bundle like metric.
Observe that if we have a Riemannian submersion , then 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 be a -dimensional smooth contact manifold. On there is a unique smooth vector field , the so-called Reeb vector field, that satisfies
where denotes the Lie derivative with respect to . On there is a foliation, the Reeb foliation, whose leaves are the orbits of the vector field . As it is well-known, it is always possible to find a Riemannian metric and a -tensor field on so that for every vector fields $X, Y$
The triple 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 is a Killing field, that is,
In that case 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 . This allows to define horizontal Brownian motions.
Let be a smooth and connected manifold with dimension . In the sequel, we assume that is equipped with a Riemannian foliation with bundle-like metric and totally geodesic -dimensional leaves.
The sub-bundle formed by vectors tangent to the leaves is referred to as the set of vertical directions. The sub-bundle which is normal to is referred to as the set of horizontal directions. The metric can be split as
On the Riemannian manifold there is the Levi-Civita connection that we denote by , 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 such that:
- is metric, that is, ;
- For , ;
- For ,
- For , and for , , where denotes the torsion tensor of
- For , .
In terms of the Levi-Civita connection, the Bott connection writes
Observe that for horizontal vector fields the torsion is given by
Also observe that for we actually have 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 . Around , there exist a local orthonormal horizontal frame and a local orthonormal vertical frame such that the following structure relations hold
where are smooth functions such that:
Moreover, at , we have
For later use, we record the fact that in this frame the Christofell symbols of the Bott connection are given by
Also observe that, since the foliation is totally geodesic, the horizontal Laplacian is locally given by
A horizontal orthonormal map at is an isometry . The horizontal orthonormal map bundle will be denoted by .
The Bott connection allows to lift vector fields on into vector fields on . Let be the canonical basis of . We denote by the vector field on such that is the lift of .
We can locally write the vector fields ‘s in terms of the normal frames constructed in the previous subsection. We consider and a normal horizontal orthonormal frame around as in the previous section.
If is an isometry, we can find an orthogonal matrix such that . It is then easy to prove that
where the ‘s are the Christoffels symbols of the Bott connection and is the vector field on defined by
In particular, at the center of the frame we have,
The main result is the following.
Proposition: Let be the bundle projection map. For a smooth ,
Proof: It is enough to prove this identity at the center of the frame . Using the fact that , we see that, at ,
The conclusion follows then easily
As a corollary, we obtain:
Corollary: Let be a -dimensional Brownian motion and let be a solution of the stochastic differential equation
then is a horizontal Brownian motion on , that is a diffusion process with generator .