Lecture 2. Riemannian foliations. Model spaces

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 \pi: (\mathbb M , g)\to (\mathbb B,j) be a Riemannian submersion. If f \in C^\infty(\mathbb M) we define its vertical gradient \nabla_{\mathcal{V}} as the projection of its gradient onto the vertical distribution and its horizontal gradient \nabla_{\mathcal{H}} as the projection of the gradient onto the horizontal distribution. We define then the vertical Laplacian \Delta_{\mathcal{V}} as the generator of the Dirichlet form

\mathcal{E}_{\mathcal{V}}(f,g)=-\int_{\mathbb M} \langle \nabla_{\mathcal{V}} f , \nabla_{\mathcal{V}} g \rangle d\mu
where \mu is the Riemannian volume measure on \mathbb M. Similarly, we define the horizonal Laplacian \Delta_{\mathcal{H}} as the generator of the Dirichlet form

\mathcal{E}_{\mathcal{H}}(f,g)=-\int_{\mathbb M} \langle \nabla_{\mathcal{H}} f , \nabla_{\mathcal{V}} g \rangle d\mu.
If X_1,\cdots,X_n is a local orthonormal frame of basic vector fields and Z_1,\cdots,Z_m a local orthonormal frame of the vertical distribution, then we have

\Delta_{\mathcal{H}}=-\sum_{i=1}^n X_i^* X_i
and

\Delta_{\mathcal{V}}=-\sum_{i=1}^m Z_i^* Z_i,
where the adjoints are understood in L^2(\mu). Classically, we have

X_i^*=-X_i+\sum_{k=1}^n \langle D_{X_k} X_k, X_i\rangle +\sum_{k=1}^m \langle D_{Z_k} Z_k, X_i\rangle,
where D is the Levi-Civita connection. As a consequence, we obtain

\Delta_{\mathcal{H}}=\sum_{i=1}^n X_i^2 -\sum_{i=1}^n (D_{X_i}X_i)_{\mathcal{H}} -\sum_{i=1}^m (D_{Z_i}Z_i)_{\mathcal{H}},
where (\cdot)_{\mathcal{H}} denotes the horizontal part of the vector. In a similar way we obviously have

\Delta_{\mathcal{V}}=\sum_{i=1}^m Z_i^2 -\sum_{i=1}^n (D_{X_i}X_i)_{\mathcal{V}} -\sum_{i=1}^m (D_{Z_i}Z_i)_{\mathcal{V}}.
We can observe that the Laplace-Beltrami operator \Delta of \mathbb M can be written

\Delta=\Delta_{\mathcal{H}}+\Delta_{\mathcal{V}}.
It is worth noting that, in general, \Delta_{\mathcal{H}} is not the lift of the Laplace-Beltrami operator \Delta_\mathbb{B} on \mathbb B. Indeed, let us denote by \overline{X}_1,\cdots,\overline{X}_n the vector fields on \mathbb B which are \pi-related to X_1,\cdots,X_n .

We have
\Delta_{\mathbb B}=\sum_{i=1}^n \overline{X}_i^2 -\sum_{i=1}^n D_{\overline{X}_i}\overline{X}_i.
Since it is easy to check that D_{\overline{X}_i}\overline{X}_i is \pi-related to (D_{X_i}X_i)_{\mathcal{H}}, we deduce that \Delta_{\mathcal{H}} lies above \Delta_{\mathbb B}, i.e. for every f \in C^\infty(\mathbb B), \Delta_{\mathcal{H}} (f \circ \pi)=(\Delta_{\mathbb B} f )\circ \pi , if and only if the vector

T=\sum_{i=1}^m D_{Z_i}Z_i
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 \mathbb M. This happens for instance for submersions with totally geodesic fibers.

We also note that from Hormander’s theorem, the operator \Delta_{\mathcal{H}} 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 \pi has totally geodesic fibers if and only if for every f \in C^\infty(\mathbb M),
\langle \nabla_{\mathcal{H}} f , \nabla_{\mathcal{H}} \| \nabla_{\mathcal{V}} f \|^2 \rangle=\langle \nabla_{\mathcal{V}} f , \nabla_{\mathcal{V}} \| \nabla_{\mathcal{H}} f \|^2 \rangle

Proof: If X_1,\cdots,X_n is a local orthonormal frame of basic vector fields and Z_1,\cdots,Z_m a local orthonormal frame of the vertical distribution, then we easily compute that

\langle \nabla_{\mathcal{H}} f , \nabla_{\mathcal{H}} \| \nabla_{\mathcal{V}} f \|^2 \rangle-\langle \nabla_{\mathcal{V}} f , \nabla_{\mathcal{V}} \| \nabla_{\mathcal{H}} f \|^2 \rangle=2\sum_{i=1}^n \sum_{j=1}^m (X_i f) (Z_j f) ([X_i,Z_j] f).

As a consequence,

\langle \nabla_{\mathcal{H}} f , \nabla_{\mathcal{H}} \| \nabla_{\mathcal{V}} f \|^2 \rangle=\langle \nabla_{\mathcal{V}} f , \nabla_{\mathcal{V}} \| \nabla_{\mathcal{H}} f \|^2 \rangle
if and only if for every basic vector field X,

\sum_{j=1}^m (Z_j f) ([X,Z_j] f)=0.
This condition is equivalent to the fact that the flow generated by X induces an isometry between the fibers, and so from Hermann’s Theorem this is equivalent to the fact that the fibers are totally geodesic \square

The second commutation result that characterizes totally geodesic submersions is due to Berard-Bergery and Bourguignon.

Theorem: The Riemannian submersion \pi has totally geodesic fibers if and only if any basic vector field X commutes with the vertical Laplacian \Delta_{\mathcal{V}}. In particular, if \pi has totally geodesic fibers, then for every f \in C^\infty(\mathbb M),

\Delta_{\mathcal{H}} \Delta_{\mathcal{V}} f=\Delta_{\mathcal{V}} \Delta_{\mathcal{H}} f.

Proof. Assume that the submersion is totally geodesic. Let X be a basic vector field and \xi_t be the flow it generates. Since \xi induces an isometry between the fibers, we have

\xi_t^* ( \Delta_{\mathcal{V}})= \Delta_{\mathcal{V}}.
Differentiating at t=0 yields [X,\Delta_{\mathcal{V}}]=0.

Conversely, assume that for every basic field X, [X,\Delta_{\mathcal{V}}]=0. Let X_1,\cdots,X_n be a local orthonormal frame of basic vector fields and Z_1,\cdots,Z_m be a local orthonormal frame of the vertical distribution. The second order part of the operator [X,\Delta_{\mathcal{V}}] must be zero. Given the expression of the vertical Laplacian, this implies

\sum_{i=1}^m [X,Z_i] Z_i=0.
So X leaves the symbol of \Delta_{\mathcal{V}} invariant which is the metric on the vertical distribution. This implies that the flow generated by X induces isometries between the fibers.

Finally, as we have seen, if the submersion is totally geodesic then in a local basic orthornomal frame

\Delta_{\mathcal{H}}=\sum_{i=1}^n X_i^2 -\sum_{i=1}^n (D_{X_i}X_i)_{\mathcal{H}}.
Since the vectors (D_{X_i}X_i)_{\mathcal{H}} are basic, from the previous result \Delta_{\mathcal{H}} commutes with \Delta_{\mathcal{V}} \square

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 \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 $mathbb n+m$ 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.

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} and a vertical Laplacian \Delta_\mathcal{V}. 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

\mathbb{H}^{2n+1}=\left\{ (x,y,z), x \in \mathbb{R}^n, y \in \mathbb{R}^n, z\in\mathbb{R} \right\}
endowed with the group law

(x_1,y_1,z_1) \star (x_2,y_2,z_2)=(x_1+x_2,y_1+y_2,z_1+z_2+\langle x_1,y_2 \rangle_{\mathbb R^n} -\langle x_2,y_1 \rangle_{\mathbb R^n}).

The vector fields

X_i=\frac{\partial}{\partial x_i} -y_i \frac{\partial}{\partial z}
Y_i=\frac{\partial}{\partial y_i} +x_i \frac{\partial}{\partial z}
and

Z=\frac{\partial}{\partial z}
form an orthonormal frame of left invariant vector fields for the left invariant metric on \mathbb{H}^{2n+1}. Note that the following commutations hold

[X_i,Y_j]=2\delta_{ij} Z, \quad [X_i,Z]=[Y_i,Z]=0.
The map
\pi :  \begin{array}{lll}  \mathbb{H}^{2n+1} &\to& \mathbb{R}^{2n} \\  (x,y,z) & \to & (x,y)  \end{array}
is then a Riemannian submersion with totally geodesic fibers. The horizontal Laplacian is the left invariant operator

\Delta_{\mathcal{H}} =\sum_{i=1}^n (X_i^2+Y_i^2)

=\sum_{i=1}^n \frac{\partial^2}{\partial x^2_i} +\frac{\partial^2}{\partial y^2_i} + 2\sum_{i=1}^n \left( x_i \frac{\partial}{\partial y_i}-y_i \frac{\partial}{\partial x_i}\right) \frac{\partial}{\partial z}+ (\| x\|^2+\| y \|^2) \frac{\partial^2}{\partial z^2}
and the vertical Laplacian is the left invariant operator

\Delta_\mathcal{V}=\frac{\partial^2}{\partial z^2} .
The horizontal distribution

\mathcal{H}=\mathbf{span} \{X_1, \cdots,X_n,Y_1,\cdots, Y_n\}
is bracket generating at every point, so \Delta_{\mathcal{H}} is a subelliptic operator. The operator \Delta_{\mathcal{H}} is invariant by the action of the orthogonal group of \mathbb{R}^{2n} on the variables (x,y). Introducing the variable r^2=\| x\|^2 +\|y\|^2, we see then that the radial part of \Delta_{\mathcal{H}} is given by

\overline{\Delta}_{\mathcal{H}}=\frac{\partial^2}{\partial r^2}+\frac{2n-1}{r} \frac{\partial}{\partial r} +r^2 \frac{\partial^2}{\partial z^2}.
This means that if f: \mathbb{R}_{\ge 0} \times \mathbb R \to \mathbb R is a smooth map and \rho is the submersion (x,y,z)\to (\sqrt{\| x\|^2+\|y\|^2},z) then

\Delta_{\mathcal{H}} (f \circ \rho)=(\overline{\Delta}_{\mathcal{H}} f) \circ \rho.
From this invariance property in order to study the heat kernel and fundamental solution of \Delta_{\mathcal{H}}  at 0 it suffices to study the heat kernel and the fundamental solution of \overline{\Delta}_{\mathcal{H}} at 0.

We denote by \overline{p}_t(r,z) the heat kernel at 0 of \overline{\Delta}_{\mathcal{H}}. It was first computed explicitly by Gaveau  building on previous works by Paul Levy.

Proposition: For r \ge 0 and z \in \mathbb{R},

\overline{p}_t (r,z)=\frac{1}{(2\pi)^{n+1}} \int_\mathbb R e^{i \lambda z} \left( \frac{\lambda}{\sinh (2\lambda t)} \right)^n e^{-\frac{\lambda r^2}{ 2} \coth (2\lambda t) } d\lambda

Proof: Since \frac{\partial}{\partial z} commutes with \overline{\Delta}_{\mathcal{H}}, the idea is to use a Fourier transform in z. We see then that

\overline{p}_t(r,z)=\frac{1}{2\pi} \int_{\mathbb R} e^{i \lambda z} \Phi_t (r,\lambda) d\lambda,
where \Phi_t (r,z,\lambda) is the fundamental solution at 0 of the parabolic partial differential equation

\frac{\partial \Phi}{ \partial t}=\frac{\partial^2 \Phi }{\partial r^2}+\frac{2n-1}{r} \frac{\partial \Phi }{\partial r} -\lambda^2 r^2 \Phi .

We thus want to compute the semigroup generated by the Schrodinger operator

\mathcal{L}_\lambda=\frac{\partial^2 }{\partial r^2}+\frac{2n-1}{r} \frac{\partial }{\partial r} -\lambda^2 r^2.
The trick is now to observe that for every f,

\mathcal{L}_\lambda \left( e^{\frac{\lambda r^2}{2}} f \right)= e^{\frac{\lambda r^2}{2}} \left( 2n\lambda +\mathcal{G}_\lambda \right)f,
where

\mathcal{G}_\lambda=\frac{\partial^2 }{\partial r^2}+\left( 2 \lambda r+\frac{2n-1}{r} \right)\frac{\partial }{\partial r}.
The operator \mathcal{G}_\lambda turns out to be the radial part of the Ornstein-Uhlenbeck operator \Delta_{\mathbb{R}^{2n}} +2 \lambda \langle x , \nabla_{\mathbb{R}^{2n}} \rangle whose heat kernel at 0 is a Gaussian density with mean 0 and variance \frac{1}{2\lambda}(e^{4\lambda t}-1). This means that the heat kernel at 0 of \mathcal{G}_\lambda is given by

q_t (r)=\frac{1}{(2\pi)^{n}} \left( \frac{2\lambda}{e^{4\lambda t}-1} \right)^n e^{-\frac{\lambda r^2}{ e^{4\lambda t}-1}}.
We conclude

\Phi_t (r,z,\lambda)=\frac{e^{2n\lambda t}}{(2\pi)^n} \left( \frac{2\lambda}{e^{4\lambda t}-1} \right)^n e^{-\frac{\lambda r^2}{2}} e^{-\frac{\lambda r^2}{ e^{4\lambda t}-1}}

\square

The second simplest and geometrically relevant example is given by the celebrated Hopf fibration. Let us consider the odd dimensional unit sphere

\mathbb{S}^{2n+1}=\lbrace z=(z_1,\cdots,z_{n+1})\in \mathbb{C}^{n+1}, \| z \| =1\rbrace.
There is an isometric group action of \mathbb{S}^1=\mathbf{U}(1) on \mathbb{S}^{2n+1} which is defined by

(z_1,\cdots, z_n) \rightarrow (e^{i\theta} z_1,\cdots, e^{i\theta} z_n).

The generator of this action shall be denoted by T. We thus have for every f \in C^\infty(\mathbb S^{2n+1})

Tf(z)=\frac{d}{d\theta}f(e^{i\theta}z)\mid_{\theta=0},
so that

T=i\sum_{j=1}^{n+1}\left(z_j\frac{\partial}{\partial z_j}-\overline{z_j}\frac{\partial}{\partial \overline{z_j}}\right).
The quotient space \mathbb S^{2n+1} / \mathbf{U}(1) is the projective complex space \mathbb{CP}^n and the projection map \pi : \mathbb S^{2n+1} \to \mathbb{CP}^n is a Riemannian submersion with totally geodesic fibers isometric to \mathbf{U}(1). The fibration

\mathbf{U}(1) \to \mathbb S^{2n+1} \to \mathbb{CP}^n

is called the Hopf fibration.

 

To study the geometry of the Hopf fibration, in particular the horizontal Laplacian \Delta_\mathcal{H}, it is convenient to introduce a set of coordinates that reflects the action of the isometry group of \mathbb{CP}^n on \mathbb{S}^{2n+1}.
Let (w_1,\cdots, w_n,\theta) be the local inhomogeneous coordinates for \mathbb{CP}^n given by w_j=z_j/z_{n+1}, and \theta be the local fiber coordinate. i.e., (w_1, \cdots, w_n) parametrizes the complex lines passing through the north pole, while \theta determines a point on the line that is of unit distance from the north pole. More explicitly, these coordinates are given by the map

(w,\theta)\longrightarrow \left(w e^{i\theta}\cos r ,e^{i\theta}\cos r \right),
where r=\arctan \sqrt{\sum_{j=1}^{n}|w_j|^2} \in [0,\pi /2), \theta \in \mathbb R/2\pi\mathbb{Z}, and w \in \mathbb{CP}^n. In these coordinates, it is clear that T=\frac{\partial}{\partial \theta} and that the vertical Laplacian is

\Delta_\mathcal{V}=\frac{\partial^2 }{\partial \theta^2}.
Our first  goal is now to compute the horizontal Laplacian \Delta_\mathcal{H}. This operator is invariant by the action on the variables (w_1,\cdots,w_n) of the group of isometries of \mathbb{CP}^n that fix the north pole of \mathbb{S}^{2n+1} (this group is \mathbf{SU}(n)). Therefore the heat kernel at the north pole only depends on the variables (r,\theta).

Proposition:  The radial part of the horizontal Laplacian is the operator

\frac{\partial^2}{\partial r^2}+((2n-1)\cot r-\tan r)\frac{\partial}{\partial r}+\tan^2r\frac{\partial^2}{\partial \theta^2}

Proof: The easiest route is to compute first the radial part of the Laplace-Beltrami operator \Delta and then to use the formula

\Delta_\mathcal{H}=\Delta-\Delta_\mathcal{V}=\Delta-\frac{\partial}{\partial \theta^2}.
In our parametrization of \mathbb{S}^{2n+1} we have,

z_{n+1}=e^{i\theta}\cos r.
Therefore if \delta_1 denotes the Riemannian distance based at the north pole, we have \cos \delta_1 =\cos r \cos \theta and if \delta_2 denotes the Riemannian distance based at the point with real coordinates (0,\cdots,0,1) then we have \cos \delta_2=\cos r \sin \theta. 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 \Delta acts on functions depending only on \delta_1,\delta_2 as

\frac{\partial^2}{\partial \delta_1^2} + 2n \cot \delta_1 \frac{\partial }{\partial \delta_1} +\frac{\partial^2}{\partial \delta_2^2} + 2n \cot \delta_2 \frac{\partial }{\partial \delta_2}
In the variables (r,\theta) this last operator writes

\frac{\partial^2}{\partial r^2}+((2n-1)\cot r-\tan r)\frac{\partial}{\partial r}+\frac{1}{\cos^2 r}\frac{\partial^2}{\partial \theta^2}.

This concludes the proof \square

 

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