In this lecture we continue the study of the Hopf fibration and compute the horizontal heat kernel of this fibration. In the second part of the lecture, we come back to the general framework of Riemannian foliations and introduce a canonical sub-Laplacian on one-form for which we prove Bochner-Weitzenbock type identities.
As we have seen in the previous lecture, the radial part of the Laplacian of the Hopf fibration is given
We can observe that is symmetric with respect to the measure
where the normalization is chosen in such a way that
As mentioned above, the heat kernel at the north pole of only depends on , that is , where is the heat kernel at 0 of .
Proposition: For , , :
is a Jacobi polynomial.
Proof: Similarly to the Heisenberg group case, we observe that commutes with , so the idea is to expand as a Fourier series in . We can write
where is the fundamental solution at 0 of the parabolic equation
By writing in the form
The eigenvectors of solve the Jacobi differential equation, and are thus given by the Jacobi polynomials
By using the fact that the family is an orthogonal basis of , such that
we easily compute the fundamental solution of the operator
Note that as a by-product of the previous result we obtain that the spectrum of is given by
We can give another representation of the heat kernel which is easier to handle analytically. The key idea is to observe that since and commute, we formally have
This gives a way to express the horizontal heat kernel in terms of the Riemannian one.
Let us recall that the Riemannian heat kernel on the sphere is given by
where, is the Riemannian distance based at the north pole and
is a Gegenbauer polynomial. Another expression of is
where is a theta function.
Using the commutation and the formula , we then infer the following proposition which is easy to prove
Proposition: For , , ,
Applications of this formula are given in my paper with Jing Wang. We can, in particular, deduce from it small asymptotics of the kernel when . Interestingly, these small-time asymptotics allow to compute explicitly the sub-Riemannian distance.
We now come back to the general framework of a Riemannian foliation.
Let be a smooth, connected manifold with dimension . We assume that is equipped with a Riemannian foliation with bundle like metric and totally geodesic -dimensional leaves.
As usual, the sub-bundle formed by vectors tangent to the leaves will be referred to as the set of vertical directions and the sub-bundle which is normal to will be referred to as the set of horizontal directions. The metric can be split as
We define the canonical variation of as the one-parameter family of Riemannian metrics:
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 connection that we now define. In terms of the Levi-Civita connection, the Bott connection writes
where the subscript (resp. ) denotes the projection on (resp. ). 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. Finally, it is easy to check that for every , the Bott connection satisfies .
Example: Let be a K-contact Riemannian manifold. The Bott connection coincides with the Tanno’s connection, which is the unique connection that satisfies:
We now introduce some tensors and definitions that will play an important role in the sequel.
For , there is a unique skew-symmetric endomorphism such that for all horizontal vector fields and ,
where is the torsion tensor of . We then extend to be 0 on . If is a local vertical frame, the operator does not depend on the choice of the frame and shall concisely be denoted by . For instance, if is a K-contact manifold equipped with the Reeb foliation, then is an almost complex structure, .
The horizontal divergence of the torsion is the tensor which is defined in a local horizontal frame by
The -adjoint of will be denoted .
Definition: We say that the Riemannian foliation is of Yang-Mills type if .
Example: Let be a K-contact Riemannian manifold. It is easy to see that the Reeb foliation is of Yang-Mills type if and only if . Equivalently this condition writes . If is a strongly pseudo convex CR manifold with pseudo-Hermitian form , then the Tanno’s connection is the Tanaka-Webster connection. In that case, we have then and thus . CR manifold of K-contact type are called Sasakian manifolds. Thus the Reeb foliation on any Sasakian manifold is of Yang-Mills type.
In the sequel, we shall need to perform computations on one-forms. For that purpose we introduce some definitions and notations on the cotangent bundle.
We say that a one-form to be horizontal (resp. vertical) if it vanishes on the vertical bundle (resp. on the horizontal bundle ). We thus have a splitting of the cotangent space
The metric induces then a metric on the cotangent bundle which we still denote . By using similar notations and conventions as before we have for every in ,
By using the duality given by the metric , tensors can also be seen as linear maps on the cotangent bundle . More precisely, if is a tensor, we will still denote by the fiberwise linear map on the cotangent bundle which is defined as the -adjoint of the dual map of . The same convention will be made for any tensor.
We define then the horizontal Ricci curvature as the fiberwise symmetric linear map on one-forms such that for every smooth functions ,
where is the Ricci curvature of the connection .
If is a horizontal vector field and , we consider the fiberwise linear map from the space of one-forms into itself which is given for and by
We observe that is skew-symmetric for the metric so that is a -metric connection.
If is a one-form, we define the horizontal gradient of in a local frame as the tensor
We denote by the symmetrization of .
Similarly, we will use the notation
Finally, we will still denote by the covariant extension on one-forms of the horizontal Laplacian. In a local horizontal frame, we have thus
For , we consider the following operator which is defined on one-forms by
where the adjoint is understood with respect to the metric . It is easily seen that, in a local horizontal frame,
Observe that if the foliation is of Yang-Mills type then
As a consequence, in the Yang-Mills case the operator is seen to be symmetric for the metric .
Theorem: For every , we have
Proof: We only sketch the proof and refer to this paper for the details. If is a local vertical frame of the leaves, we denote
where is the the projection of to the vertical cotangent bundle. It does not depend on the choice of the frame and therefore defines a globally defined tensor.
Also, let us consider the map which is given in a local coframe ,
A direct computation shows then that
Thus, we just need to prove that if is the operator defined on one-forms by
then for any ,
A computation in local frame shows that
which completes the proof
We also can prove the following Bochner’s type identity whose proof can be found in the paper.
Theorem: For any ,