Lecture 1. Introduction and Riemannian submersions

In this first lecture, I give a general introduction about the plan of the lectures and the materials to be covered and start with the study of Riemannian submersions. Thanks to  Ugo Boscain, the videos of the lectures are available and I greatly thank him for that.

It is a fact that many interesting hypoelliptic diffusion operators may be studied by introducing a well-chosen Riemannian foliation. In particular, several sub-Laplacians on sub-Riemannians manifolds often appear as horizontal Laplacians of a foliation and several of the Kolmogorov type hypoelliptic diffusion operators which are used in the theory of kinetic equations appear as the sum of the vertical Laplacian of a foliation and of a first order term.

The goal of the lectures is to survey some geometric analysis tools to study this kind of diffusion operators. We specially would like to stress the importance of subelliptic Bochner’s type identities in this framework and show how they can be used to deduce a variety of results ranging from topological informations on a sub-Riemannian manifold to hypocoercive estimates and convergence to equilibrium for kinetic Fokker-Planck equations. As an illustration of those methods we  will give a proof of a sub-Riemannian Bonnet-Myers type compactness theorem and, in the last lecture, study a version of the Bakry-Emery criterion for Kolmogorov type operators.

For the proof of the sub-Riemannian Bonnet-Myers theorem we will  adapt an approach developed in a joint program with Nicola Garofalo. The object of this program  has been to propose a generalized curvature dimension inequality that fits a number of interesting subelliptic situations including the ones considered in these lectures. While some of them will be discussed here, the numerous applications of the generalized curvature dimension inequality are beyond the scope of the lectures and we will only give the relevant pointers to the literature. We focus here more on the Bonnet-Myers theorem and the geometric framework in which this curvature-dimension estimate is available.

The course will be  organized as follows.

1. We introduce first  the concept of Riemannian foliation and define the horizontal and vertical Laplacians. Basic theorems like the Berard-Bergery-Bourguignon commutation theorem will be proved.
2. We will study in details some examples of Riemannian foliations with totally geodesic leaves that can be seen as model spaces. Besides the Heisenberg group, these examples are associated to the Hopf fibrations on the sphere. We give explicit expressions for the radial parts of the horizontal and vertical Laplacians and for the horizontal heat kernels of these model spaces.
3. We will prove a transverse Weitzenbock formula for the horizontal Laplacian of a Riemannian foliation with totally geodesic leaves. It is the main geometric analysis tool for the study of the horizontal Laplacian. As a first consequence of this Weitzenbock formula, we prove that if natural assumptions are satisfied, then the horizontal Laplacian satisfies the generalized curvature dimension inequality. As a second consequence, we will prove sharp lower bounds for the first eigenvalue of the horizontal Laplacian.
4. We will introduce the horizontal semigroup of a Riemannian foliation with totally geodesic leaves and discuss fundamental questions like essential self-adjointness for the horizontal Laplacian and stochastic completeness. We also prove Li-Yau gradient bounds for this horizontal semigroup.
5. By using semigroup methods, we will prove a sub-Riemannian Bonnet-Myers theorem in the context of Riemannian foliations with totally geodesic leaves.
6. The last course will be an introduction to the analysis of hypoelliptic Kolmogorov type operators on Riemannian foliations. We mainly focus on the problem of convergence to equilibrium for the parabolic equation associated to the operator and on methods to prove hypocoercive estimates. The example of the kinetic Fokker-Planck equation is given as an illustration.

Let $(\mathbb{M} , g)$ and $(\mathbb{B},j)$ be smooth and connected Riemannian manifolds.

Definition:  A smooth surjective map $\pi:(\mathbb{M} , g) \to(\mathbb{B},j)$ is called a Riemannian submersion if its derivative maps $T_x\pi : T_x \mathbb{M} \to T_{\pi(x)} \mathbb{B}$ are orthogonal projections.

Example: (Warped products) Let $(\mathbb{M}_1 , g_1)$ and $(\mathbb{M}_2 , g_2)$ be Riemannian manifolds and $f$ be a smooth and positive function on $\mathbb{M}_1$. Then the first projection $(\mathbb{M}_1 \times \mathbb{M}_2,g_1 \oplus f g_2) \to (\mathbb{M}_1, g_1)$ is a Riemannian submersion.

Example: (Quotient by an isometric action) Let $(\mathbb{M} , g)$ be a Riemannian manifold and $\mathbb G$ be a closed subgroup of the isometry group of $(\mathbb M , g)$. Assume that the projection map $\pi$ from $\mathbb M$ to the quotient space $\mathbb M /\mathbb{G}$ is a smooth submersion. Then there exists a unique Riemannian metric $j$ on $\mathbb M /\mathbb{G}$ such that $\pi$ is a Riemannian submersion.

If $\pi$ is a Riemannian submersion and $b \in \mathbb B$, the set $\pi^{-1}(\{ b \})$ is called a fiber.

For $x \in \mathbb{M}$, $\mathcal{V}_x =\mathbf{Ker} (T_x\pi)$ is called the vertical space at $x$. The orthogonal complement of $\mathcal{H}_x$ shall be denoted $\mathcal{H}_x$ and will be referred to as the horizontal space at $x$.

We have an orthogonal decomposition

$T_x \mathbb M=\mathcal{H}_x \oplus \mathcal{V}_x$

and a corresponding splitting of the metric

$g=g_{\mathcal{H}} \oplus g_{\mathcal{V}}.$
The vertical distribution $\mathcal V$ is of course integrable since it is the tangent distribution to the fibers, but the horizontal distribution is in general not integrable. Actually, in all the situations we will consider the horizontal distribution is everywhere bracket-generating in the sense that for every $x \in \mathbb M$, $\mathbf{Lie} (\mathcal{H}) (x)=T_x \mathbb M$. In that case it is natural to study the sub-Riemannian geometry of the triple $(\mathbb M, \mathcal{H}, g_{\mathcal{H}})$. As we will see, many interesting examples of sub-Riemannian structures arise in this framework and this is really the situation which is interesting for us.

We shall mainly be interested in submersion with totally geodesic fibers.

Definition: A Riemannian submersion $\pi: (\mathbb M , g)\to (\mathbb B,j)$ is said to have totally geodesic fibers if for every $b \in \mathbb B$, the set $\pi^{-1}(\{ b \})$ is a totally geodesic submanifold of $\mathbb M$.

Example: (Quotient by an isometric action) Let $(\mathbb M , g)$ be a Riemannian manifold and $\mathbb G$ be a closed one-dimensional subgroup of the isometry group of $(\mathbb M , g)$ which is generated by a complete Killing vector field $X$. Assume that the projection map $\pi$ from $\mathbb M$ to $\mathbb M /\mathbb{G}$ is a smooth submersion. Then the fibers are totally geodesic if and only if the integral curves of $X$ are geodesics, which is the case if and only if $X$ has a constant length.

Example: (Principal bundle) Let $\mathbb M$ be a principal bundle over $\mathbb B$ with fiber $\mathbf F$ and structure group $\mathbb G$. Then, given a Riemannian metric $j$ on $\mathbb B$, a $\mathbb G$-invariant metric $k$ on $\mathbf{F}$ and a $\mathbb G$ connection form $\theta$, there exists a unique Riemannian metric $g$ on $\mathbb M$ such that the bundle projection map $\pi: \mathbb M \to \mathbb B$ is a Riemannian submersion with totally geodesic fibers isometric to $(\mathbf{F},k)$ and such that the horizontal distribution of $\theta$ is the orthogonal complement of the vertical distribution. In the case of the tangent bundle of a Riemannian manifold, the construction yields the Sasaki metric on the tangent bundle.

As we will see, for a Riemannian submersion with totally geodesic fibers, all the fibers are isometric. The argument, due to Hermann  relies on the notion of basic vector field that we now introduce.

Let $\pi: (\mathbb M , g)\to (\mathbb B,j)$ be a Riemannian submersion. A vector field $X \in \Gamma^\infty(T\mathbb M)$ is said to be projectable if there exists a smooth vector field $\overline{X}$ on $\mathbb B$ such that for every $x \in \mathbb M$, $T_x \pi ( X(x))= \overline {X} (\pi (x))$. In that case, we say that $X$ and $\overline{X}$ are $\pi$-related.

Definition: A vector field $X$ on $\mathbb M$ is called basic if it is projectable and horizontal.

If $\overline{X}$ is a smooth vector field on $\mathbb B$, then there exists a unique basic vector field $X$ on $\mathbb M$ which is $\pi$-related to $\overline{X}$. This vector is called the lift of $\overline{X}$.

Notice that if $X$ is a basic vector field and $Z$ is a vertical vector field, then $T_x\pi ( [X,Z](x))=0$ and thus $[X,Z]$ is a vertical vector field. The following result is due to Hermann.

Proposition: The submersion $\pi$ has totally geodesic fibers if and only if the flow generated by any basic vector field induces an isometry between the fibers.

Proof: We denote by $D$ the Levi-Civita connection on $\mathbb M$. Let $X$ be a basic vector field. If $Z_1,Z_2$ are vertical fields, the Lie derivative of $g$ with respect to $X$ can be computed as

$(\mathcal{L}_X g)(Z_1,Z_2)=\langle D_{Z_1} X ,Z_2 \rangle +\langle D_{Z_2} X ,Z_1 \rangle.$
Because $X$ is orthogonal to $Z_2$, we now have $\langle D_{Z_1} X ,Z_2 \rangle=-\langle X ,D_{Z_1} Z_2 \rangle$. Similarly $\langle D_{Z_2} X ,Z_1 \rangle=-\langle X ,D_{Z_2} Z_1 \rangle$. We deduce

$(\mathcal{L}_X g)(Z_1,Z_2) =-\langle X ,D_{Z_1} Z_2 +D_{Z_2} Z_1 \rangle$

$=-2 \langle X ,D_{Z_1} Z_2 \rangle.$

Thus the flow generated by any basic vector field induces an isometry between the fibers if and only if $D_{Z_1} Z_2$ is always vertical which is equivalent to the fact that the fibers are totally geodesic submanifolds.

This entry was posted in Hypoelliptic operators. Bookmark the permalink.