Lecture 3. Hopf fibration. Transverse Weitzenbock formulas

Posted on October 11, 2014 by Fabrice Baudoin

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

\overline{\Delta}_{\mathcal{H}}=\frac{\partial^2}{\partial r^2}+((2n-1)\cot r-\tan r)\frac{\partial}{\partial r}+\tan^2r\frac{\partial^2}{\partial \theta^2}

We can observe that \overline{\Delta}_{\mathcal{H}} is symmetric with respect to the measure

d\overline{\mu}=\frac{2\pi^n}{\Gamma(n)}(\sin r)^{2n-1}\cos r drd\theta,

where the normalization is chosen in such a way that

\int_{-\pi}^{\pi}\int_0^{\frac{\pi}{2}}d\overline{\mu}=\mu(\mathbb S^{2n+1})=\frac{2\pi^{n+1}}{\Gamma (n+1)}.

As mentioned above, the heat kernel at the north pole of \Delta_{\mathcal{H}} only depends on (r, \theta), that is p\left(w e^{i\theta}\cos r ,e^{i\theta}\cos r \right)=\overline{p}_t(r, \theta), where \overline{p}_t is the heat kernel at 0 of \overline{\Delta}_{\mathcal{H}}.

Proposition: For t>0, r\in[0,\frac{\pi}{2}), \theta\in[-\pi,\pi]:

\overline{p}_t(r, \theta)=\frac{\Gamma(n)}{2\pi^{n+1}}\sum_{k=-\infty}^{+\infty}\sum_{m=0}^{+\infty} (2m+|k|+n){m+|k|+n-1\choose n-1}e^{-\lambda_{m,k}t+ik \theta}(\cos r)^{|k|}P_m^{n-1,|k|}(\cos 2r),

where \lambda_{m,k}=4m(m+|k|+n)+2|k|n and

P_m^{n-1,|k|}(x)=\frac{(-1)^m}{2^m m!(1-x)^{n-1}(1+x)^{|k|}}\frac{d^m}{dx^m}((1-x)^{n-1+m}(1+x)^{|k|+m})

is a Jacobi polynomial.

Proof: Similarly to the Heisenberg group case, we observe that \overline{\Delta}_{\mathcal{H}} commutes with \frac{\partial}{\partial \theta}, so the idea is to expand p_t(r, \theta) as a Fourier series in \theta. We can write

\overline{p}_t(r, \theta)=\frac{1}{2\pi} \sum_{k=-\infty}^{+\infty} e^{ik\theta}\phi_k(t,r),

where \phi_k is the fundamental solution at 0 of the parabolic equation

\frac{\partial\phi_k}{\partial t}=\frac{\partial^2\phi_k}{\partial r^2}+((2n-1)\cot r-\tan r)\frac{\partial\phi_k}{\partial r}-k^2\tan^2 r\phi_k.

By writing \phi_k(t,r) in the form

\phi_k(t,r)=e^{-2n|k|t}(\cos r)^{|k|}g_k(t, \cos 2r),

we get

\frac{\partial g_k}{\partial t}=4\mathcal{L}_k(g_k),

where

\mathcal{L}_k=(1-x^2)\frac{\partial^2}{\partial x^2}+[(|k|+1-n)-(|k|+1+n)x]\frac{\partial}{\partial x}.

The eigenvectors of \mathcal{L}_k solve the Jacobi differential equation, and are thus given by the Jacobi polynomials

P_m^{n-1,|k|}(x)=\frac{(-1)^m}{2^m m!(1-x)^{n-1}(1+x)^{|k|}}\frac{d^m}{dx^m}((1-x)^{n-1+m}(1+x)^{|k|+m}),

which satisfy

\mathcal{L}_k(P_m^{n-1,|k|})(x)=-m(m+n+|k|)P_m^{n-1,|k|}(x).

By using the fact that the family (P_m^{n-1,|k|}(x)(1+x)^{|k|/2})_{m\geq0} is an orthogonal basis of L^2([-1,1],(1-x)^{n-1}dx), such that

\int_{-1}^1 P_m^{n-1,|k|}(x)P_l^{n-1,|k|}(x)(1-x)^{n-1}(1+x)^{|k|}dx=\frac{2^{n+|k|}}{2m+|k|+n}\frac{\Gamma(m+n)\Gamma(m+|k|+1)}{\Gamma(m+1)\Gamma(m+n+|k|)}\delta_{ml},

we easily compute the fundamental solution of the operator \frac{\partial }{\partial t}- 4\mathcal{L}_k \square

Note that as a by-product of the previous result we obtain that the L^2 spectrum of -\Delta_\mathcal{H} is given by

\mathbf{Sp} (-\Delta_\mathcal{H}) =\left\{ 4m(m+k+n)+2kn, k \in \mathbb{N}, m \in \mathbb{N} \right\}.

We can give another representation of the heat kernel \overline{p}_t(r,\theta) which is easier to handle analytically. The key idea is to observe that since \Delta and \frac{\partial}{\partial \theta} commute, we formally have

e^{t\Delta_{\mathcal{H}}}=e^{-t\frac{\partial^2}{\partial\theta^2}}e^{t\Delta}.

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 \mathbb{S}^{2n+1} is given by

q_t(\cos\delta)=\frac{\Gamma(n)}{2\pi^{n+1}}\sum_{m=0}^{+\infty}(m+n)e^{-m(m+2n)t}C_m^n(\cos \delta),

where, \delta is the Riemannian distance based at the north pole and

C_m^n(x)=\frac{(-1)^m}{2^m}\frac{\Gamma(m+2n)\Gamma(n+1/2)}{\Gamma(2n)\Gamma(m+1)\Gamma(n+m+1/2)}\frac{1}{(1-x^2)^{n-1/2}}\frac{d^m}{dx^m}(1-x^2)^{n+m-1/2},

is a Gegenbauer polynomial. Another expression of q_t (\cos \delta) is

q_t (\cos \delta)= e^{n^2t} \left( -\frac{1}{2\pi \sin \delta} \frac{\partial}{\partial \delta} \right)^n V

where V(t,\delta)=\frac{1}{\sqrt{4\pi t}} \sum_{k \in \mathbb{Z}} e^{-\frac{(\delta-2k\pi)^2}{4t} } is a theta function.

Using the commutation and the formula \cos \delta =\cos r \cos \theta, we then infer the following proposition which is easy to prove

Proposition: For t>0, r\in[0,\pi/2), \theta\in[-\pi,\pi],

\overline{p}_t(r, \theta)=\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{+\infty}e^{-\frac{(y+i \theta)^2}{4t} }q_t(\cos r\cosh y)dy.

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 t \to 0. 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 \mathbb{M} be a smooth, connected manifold with dimension n+m. We assume that \mathbb{M} is equipped with a Riemannian foliation \mathcal{F} with bundle like metric g and totally geodesic m-dimensional leaves.

As usual, the sub-bundle \mathcal{V} formed by vectors tangent to the leaves will be referred to as the set of vertical directions and the sub-bundle \mathcal{H} which is normal to \mathcal{V} will be referred to as the set of horizontal directions. The metric g can be split as

g=g_\mathcal{H} \oplus g_{\mathcal{V}},

We define the canonical variation of g as the one-parameter family of Riemannian metrics:

g_{\varepsilon}=g_\mathcal{H} \oplus \frac{1}{\varepsilon }g_{\mathcal{V}}, \quad \varepsilon > 0.

On the Riemannian manifold (\mathbb M,g) there is the Levi-Civita connection that we denote by D, 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

\nabla_X Y = \begin{cases} ( D_X Y)_{\mathcal{H}} , \quad X,Y \in \Gamma^\infty(\mathcal{H}) \\ [X,Y]_{\mathcal{H}}, \quad X \in \Gamma^\infty(\mathcal{V}), Y \in \Gamma^\infty(\mathcal{H}) \\ [X,Y]_{\mathcal{V}}, \quad X \in \Gamma^\infty(\mathcal{H}), Y \in \Gamma^\infty(\mathcal{V}) \\ ( D_X Y)_{\mathcal{V}}, \quad X,Y \in \Gamma^\infty(\mathcal{V}) \end{cases}

where the subscript \mathcal H (resp. \mathcal{V}) denotes the projection on \mathcal{H} (resp. \mathcal{V}). Observe that for horizontal vector fields X,Y the torsion T(X,Y) is given by

T(X,Y)=-[X,Y]_\mathcal V.

Also observe that for X,Y \in \Gamma^\infty(\mathcal{V}) we actually have ( D_X Y)_{\mathcal{V}}= D_X Y because the leaves are assumed to be totally geodesic. Finally, it is easy to check that for every \varepsilon > 0, the Bott connection satisfies \nabla g_\varepsilon=0.

Example: Let (\mathbb M, \theta,g) be a K-contact Riemannian manifold. The Bott connection coincides with the Tanno’s connection, which is the unique connection that satisfies:

  • \nabla\theta=0;
  • \nabla T=0;
  • \nabla g=0;
  • T(X,Y)=d\theta(X,Y)T for any X,Y\in \Gamma^\infty(\mathcal{H});
  • {T}(T,X)=0 for any vector field X\in \Gamma^\infty(\mathcal{H}).

    • We now introduce some tensors and definitions that will play an important role in the sequel.

    For Z \in \Gamma^\infty(T\mathbb M), there is a unique skew-symmetric endomorphism J_Z:\mathcal{H}_x \to \mathcal{H}_x such that for all horizontal vector fields X and Y,

    g_\mathcal{H} (J_Z (X),Y)= g_\mathcal{V} (Z,T(X,Y)).

    where T is the torsion tensor of \nabla. We then extend J_{Z} to be 0 on \mathcal{V}_x. If Z_1,\cdots,Z_m is a local vertical frame, the operator \sum_{l=1}^m J_{Z_l}J_{Z_l} does not depend on the choice of the frame and shall concisely be denoted by \mathbf{J}^2. For instance, if \mathbb M is a K-contact manifold equipped with the Reeb foliation, then \mathbf{J} is an almost complex structure, \mathbf{J}^2=-\mathbf{Id}_{\mathcal{H}}.

    The horizontal divergence of the torsion T is the (1,1) tensor which is defined in a local horizontal frame X_1,\cdots,X_n by

    \delta_\mathcal{H} T (X)= \sum_{j=1}^n(\nabla_{X_j} T) (X_j,X), \quad X \in \Gamma^\infty(\mathbb M).

    The g-adjoint of \delta_\mathcal{H}T will be denoted \delta_\mathcal{H} T^*.

    Definition: We say that the Riemannian foliation is of Yang-Mills type if \delta_\mathcal{H} T=0.

    Example: Let (\mathbb M, \theta,g) be a K-contact Riemannian manifold. It is easy to see that the Reeb foliation is of Yang-Mills type if and only if \delta_\mathcal{H} d \theta=0. Equivalently this condition writes \delta_\mathcal{H} J =0. If \mathbb M is a strongly pseudo convex CR manifold with pseudo-Hermitian form \theta, then the Tanno’s connection is the Tanaka-Webster connection. In that case, we have then \nabla J=0 and thus \delta_\mathcal{H} J =0. 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 \mathcal{V} (resp. on the horizontal bundle \mathcal{H}). We thus have a splitting of the cotangent space

    T^*_x \mathbb M= \mathcal{H}^*(x) \oplus \mathcal{V}^*(x)

    The metric g_\varepsilon induces then a metric on the cotangent bundle which we still denote g_\varepsilon. By using similar notations and conventions as before we have for every \eta in T^*_x \mathbb M,

    \| \eta \|^2_{\varepsilon} =\| \eta \|_\mathcal{H}^2+\varepsilon \| \eta \|_\mathcal{V}^2.

    By using the duality given by the metric g, (1,1) tensors can also be seen as linear maps on the cotangent bundle T^* \mathbb M. More precisely, if A is a (1,1) tensor, we will still denote by A the fiberwise linear map on the cotangent bundle which is defined as the g-adjoint of the dual map of A. The same convention will be made for any (r,s) tensor.

    We define then the horizontal Ricci curvature \mathfrak{Ric}_{\mathcal{H}} as the fiberwise symmetric linear map on one-forms such that for every smooth functions f,g,

    \langle \mathfrak{Ric}_{\mathcal{H}} (df), dg \rangle=\mathbf{Ricci} (\nabla_\mathcal{H} f ,\nabla_\mathcal{H} g),

    where \mathbf{Ricci} is the Ricci curvature of the connection \nabla.

    If V is a horizontal vector field and \varepsilon > 0, we consider the fiberwise linear map from the space of one-forms into itself which is given for \eta \in \Gamma^\infty(T^* \mathbb M) and Y \in \Gamma^\infty(T \mathbb M) by

    \mathfrak{T}^\varepsilon_V \eta (Y) = \begin{cases} \frac{1}{\varepsilon} \eta (J_Y V), \quad Y \in \Gamma^\infty(\mathcal{V}) \\ -\eta (T(V,Y)), Y \in \Gamma^\infty(\mathcal{H}) \end{cases}

    We observe that \mathfrak{T}^\varepsilon_V is skew-symmetric for the metric g_\varepsilon so that \nabla -\mathfrak{T}^\varepsilon is a g_\varepsilon-metric connection.

    If \eta is a one-form, we define the horizontal gradient of \eta in a local frame as the (0,2) tensor

    \nabla_\mathcal{H} \eta =\sum_{i=1}^n \nabla_{X_i} \eta \otimes \theta_i.

    We denote by \nabla_\mathcal{H}^\# \eta the symmetrization of \nabla_\mathcal{H} \eta .

    Similarly, we will use the notation

    \mathfrak{T}^\varepsilon_\mathcal{H} \eta =\sum_{i=1}^n \mathfrak{T}^\varepsilon_{X_i} \eta \otimes \theta_i.

    Finally, we will still denote by \Delta_{\mathcal{H}} the covariant extension on one-forms of the horizontal Laplacian. In a local horizontal frame, we have thus

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

    For \varepsilon > 0, we consider the following operator which is defined on one-forms by

    \square_\varepsilon=-(\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)^* (\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)-\frac{1}{ \varepsilon}\mathbf{J}^2+\frac{1}{\varepsilon} \delta_\mathcal{H} T- \mathfrak{Ric}_{\mathcal{H}},

    where the adjoint is understood with respect to the metric g_{\varepsilon}. It is easily seen that, in a local horizontal frame,

    -(\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)^* (\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon) =\sum_{i=1}^n (\nabla_{X_i} -\mathfrak{T}^\varepsilon_{X_i})^2 - ( \nabla_{\nabla_{X_i} X_i}- \mathfrak{T}^\varepsilon_{\nabla_{X_i} X_i}),

    Observe that if the foliation is of Yang-Mills type then

    \square_\varepsilon=-(\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)^* (\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)-\frac{1}{ \varepsilon}\mathbf{J}^2- \mathfrak{Ric}_{\mathcal{H}}.

    As a consequence, in the Yang-Mills case the operator \square_\varepsilon is seen to be symmetric for the metric g_\varepsilon.

    Theorem: For every f \in C^\infty(\mathbb M), we have

    d \Delta_{\mathcal{H}} f=\square_\varepsilon df.

    Proof: We only sketch the proof and refer to this paper for the details. If Z_1,\cdots,Z_m is a local vertical frame of the leaves, we denote

    \mathbf J(\eta)=\sum_{l=1}^mJ_{Z_l}(\iota_{Z_l}d\eta_\mathcal V),

    where \eta_\mathcal V is the the projection of \eta 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 \mathcal{T} \colon \Gamma^\infty(\wedge^2 T^*\mathbb M)\to \Gamma^\infty( T^*\mathbb M) which is given in a local coframe \theta_i \in \Gamma^\infty(\mathcal{H}^*), \nu_k \in \Gamma^\infty(\mathcal{V}^*)

    \mathcal{T}(\theta_i\wedge\theta_j)=-\gamma_{ij}^l\nu_l,\quad \mathcal{T}(\theta_i\wedge\nu_k)=\mathcal{T}(\nu_k\wedge\nu_l)=0.

    A direct computation shows then that

    -(\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)^* (\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon) = \Delta_{\mathcal{H}} +2\mathbf{J}-\frac{2}{\varepsilon}\mathcal{T}\circ d+\delta_\mathcal{H} T^*-\frac{1}{\varepsilon}\delta_\mathcal{H} T+\frac{1}{\varepsilon}\mathbf{J}^2.

    Thus, we just need to prove that if \square_\infty is the operator defined on one-forms by

    \square_\infty=\Delta_{\mathcal{H}}+2\mathbf J-\mathbf{Ric}_\mathcal{H}+\delta_\mathcal{H} T^* ,

    then for any f\in C^\infty(\mathbb M),

    d\Delta_\mathcal{H} f=\square_\infty df.

    A computation in local frame shows that

    d\Delta_{\mathcal{H}} f - \Delta_{\mathcal{H}} df = 2\mathbf{J}(df) -\mathbf{Ric}_\mathcal{H} (df) +\delta_\mathcal{H} T^* (df),

    which completes the proof \square

    We also can prove the following Bochner’s type identity whose proof can be found in the paper.

    Theorem: For any \eta \in \Gamma^\infty(T^* \mathbb M),

    \frac{1}{2} \Delta_{\mathcal{H}} \| \eta \|_{\varepsilon}^2 -\langle \square_\varepsilon \eta , \eta \rangle_{\varepsilon} = \| \nabla_{\mathcal{H}} \eta -\mathfrak{T}^\varepsilon_{\mathcal{H}} \eta \|_{\varepsilon}^2 + \left\langle \mathfrak{Ric}_{\mathcal{H}} (\eta), \eta \right\rangle_\mathcal{H} -\left \langle \delta_\mathcal{H} T (\eta) , \eta \right\rangle_\mathcal{V} +\frac{1}{\varepsilon} \langle \mathbf{J}^2 (\eta) , \eta \rangle_\mathcal{H}.

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