In this lecture, we first quickly review what was done in the previous one. We then introduce the horizontal and vertical Laplacians and prove basic commutations results. We explain them the notion of Riemannian foliation. In the second part of the lecture we study the Heisenberg group and show how to compute its heat kernel.
Let be a Riemannian submersion. If we define its vertical gradient as the projection of its gradient onto the vertical distribution and its horizontal gradient as the projection of the gradient onto the horizontal distribution. We define then the vertical Laplacian as the generator of the Dirichlet form
where is the Riemannian volume measure on . Similarly, we define the horizonal Laplacian as the generator of the Dirichlet form
If is a local orthonormal frame of basic vector fields and a local orthonormal frame of the vertical distribution, then we have
where the adjoints are understood in . Classically, we have
where is the Levi-Civita connection. As a consequence, we obtain
where denotes the horizontal part of the vector. In a similar way we obviously have
We can observe that the Laplace-Beltrami operator of can be written
It is worth noting that, in general, is not the lift of the Laplace-Beltrami operator on . Indeed, let us denote by the vector fields on which are -related to .
Since it is easy to check that is -related to , we deduce that lies above , i.e. for every , , if and only if the vector
is vertical. This condition is equivalent to the fact that the mean curvature of each fiber is zero, or in other words that the fibers are minimal submanifolds of . This happens for instance for submersions with totally geodesic fibers.
We also note that from Hormander’s theorem, the operator is subelliptic if the horizontal distribution is bracket generating. Of course, the vertical Laplacian is never subelliptic because the vertical distribution is always integrable.
The following result, though simple, will turn out to be extremely useful in the sequel when dealing with curvature dimension estimates and functional inequalities.
Theorem: The Riemannian submersion has totally geodesic fibers if and only if for every ,
Proof: If is a local orthonormal frame of basic vector fields and a local orthonormal frame of the vertical distribution, then we easily compute that
As a consequence,
if and only if for every basic vector field ,
This condition is equivalent to the fact that the flow generated by induces an isometry between the fibers, and so from Hermann’s Theorem this is equivalent to the fact that the fibers are totally geodesic
The second commutation result that characterizes totally geodesic submersions is due to Berard-Bergery and Bourguignon.
Theorem: The Riemannian submersion has totally geodesic fibers if and only if any basic vector field commutes with the vertical Laplacian . In particular, if has totally geodesic fibers, then for every ,
Proof. Assume that the submersion is totally geodesic. Let be a basic vector field and be the flow it generates. Since induces an isometry between the fibers, we have
Differentiating at yields .
Conversely, assume that for every basic field , . Let be a local orthonormal frame of basic vector fields and be a local orthonormal frame of the vertical distribution. The second order part of the operator must be zero. Given the expression of the vertical Laplacian, this implies
So leaves the symbol of invariant which is the metric on the vertical distribution. This implies that the flow generated by induces isometries between the fibers.
Finally, as we have seen, if the submersion is totally geodesic then in a local basic orthornomal frame
Since the vectors are basic, from the previous result commutes with
We now turn to the notion of Riemannian foliation.
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 n+m 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 $mathbb n+m$ 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.
Since Riemannian foliations with a bundle like metric can locally be desribed by a Riemannian submersion, we can define a horizontal Laplacian and a vertical Laplacian . Observe that they commute on smooth functions if the foliation is totally geodesic. More generally all the local properties of a Riemannian submersion extend to Riemannian foliations.
In the second part of the lecture, we deal with model spaces.
One of the simplest non trivial Riemannian submersions with totally geodesic fibers and bracket generating horizontal distribution is associated to the Heisenberg group. The Heisenberg group is the set
endowed with the group law
The vector fields
form an orthonormal frame of left invariant vector fields for the left invariant metric on . Note that the following commutations hold
is then a Riemannian submersion with totally geodesic fibers. The horizontal Laplacian is the left invariant operator
and the vertical Laplacian is the left invariant operator
The horizontal distribution
is bracket generating at every point, so is a subelliptic operator. The operator is invariant by the action of the orthogonal group of on the variables . Introducing the variable , we see then that the radial part of is given by
This means that if is a smooth map and is the submersion then
From this invariance property in order to study the heat kernel and fundamental solution of at it suffices to study the heat kernel and the fundamental solution of at .
We denote by the heat kernel at 0 of . It was first computed explicitly by Gaveau building on previous works by Paul Levy.
Proposition: For and ,
Proof: Since commutes with , the idea is to use a Fourier transform in . We see then that
where is the fundamental solution at 0 of the parabolic partial differential equation
We thus want to compute the semigroup generated by the Schrodinger operator
The trick is now to observe that for every ,
The operator turns out to be the radial part of the Ornstein-Uhlenbeck operator whose heat kernel at 0 is a Gaussian density with mean 0 and variance . This means that the heat kernel at 0 of is given by
The second simplest and geometrically relevant example is given by the celebrated Hopf fibration. Let us consider the odd dimensional unit sphere
There is an isometric group action of on which is defined by
The generator of this action shall be denoted by . We thus have for every
The quotient space is the projective complex space and the projection map is a Riemannian submersion with totally geodesic fibers isometric to . The fibration
is called the Hopf fibration.
To study the geometry of the Hopf fibration, in particular the horizontal Laplacian , it is convenient to introduce a set of coordinates that reflects the action of the isometry group of on .
Let be the local inhomogeneous coordinates for given by , and be the local fiber coordinate. i.e., parametrizes the complex lines passing through the north pole, while determines a point on the line that is of unit distance from the north pole. More explicitly, these coordinates are given by the map
where , , and . In these coordinates, it is clear that and that the vertical Laplacian is
Our first goal is now to compute the horizontal Laplacian . This operator is invariant by the action on the variables of the group of isometries of that fix the north pole of (this group is ). Therefore the heat kernel at the north pole only depends on the variables .
Proposition: The radial part of the horizontal Laplacian is the operator
Proof: The easiest route is to compute first the radial part of the Laplace-Beltrami operator and then to use the formula
In our parametrization of we have,
Therefore if denotes the Riemannian distance based at the north pole, we have and if denotes the Riemannian distance based at the point with real coordinates then we have . The formula for the Laplace-Beltrami operator acting on functions depending on the Riemannian distance based at a point is well-known and we deduce from it that acts on functions depending only on as
In the variables this last operator writes
This concludes the proof