January 01, 1970
We study eigenvalues and eigenfunctions of the “div-grad type" sub-Laplacian with respect to Popp’s volume on a compact equiregular sub-Riemannian manifold \(M\). Since Popp’s volume is canonically determined by the
sub-Riemannian structure of \(M\), the spetra of the sub-Laplacian carry geometric meanings. In this paper, we first embed \(M\) into the Hilbert space of square-summable sequences using
eigenfunctions and then define a spectral distance between two compact equiregular sub-Riemannian manifolds. Our result is a sub-Riemannian analogue of Bérard-Besson-Gallot’s classical work in the Riemannian case.
Mathematics Subject Classification (2020): 53C17, 58C40, 35K08, 58J65, 60H07.
Keywords: sub-Riemannian geometry, sub-Laplacian, spectral embedding, spectral distance, stochastic differential equation
The study of eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian manifold is one of the most beautiful topics in Riemannian geometry. These spectral problems have been studied extensively and intensively. One of many significant works on this topic is Bérard-Besson-Gallot’s spectral embedding theorem. Using this, they also defined the spectral distance between two compact Riemannian manifolds. (See [1]. Recently, this result was generalized to the case of RCD spaces by [2], [3].) Roughly speaking, these results can be summarized in the following way.
Denote by \(0 =\lambda_0 < \lambda_1 \le \lambda_2 \le \cdots\) all the eigenvalues of the Laplace-Beltrami operator \(\Delta_{\mathcal{M}}\) on a connected compact Riemannian manifold \({\mathcal{M}}\) in non-decreasing order counting the multiplicities. Let the sequence \(\{\varphi_i\}_{i=0}^\infty\) of real-valued functions be an orthonormal basis (ONB) of \(L^2 ({\mathcal{M}})\) such that \(\Delta_{\mathcal{M}}\varphi_i =\lambda_i \varphi_i\) for all \(i \ge 0\). They showed that the map \[{\mathcal{M}}\ni x \,\,\mapsto \,\, Z_M \{ e^{-\lambda_i t/2} \varphi_i (x)\}_{i=1}^\infty \,\,\in \ell^2\] is an embedding for every \(t>0\). Here, \(Z_{\mathcal{M}}>0\) is a suitable canstant and \(\ell^2\) is the Hilbert space of square-summable sequences. (Moreover, this embedding is isometric if \(Z_{\mathcal{M}}\) is suitably chosen.) By taking the Hausdorff distance of images of the embeddings and then varying all ONB’s of eigenfunctions as above, they defined a distance \(\mathbf{dist}_t ({\mathcal{M}}, \hat{{\mathcal{M}}})\) for each \(t>0\). Since they proved that \(\mathbf{dist}_t ({\mathcal{M}}, \hat{{\mathcal{M}}})=0\) if and only if \({\mathcal{M}}\) and \(\hat{{\mathcal{M}}}\) are isometric as Riemannian manifolds, \(\mathbf{dist}_t\) is actually a distance and called the spectral distance.
In this paper, we generalize the above-mentioned results to the case of equiregular sub-Riemannian manifolds. An equiregular sub-Riemannian manifold has a natural measure which is determined by its geometric structure only. Hence, the spectra of the “div-grad type" sub-Laplacian are supposed to have valuable geometric information. This is why spectral geometry for equiregular sub-Riemannian manifolds looks quite intriguing. Of course, Riemannian manifolds are very special examples of equiregular sub-Riemannian manifolds.
Our main results in this paper are twofold. Firstly, we prove that Bérard-Besson-Gallot’s embedding theorem still holds for compact equiregular sub-Riemannian manifolds. The most difficult part of the proof (to the author) is an \(L^\infty\)-type estimate of the first-order derivatives of eigenfunctions, which is shown by probabilistic methods. Secondly, we generalize the spectral distance to the case of compact connected equiregular sub-Riemannian manifolds and then prove that the distance of two such manifolds equals zero if and only if they are isometrically isomorphic as sub-Riemannian manifolds. Our proofs basically follow those in [1]. It should be noted that we do not use Varadhan’s asymptotics.
The organization of this paper is as follows. In Section 2, we recall the basics of sub-Riemannian geometry, which include the equiregularity, Popp’s volume, the div-grad type sub-Laplacian and the cometric. In Section 3, we prove several technical lemmas for later use. In Section 4, we introduce the spectral embedding and the spectral distance in an analogous way to the classical Riemannian case. In Section 5, the non-degeneracy of the spectral distance is shown. Appendix is devoted to proving a key technical proposition (Proposition 2). Our proof is probabilistic.
In this section we recall the basics of sub-Riemannian geometry, following nice textbooks such as [4]–[7]. We say that \((M,{\mathcal{D}},g)\) is a sub-Riemannian manifold if
\(M\) is a connected, smooth manifold of dimension \(d\),
\({\mathcal{D}}\subset TM\), \(TM\) being the tangent bundle of \(M\), is a smooth distribution of constant rank \(n~(1 \le n \le d)\) which satisfies the Hörmander condition at every \(x \in M\),
\(g=(g_x)_{x\in M}\), where each \(g_x\) is an inner product on the fiber \({\mathcal{D}}_x\) and \(x\mapsto g_x\) is smooth as a function of \(x\).
When there is no risk of confusion, we simply say that \(M\) is a sub-Riemannian manifold. Throughout this paper, \(M\) is assumed to be compact.
The precise statement of the Hörmander condition on \({\mathcal{D}}\) at \(x \in M\) is as follows: Define \({\mathcal{D}}_0(x)=\{0\}\), \({\mathcal{D}}_1 (x) ={\mathcal{D}}(x)\) and \[{\mathcal{D}}_{k} (x) = linear span of\Bigl\{ \underbrace{ [A_1, [\ldots, [A_{l-1}, A_l]] ] }_{(l-1) {\rm brackets}} (x) \, \Big\vert\, 1 \le l \le k, \, A_{1}, \ldots, A_l \in \Gamma ({\mathcal{D}}) \Bigr\}\] for \(k \ge 2\). Here, \(\Gamma ({\mathcal{D}})\) stands for the \(C^\infty\)-module of smooth sections of \({\mathcal{D}}\) over \(M\). We say that \({\mathcal{D}}\) satisfies the Hörmander condition at \(x\) if there exists \(N =N(x)\) such that \({\mathcal{D}}_N (x)= T_x M\).
A sub-Riemannian manifold \((M,{\mathcal{D}},g)\) is said to be equiregular if \(\dim {\mathcal{D}}_k(x)\) is constant in \(x \in M\) for all \(k \ge 1\). The smallest constant \(N_0\) such that \({\mathcal{D}}_{N_0} (x)= T_x M\) is called the step of the Hörmander condition. In this case, \[Q :=\sum_{k=1}^{N_0} k(\dim {\mathcal{D}}_k(x)-\dim{\mathcal{D}}_{k-1}(x))\] is also constant in \(x\) and equals the Hausdorff dimension of \(M\) with respect to the usual sub-Riemannian distance on \(M\). Denote Popp’s volume on \(M\) by \(\mu\). This is determined by the equiregular sub-Riemannian structure only and is a smooth measure on \(M\) in the sense that its restriction to every local coordinate chart is written as a strictly positive smooth density function times the Lebesgue measure on the chart. (For Popp’s volume, see [4] or [6].)
We study the second-order differential operator of the form \(\Delta =(\nabla^{{\mathcal{D}}})^* \nabla^{{\mathcal{D}}}\), where \(\nabla^{{\mathcal{D}}}\) is the horizontal gradient in the direction of \({\mathcal{D}}\) and \((\nabla^{{\mathcal{D}}})^*\) is the adjoint of \(\nabla^{{\mathcal{D}}}\) with respect to \(\mu\). (In our sign convention, the sub-Laplacian \(\Delta\) is a non-negative operator on \(L^2 (M)\).) By the way it is defined, \(\Delta\) with its domain \(C_0^{\infty} (M)\) is clearly symmetric on \(L^2 (M)\). Since \(M\) is compact, \(\Delta\) is known to be essentially self-adjoint on \(C^{\infty} (M)\) (whose unique self-adjoint extension will be denoted by the same symbol again) and \(e^{-t \Delta}\) is of trace class for every \(t >0\), where \((e^{-t \Delta})_{t \ge 0}\) is the heat semigroup associated with \(\Delta\).
Needless to say, a connected Riemannian manifold is a special example of equiregular sub-Riemannian manifold. In that case, Popp’s volume \(\mu\) coincides with the usual Riemannian measure and the associated sub-Laplacian \(\Delta\) coincides with the Laplace-Beltrami operator.
Denote by \(g^*\) the cometric of \((M,{\mathcal{D}},g)\). At \(x\in M\), the cometric \(g^*_x \in (T_x^* M)^* \otimes (T_x^* M)^* =T_xM \otimes T_xM\) is defined by \[g^*_x \langle\xi, \eta\rangle= \sum_{i=1}^n \langle\xi, v_i\rangle\langle\eta, v_i\rangle, \qquad \xi,\eta \in T^*_x M,\] where \(\{v_i\}_{i=1}^n\) is an (or any) ONB of \({\mathcal{D}}_x\subset T_xM\). Hence, \(g^* \in \Gamma (T_xM \otimes T_xM)\).
It is well-known that one can recover \(({\mathcal{D}}_x, g_x)\) from \(g^*_x\) as follows. First, \({\mathcal{D}}_x = (\ker g^*_x)^\perp\) holds, where \(\ker g^*_x := \{ \xi \in T_x^*M \mid g^*_x \langle\xi, \xi\rangle=0\}\). Moreover, one can easily see that \[g_x(v,v) = \sup \left\{ 2 \langle\xi, v\rangle-g^*_x \langle\xi, \xi\rangle\mid \xi \in T^*_x M \right\}, \qquad v \in {\mathcal{D}}_x.\] Therefore, identifying the sub-Riemannian structure is equivalent to identifying the cometric.
One can obtain the cometric from the sub-Laplacian \(\Delta\). Let \(\{V_i\}_{i=1}^n\) is a local orthonormal frame of \({\mathcal{D}}\) around \(x_0\in M\). Then, the sub-Laplacian \(\Delta\) writes \[-\Delta =\sum_{i=1}^n V_i^2 + (1st order differential operator).\] Hence, for any \(\psi, \chi \in C^\infty (M)\) vanishing at \(x_0\), we have \[-\Delta (\psi \chi) \vert_{x=x_0} = \sum_{i=1}^n \langle d\psi (x_0), V_i (x_0)\rangle\langle d\chi (x_0), V_i (x_0)\rangle, = g^*_{x_0} \langle d\psi (x_0), d\chi (x_0)\rangle,\] where \(d\psi\) and \(d\chi\) are the exterior derivative of \(\psi\) and \(\chi\), respectively. Since any covector \(\xi\) at \(x_0\) can be written as \(d\psi(x_0)\) for some \(\psi\), we can obtain \(g^*\) from \(\Delta\).
Now let us introduce the heat kernel \(p=p (t,x,y)\), which is a smooth function on \((0,\infty)\times M \times M\) and satisfies \[e^{-t \Delta}f (x) = \int_M p (t,x,y) f(y) \mu (dy), \qquad f\in L^2 (M).\] Since \(\Delta\) is self-adjoint, \(p (t,x,y) = p(t,y,x)\). By Mercer’s theorem, we have \({\rm Trace} (e^{t \Delta}) = \int_M p (t,x,x) \mu (dx)\) for all \(t>0\). The following Weyl-type asymptotics for the heat trace is known: \[\label{asy46trc} {\rm Trace} (e^{-t \Delta}) \sim \frac{c_0}{t^{Q/2}} \qquad as t \searrow 0\tag{1}\] for some constant \(c_0 >0\). (See [8], [9] for example.) Here, the asymptotic symbol “\(\sim\)" means that the quotient converges to \(1\).
Denote by \(0 =\lambda_0 < \lambda_1 \le \lambda_2 \le \cdots \nearrow \infty\) be all the eigenvalues of \(\Delta\) in non-decreasing order counting the multiplicities. Due to Chow-Reshevskii’s theorem, the lowest eigenvalue \(\lambda_0=0\) is necessarily simple and the corresponding eigenspace consists of constant functions only. Set the eigenvalue counting function \(\mathcal{N}_{\Delta} (\lambda) :=\# \{i \ge 0 \mid \lambda_i \le \lambda\}\) for \(\lambda \ge 0\). By Karamata’s Tauberian theorem, we have from 1 that \[\label{asy46numb95e46v46} \mathcal{N}_{\Delta} (\lambda) \sim \frac{c_0}{\Gamma (Q/2 +1)} \lambda^{Q/2} \qquad as \lambda \to\infty.\tag{2}\] Here, \(\Gamma\) stands for the usual Gamma function.
Let the sequence \(\{\varphi_i\}_{i=0}^\infty\) of real-valued functions be an ONB of \(L^2 (M)\) such that \(\Delta\varphi_i =\lambda_i \varphi_i\) for all \(i \ge 0\) and \(\varphi_0 \equiv \mu (M)^{-1/2}\). Thanks to the hypoellipticity, \(\varphi_i\)’s are necessarily smooth. The totality of such ONB’s is denoted by \(\mathcal{B} (M, \Delta)\). Then, we have \[\label{hk95L94246eq} p(t,x,y) = \sum_{i=0}^\infty e^{-\lambda_i t} \varphi_i (x)\varphi_i (y)\tag{3}\] At the moment, the convergence on the right hand side (RHS) of 3 takes place in \(L^2 (M\times M)\) for every fixed \(t>0\). So, it is not so clear a priori whether RHS makes sense for a fixed \((x,y)\). In the next section, we will check that this sum actually converges uniformly and therefore 3 makes pointwise sense.
In this section we provide several lemmas on eigenvalues, eigenfunctions and the heat kernel associated with the sub-Laplancian.
Lemma 1. For every \(r>0\) and \(t >0\), we have \[\sum_{i=0}^\infty \lambda_i^r e^{-\lambda_i t} <\infty.\] Similarly, if \(q>Q/2 +1\), we then have \[\sum_{i=1}^\infty \lambda_i^{-q} <\infty.\]
Proof. We prove the first assertion. One can easily see from 2 that \[\begin{align} \sum_{i=0}^\infty \lambda_i^r e^{-\lambda_i t} &= \sum_{\lambda_i \le 1} \lambda_i^r e^{-\lambda_i t} + \sum_{n=1}^\infty \sum_{n<\lambda_i \le n+1} \lambda_i^r e^{-\lambda_i t} \\ &\le \mathcal{N}_{\Delta} (1) + \sum_{n=1}^\infty (n+1)^re^{-n t}\mathcal{N}_{\Delta} (n+1) \\ &\le \mathcal{N}_{\Delta} (1) +C \sum_{n=1}^\infty (n+1)^{r +Q/2}e^{-n t}<\infty \end{align}\] for some constant \(C>0\) independent of \(n\).
We prove the second assertion. By similar computation as above, we have \[\begin{align} \sum_{i=1}^\infty \lambda_i^{-q} &= \sum_{\lambda_i \le 1} \lambda_i^{-q} + \sum_{n=1}^\infty \sum_{n<\lambda_i \le n+1} \lambda_i^{-q} \\ &\le \lambda_1^{-q}\mathcal{N}_{\Delta} (1) + C \sum_{n=1}^\infty n^{-q} (n+1)^{Q/2}<\infty. \end{align}\] Here we used that \(q -Q/2 >1\). ◻
Proposition 1. Let \(i \ge 1\) and \(\psi_i\) be an (real-valued) eigenfunction of \(\Delta\) associated with the eigenvalue \(\lambda_i\) with \(\| \psi_i\|_{L^2}=1\). Then, there exists a constant \(C>0\) such that \[\|\psi_i\|_\infty \le C \lambda_i^{Q/4}, \qquad i \ge 1.\] Here, \(\|\,\cdot\,\|_\infty\) is the usual sup-norm and \(C\) does not depend on \(i\) or \(\psi_i\).
Proof. Since \(\psi_i\) is an eigenfunction of \(e^{-t \Delta}\) with eigenvalue \(e^{-\lambda_i t}\), we have \[\begin{align} |e^{-\lambda_i t}\psi_i (x)| &= \left| \int_M p (t,x,y) \psi_i (y) \mu (dy) \right| \\ &\le \| \psi_i\|_{L^2} \left\{ \int_M p (t,x,y)^2 \mu (dy)\right\}^{1/2} \le p (2t,x,x)^{1/2}. \end{align}\] Here, we used the symmetry of \(p\) and the Chapman-Kolmogorov formula.
Uniform on-diagonal short-time asymptotics of the heat kernel was proved in [8], [9]. It claims that there exists a constant \(c_1 >0\), which is independent of \(t\) and \(x\), such that \[p (t,x,x) \le \frac{c_1}{t^{Q/2}}, \qquad (t, x) \in (0,1] \times M.\] By putting \(t =1/\lambda_i\), we have \[|\psi_i (x)| \le e \sqrt{c_1} (2\lambda_i)^{Q/4} \qquad if \lambda_i \ge 1.\] Since there are only finitely many \(\lambda_i\)’s less than \(1\), the proof is finished. ◻
Lemma 2. The series on RHS of 3 converges absolutely and uniformly on \([s, \infty) \times M\times M\) for every \(s>0\).
Proof. From Proposition 1, we can see that \[\sum_{i=1}^\infty \left| e^{-\lambda_i t} \varphi_i (x)\varphi_i (y)\right| \le C^2\sum_{i=1}^\infty e^{-\lambda_i s} \lambda_i^{Q/2}, \qquad (t,x,y) \in [s, \infty) \times M\times M.\] Here, \(C>0\) is the constant in Proposition 1. By Lemma 1, RHS is finite and independent of \((t,x,y)\), from which our assertion follows. ◻
Since \(\{\varphi_i\}_{i=0}^\infty \in \mathcal{B} (M, \Delta)\) is an ONB of \(L^2 (M)\), every \(f \in L^2 (M)\) admits the following expansion in the \(L^2\)-topology: \[\label{eq46Fourier} f (x) = \sum_{i=0}^\infty \langle f, \varphi_i\rangle \varphi_i (x)\tag{4}\] where \(\langle f, \varphi_i\rangle =\int_M f \varphi_i d\mu\) stands for the inner product of \(L^2 (M)\). When \(f\) is regular enough, the series in 4 converges absolutely and uniformly and 4 makes sense at every point \(x\).
Lemma 3. Suppose that \(f \in C^\infty (M)\) and \(k \in {\mathbb{N}}:=\{1,2, \ldots\}\). Then, we have \[|\langle f, \varphi_i\rangle| \le \lambda_i^{-k} \| \Delta^k f\|_{L^2}, \qquad i \ge 1.\] In particular, the series in 4 converges absolutely and uniformly on \(M\).
Proof. It is easy to see from the self-adjointness of \(\Delta\) that \[\lambda_i^{k}\langle f, \varphi_i\rangle = \langle f, \Delta^k \varphi_i\rangle =\langle \Delta^k f, \varphi_i\rangle.\] Then, \[|\lambda_i^{k}\langle f, \varphi_i\rangle|^2 \le \sum_{j=0}^\infty \langle \Delta^k f, \varphi_j\rangle^2 = \| \Delta^k f\|_{L^2}^2.\] Thus, we obtained the first assertion.
We prove the second assertion. It follows from the first assertion and Proposition 1 that \[\sum_{i=1}^\infty |\langle f, \varphi_i\rangle| \, \|\varphi_i \|_{\infty} \le C\| \Delta^k f\|_{L^2} \sum_{i=1}^\infty \lambda_i^{ -k +Q/4}<\infty\] if \(k > Q+1\). Here, we used Lemma 1. ◻
Corollary 1. For every \(f \in C^\infty (M)\), it holds that \[\label{eq46Fourier2} \Delta f (x) = \sum_{i=0}^\infty \lambda_i \langle f, \varphi_i\rangle \varphi_i (x)\tag{5}\] and that the series in 5 converges absolutely and uniformly on \(M\).
Proof. The absolute and uniform convergence can be shown in the same way as in Lemma 3. Hence, it also converges in \(L^2 (M)\). Obviously, we have \[\Delta \left[ \sum_{i=0}^m \langle f, \varphi_i\rangle \varphi_i \right] = \sum_{i=0}^m \lambda_i \langle f, \varphi_i\rangle \varphi_i\] for all \(m\in {\mathbb{N}}\). Since \(\Delta\) is a closed operator on \(L^2 (M)\), we may let \(m\to\infty\) to obtain \[\Delta f= \Delta \left[ \sum_{i=0}^\infty \langle f, \varphi_i\rangle \varphi_i \right] = \sum_{i=0}^\infty \lambda_i \langle f, \varphi_i\rangle \varphi_i\] in \(L^2 (M)\). Since both sides are continuous in \(x\), 5 was proved. ◻
Corollary 2. Let \(\{\varphi_i\}_{i=0}^\infty \in \mathcal{B} (M, \Delta)\). Then, any two points \(x, y \in M~(x\neq y)\) are separated by \(\{\varphi_i\}_{i=0}^\infty\).
Proof. By Lemma 3, the expansion 4 makes pointwise sense. Therefore, if \(\varphi_i (x) = \varphi_i (y)\) for all \(i \ge 0\), we have \(f (x) =f(y)\) for all \(f\in C^\infty (M)\). This is a contradiction. ◻
Now we estimate the first-order derivative of eigenfunctions of \(\Delta\). Technically, this lemma is quite important. Unfortunately, the author has no precise information on the exponent \(\nu\).
Proposition 2. Let \(i \ge 1\) and \(\psi_i\) be an (real-valued) eigenfunction of \(\Delta\) associated with the eigenvalue \(\lambda_i\) with \(\| \psi_i\|_{L^2}=1\). Let \(A\) be a smooth vector field on \(M\). Then, there exist constants \(C>0\) and \(\nu >0\) such that \[\|A \psi_i\|_\infty \le C \lambda_i^{\nu}, \qquad i \ge 1.\] Here, (i) \(C\) does not depend on \(i\) or \(\psi_i\) and (ii) \(\nu\) does not depend on \(A\), \(i\) or \(\psi_i\).
Proof. We use Proposition 5, which provides an \(L^\infty\)-estimate of the first-order derivative of the heat semigroup.
Since \(\psi_i\) is an eigenfunction of \(e^{-t \Delta}\) with eigenvalue \(e^{-\lambda_i t}\), we have \(e^{-\lambda_i t}\psi_i = e^{-t \Delta}\psi_i\). Apply \(A\) to both sides and use Proposition 5. Then, we have \[e^{-\lambda_i t}\|A\psi_i \|_\infty = \| A e^{-t \Delta}\psi_i \|_\infty \le C_1 t^{-\nu_1}\|\psi_i \|_\infty \le C_2 t^{-\nu_1} \lambda_i^{Q/4}, \qquad t \in (0,1], \, i \ge 1\] where \(C_1, C_2\) an \(\nu_1\) are positive constants independent of \(i\), \(\psi_i\), and \(t\). For the last inequality, we used Proposition 1.
If \(\lambda_i \ge 1\), set \(t = 1/\lambda_i\) in the above inequality. Then, \(\|A\psi_i \|_\infty \le e C_2 \lambda_i^{\nu_1 +Q/4}\). Since there are only finitely many \(\lambda_i\)’s belonging to \((0,1)\), we are done. ◻
In the following corollary, \(A^{(x)}p(t,x,y)\) stands for the application of \(A\in \Gamma (TM)\) to the function \(x \mapsto p(t,x,y)\) for each fixed \((t, y)\). Also, \(A^{(x)}\tilde{A}^{(y)}p(t,x,y)\) is defined in an analogous way.
Corollary 3. Let \(A\) and \(\tilde{A}\) be smooth vector fields on \(M\). Then, for every fixed \(t>0\), we have \[\begin{align} A^{(x)} p(t,x,y) &= \sum_{i=1}^\infty e^{-\lambda_i t}( A\varphi_i) (x)\varphi_i (y), \tag{6} \\ A^{(x)}\tilde{A}^{(y)}p(t,x,y) &= \sum_{i=1}^\infty e^{-\lambda_i t} ( A\varphi_i) (x) (\tilde{A}\varphi_i) (y) \tag{7} \end{align}\] for every \((t, x, y) \in (0,\infty)\times M \times M\). Moreover, the series in 6 and that in 7 both converge absolutely and uniformly on \([s ,\infty)\times M \times M\) for every \(s>0\).
Proof. We prove 6 . Take any \(x\in M\). Then, there exists a coordinate neighborhood \(\{U, (x^1, \ldots, x^d)\}\) of \(x\) such that \(\partial/ \partial x^j\) extends to a global vector field on \(M\) for all \(j~(1\le j \le d)\). (The extended vector fields are denoted by the same symbols again.) By taking \(U\) smaller if necessary, we may also assume that \(A\) can be written as \[A (x) = \sum_{j=1}^d a_j (x) \frac{\partial}{\partial x^j}, \qquad x \in U,\] where \(a_j\)’s are certain smooth and bounded functions on \(U\) for all \(j~(1\le j \le d)\).
By Lemma 1, Propositions 1 and 2, it holds for all \(j\) that \[\begin{align} \nonumber\sum_{i=0}^\infty e^{-\lambda_i t} \left|\frac{\partial \varphi_i }{\partial x^j} (x)\right| \, |\varphi_i (y)| &\le c_1 \sum_{i=0}^\infty e^{-\lambda_i s} \lambda_i^{\nu + Q/4} <\infty, \quad (t,x,y) \in [s, \infty)\times U \times M. \end{align}\] Here, \(c_1 >0\) is a constant independent of \(i\). So, this above absolute series converges uniformly on \([s, \infty)\times U \times M\), which implies that \[\left. \frac{\partial}{\partial x^j} p(t, \,\cdot, y) \right|_{x} = \sum_{i=0}^\infty e^{-\lambda_i t} \frac{\partial \varphi_i }{\partial x^j} (x) \,\varphi_i (y) \qquad on [s, \infty)\times U\times M.\] Multiplying both side by \(a_j (x)\) and then summing over \(j\), we obtain 6 on \([s, \infty)\times U\times M\). Since \(M\) is compact, \(M\) can be covered by finitely many such \(U\)’s. Thus, we have shown 6 .
The proof of 7 is essentially the same. So, we omit it. ◻
In this section we define spectral embeddings and spectral distances in an analogous way to Bérard-Besson-Gallot [1].
Definition 1. For \(t>0\) and \(\mathbf{a} =\{\varphi_i^{\mathbf{a}}\}_{i=0}^\infty\in \mathcal{B} (M, \Delta)\), we define a map \(I_t^{\mathbf{a}}\colon M \to \ell^2\) by \[I_t^{\mathbf{a}} (x):= Z_M \{ e^{-\lambda_i t/2} \varphi_i^{\mathbf{a}} (x)\}_{i=1}^\infty,\] where we set \(Z_M := \sqrt{\mu (M)} >0\).
Note that \(i=0\) is excluded from the above definition. By Corollary 2, \(I_t^{\mathbf{a}}\) is injective. Moreover, the map \[(0, \infty)\times M \ni (t, x) \mapsto I_t^{\mathbf{a}} (x)\in \ell^2\] is continuous. Indeed, noting that \(\varphi_0^{\mathbf{a}}\) is constant, we can easily see that \[\begin{align} Z_M^{-2} \| I_{t_n}^{\mathbf{a}} (x_n)- I_t^{\mathbf{a}} (x) \|_{\ell^2}^2 &= \sum_{i=1}^\infty \{ e^{-\lambda_i t_n/2} \varphi_i^{\mathbf{a}} (x_n)-e^{-\lambda_i t/2} \varphi_i^{\mathbf{a}} (x)\}^2 \\ &= p(t_n, x_n, x_n) +p(t, x, x) -2p((t_n+t)/2, x_n, x) \to 0 \end{align}\] if \((t_n, x_n)\to (t,x)\) as \(n\to \infty\). Therefore, \(I_t^{\mathbf{a}} (M)\) is compact and \(I_t^{\mathbf{a}}\) is a homeomorphism from \(M\) to \(I_t^{\mathbf{a}} (M)\) for every \(t>0\).
Lemma 4. For every \(t>0\), the injection \(I_t^{\mathbf{a}}\colon M \hookrightarrow \ell^2\) is an embedding.
Proof. Suppose that \(v \in T_z M\) satisfies that \[\langle dI_t^{\mathbf{a}}, v\rangle = Z_M \left\{e^{-\lambda_i t /2} \langle d\varphi_i^{\mathbf{a}}, v\rangle \right\}_{i=1}^\infty =\mathbf{0},\] where \(d\) stands for the exterior derivative on \(M\). Obviously, this is equivalent to \(\langle d\varphi_i^{\mathbf{a}}, v\rangle =0\) for all \(i\ge 1\). It is enough to show \(v=0\) from this condition. (Note that the left-most side is \(\ell^2\)-valued differentiation and that the left equality can be verified by Proposition 2 and the dominated convergence theorem for the infinite sum.)
Take a smooth vector field \(V\) on \(M\) such that \(V_z =v\). Assume that \(f \in C^\infty (M)\) and \(k\) is large enough. Then, we can see from Lemma 3, Proposition 2 and Lemma 1 that \[\sum_{i=0}^\infty |\langle f, \varphi_i^{\mathbf{a}}\rangle | \|V\varphi_i^{\mathbf{a}} \|_{\infty} \le C \sum_{i=1}^\infty \lambda_i^{-k} \| \Delta^k f\|_{L^2} \cdot C \lambda_i^{\nu} <\infty\] and that \[(Vf) (x) = \sum_{i=1}^\infty \langle f, \varphi_i^{\mathbf{a}}\rangle (V\varphi_i^{\mathbf{a}}) (x), \qquad x\in M.\] Note that the above series converges absolutely and uniformly on \(M\). Hence, we may evaluate it at the given point \(z\) to obtain \(\langle df, v\rangle = \sum_{i=0}^\infty \langle f, \varphi_i^{\mathbf{a}}\rangle \langle d\varphi_i^{\mathbf{a}}, v\rangle =0\). Since \(f\) is arbitrary, we obtain \(v=0\). ◻
Now we introduce a pseudometric between two equiregular sub-Riemannian manifolds. As we will see below, this becomes a metric (distance). Therefore, we will call it the sub-Riemannian spectral distance.
Definition 2. For \(t>0\) and two compact equiregular sub-Riemannian manifolds \((M,{\mathcal{D}},g)\) and \((\hat{M}, \hat{{\mathcal{D}}}, \hat{g})\), we set \[\begin{align} \mathbf{dist}_t (M, \hat{M}) &:=\max \left\{ \sup_{\mathbf{a}\in \mathcal{B} (M, \Delta)} \inf_{\mathbf{b}\in \mathcal{B} (\hat{M}, \hat{\Delta})} \mathrm{HD} (I_t^{\mathbf{a}} (M), I_t^{\mathbf{b}} (\hat{M})) \right., \\ &\qquad\qquad\qquad\qquad \left. \sup_{\mathbf{b}\in \mathcal{B} (\hat{M}, \hat{\Delta})} \inf_{\mathbf{a}\in \mathcal{B} (M, \Delta)} \mathrm{HD} (I_t^{\mathbf{a}} (M), I_t^{\mathbf{b}} (\hat{M})) \right\}. \end{align}\] Here, \(\mathrm{HD}\) stands for the Hausdorff distance for compact subsets of \(\ell^2\).
It is trivial that \(\mathbf{dist}_t\) satisfies the triangle inequality. Obviously, \(\mathbf{dist}_t (M, \hat{M}) =0\) if \((M,{\mathcal{D}},g)\) and \((\hat{M}, \hat{{\mathcal{D}}}, \hat{g})\) are isomorphic as sub-Riemannian manifolds. In what follows, we will show that the converse is also true.
Consider a compact equiregular sub-Riemannian manifold \((M,{\mathcal{D}},g)\) and its sub-Laplacian \(\Delta\) again. Denote by \(0 =\lambda_0^\prime < \lambda_1^\prime <\lambda_2^\prime < \cdots\) be all the eigenvalues of \(\Delta\) in strictly increasing order without counting the multiplicities. We denote by \(E_j\) the eigenspace corresponding to \(\lambda_j^\prime\). Clearly, \(\dim E_j <\infty\) for all \(j \ge 0\) and \(L^2 (M, \mu) = \oplus_{j=0}^\infty E_j\). It immediately follows that \(\mathcal{B} (M, \Delta)= \{\varphi_0\}\times \prod_{j=1}^\infty \mathcal{B} (E_j)\). Here, \(\mathcal{B} (E_j)\) denotes the set of all ONB’s of \(E_j\) and therefore can naturally be identified with the orthogonal group \(O (\dim E_j)\). By Tychonoff’s theorem, \(\mathcal{B} (M, \Delta)\) is compact.
On \(O(k)\), \(k \ge 1\), the distance \(\tilde{\rho}_k\) defined by the Frobenius norm \(\tilde{\rho}_k (A,B) :=\|A-B\|_{{\rm Fr}}\) induces the usual topology and satisfies \(\tilde{\rho}_k (A,B) \le 2\sqrt{k}\) for all \(A, B\in O(k)\). Moreover, \(\tilde{\rho}_k\) is bi-invariant.
This distance viewed as one on \(\mathcal{B} (E_j)\) through the above identification is denoted by \(\rho_{E_j}\), where \(k=\dim E_j\) is assumed. We introduce a distance \(\rho\) on \(\mathcal{B} (M, \Delta)\): \[\rho (\mathbf{a}, \mathbf{b})^2 :=\sum_{j=1}^\infty (\lambda_i^\prime)^{-N} \rho_{E_j} ( \mathbf{a}\vert_{E_j}, \mathbf{b}\vert_{E_j})^2, \qquad \mathbf{a}, \mathbf{b} \in \mathcal{B} (M, \Delta).\] Now, let us check that the series on RHS converges if \(N =N_M\) is large enough. We can easily see that \[\begin{align} \sum_{j=1}^\infty (\lambda_j^\prime)^{-N} 2\sqrt{\dim E_j} \le 2\sum_{j=1}^\infty (\lambda_j^\prime)^{-N} \dim E_j \le 2\sum_{i=1}^\infty \lambda_i^{-N}, \end{align}\] which is finite if \(N > Q/2 +1\) (see Lemma 1). In what follows, we set \(N = Q/2 +2\). Then, one can easily see that \(\rho\) is a distance that induces the product topology of \(\mathcal{B} (M, \Delta)\).
Proposition 3. Let the notation be as above. Then, for all \(t, s \in (0, \infty)\), \(x, y\in M\) and \(\mathbf{a}, \mathbf{b} \in \mathcal{B} (M, \Delta)\), we have \[\begin{align} Z_M^{-2} \| I_{t}^{\mathbf{a}} (x)- I_s^{\mathbf{b}} (y) \|_{\ell^2}^2 &= p(t, x, x) +p(t, y, y) -2 p((t+s)/2, x, y) \\ &\qquad +2\rho (\mathbf{a}, \mathbf{b}) q^{(N)} (t,x,x)^{1/2}q^{(N)} (s,y,y)^{1/2}, \end{align}\] where we set \[q^{(N)} (t,x,x):=\sum_{i=1}^\infty \lambda_i^{N/2} e^{-\lambda_i t} \varphi_i^{\mathbf{a}} (x)^2.\] In particular, as a map from \((0, \infty) \times M\times \mathcal{B} (M, \Delta)\) to \(\ell^2\), \(I\) is continuous.
Proof. First, note that the series that defines \(q^{(N)}(t, x, x)\) converges absolutely and uniformly on \([t_0, \infty) \times M\) for every \(t_0 >0\), thanks to Lemma 1 and Proposition 1.
By straightforward computation, we obtain \[\begin{align} \label{eq46conti95love} Z_M^{-2} \| I_{t}^{\mathbf{a}} (x)- I_s^{\mathbf{b}} (y) \|_{\ell^2}^2 &= \sum_{i=1}^\infty \{ e^{-\lambda_i t/2} \varphi_i^{\mathbf{a}} (x)-e^{-\lambda_i s/2} \varphi_i^{\mathbf{b}} (y)\}^2 \nonumber\\ &= p(t, x, x) +p(t, y, y) -2 p((t+s)/2, x, y) +2A, \nonumber \end{align}\tag{8}\] where we set \[A:=\sum_{i=1}^\infty e^{-\lambda_i (t+s)/2} \{\varphi_i^{\mathbf{a}} (x) \varphi_i^{\mathbf{b}} (y) -\varphi_i^{\mathbf{a}} (x) \varphi_i^{\mathbf{a}} (y) \}.\]
We will calculate \(A\). Consider each eigenspace \(E_j\) and write \(\nu_j :=\dim E_j\). There exists unique \(r_j \in {\mathbb{N}}\) such that \(\lambda_i =\lambda_j^\prime\) if and only if \(r_j\le i \le r_j+\nu_j -1\). Then, there exists \(u_{ki} =u_{ki} (\mathbf{b}, \mathbf{a})\) for \(i, k \in \{r_j , \ldots, r_j+\nu_j -1\}\) which are constant in \(x\) and satisfies that \[\varphi_i^{\mathbf{b}} = \sum_{k= r_j }^{r_j +\nu_j -1} u_{ki} \varphi_k^{\mathbf{a}}, \quad\qquad r_j \le i \le r_j+\nu_j -1.\] Note that \(\{ u_{ki} \mid i, k \in \{r_j, \ldots, r_j+\nu_j -1\}\}\in O(\nu_j)\). From this we obtain \[A=\sum_{j=1}^\infty e^{-\lambda_j^\prime (t+s)/2} \sum_{i, k= r_j }^{r_j +\nu_j -1} \varphi_i^{\mathbf{a}} (x) \varphi_k^{\mathbf{a}} (y) \{ u_{ki} (\mathbf{b}, \mathbf{a}) -\delta_{ki}\},\] where \(\delta_{ki}\) denotes Kronecker’s delta. Since \(\tilde{\rho}_k\) is bi-invariant, we have \[\sum_{i, k= r_j }^{r_j +\nu_j -1} | u_{ki} (\mathbf{b}, \mathbf{a}) - \delta_{ki}|^2 \le \rho_{E_j} ( \mathbf{a}\vert_{E_j}, \mathbf{b}\vert_{E_j})^2, \qquad j\ge 1.\] It immediately follows from this and Schwarz’ inequality that \[\begin{align} |A| &\le \sum_{j=1}^\infty e^{-\lambda_j^\prime (t+s)/2} \left( \sum_{i= r_j }^{r_j +\nu_j -1} \varphi_i^{\mathbf{a}} (x)^2\right)^{\frac{1}{2}} \left( \sum_{k= r_j }^{r_j +\nu_j -1} \varphi_k^{\mathbf{a}} (y)^2\right)^{\frac{1}{2}} \rho_{E_j} ( \mathbf{a}\vert_{E_j}, \mathbf{b}\vert_{E_j}) \\ &\le \rho (\mathbf{a}, \mathbf{b}) \sum_{j=1}^\infty (\lambda_j^\prime)^{N/2} e^{-\lambda_j^\prime (t+s)/2} \left( \sum_{i= r_j }^{r_j +\nu_j -1} \varphi_i^{\mathbf{a}} (x)^2\right)^{\frac{1}{2}} \left( \sum_{k= r_j }^{r_j +\nu_j -1} \varphi_k^{\mathbf{a}} (y)^2\right)^{\frac{1}{2}} \\ &\le \rho (\mathbf{a}, \mathbf{b}) \left( \sum_{i=1}^\infty \lambda_i^{N/2} e^{-\lambda_i t}\varphi_i^{\mathbf{a}} (x)^2 \right)^{\frac{1}{2}} \left( \sum_{i=1}^\infty \lambda_i^{N/2} e^{-\lambda_i s}\varphi_i^{\mathbf{a}} (y)^2 \right)^{\frac{1}{2}} \\ &\le \rho (\mathbf{a}, \mathbf{b}) q^{(N)} (t,x,x)^{1/2}q^{(N)} (s,y,y)^{1/2}. \end{align}\] Thus, we have obtained the desired estimate, from which the continuity of \(I\) immediately follows. ◻
Now we state our main theorem of this paper.
Theorem 4. Let \(t>0\) and let \((M,{\mathcal{D}},g)\) and \((\hat{M}, \hat{{\mathcal{D}}}, \hat{g})\) be two compact equiregular sub-Riemannian manifolds. Then, \(\mathbf{dist}_t (M, \hat{M})=0\) if and only if these two manifolds are isometrically isomorphic as sub-Riemannian manifolds, that is, there exists a \(C^\infty\)-diffeomorphism \(\psi =\psi_t\colon M \to \hat{M}\) such that
\(\psi_* ({\mathcal{D}}_x) = \hat{{\mathcal{D}}}_{\psi (x)}\) for every \(x\in M\).
\(\hat{g}_{\psi (x)} (\psi_* v, \psi_* v) = g_x (v, v)\) for every \(x\in M\) and \(v\in {\mathcal{D}}_x\).
In particular, \(\mathbf{dist}_t\) is a distance on the isometrically isomorphic class of compact equiregular sub-Riemannian manifolds for each fixed \(t >0\).
Proof. The “if part" is trivial. We will prove”only if part" below.
Suppose that \(\sup_{\mathbf{b}\in \mathcal{B} (\hat{M}, \hat{\Delta})} \inf_{\mathbf{a}\in \mathcal{B} (M, \Delta)} \mathrm{HD} (I_t^{\mathbf{a}} (M), I_t^{\mathbf{b}} (\hat{M})) =0\). This implies that for every \(\mathbf{b}\), there exists a sequence \(\{\mathbf{a}_k\}_{k\in {\mathbb{N}}}\subset \mathcal{B} (M, \Delta)\) such that \(\mathrm{HD} (I_t^{\mathbf{a}_k} (M), I_t^{\mathbf{b}} (\hat{M})) \to 0\) as \(k\to\infty\). Since \(\mathcal{B} (M, \Delta)\) is compact, a subsequence, which will be denoted by the same symbol again, converges to a certain element \(\mathbf{a}_\infty \in\mathcal{B} (M, \Delta)\).
From Proposition 3, we can easily see that \[\sup_{x\in M}\inf_{y\in M}\| I_{t}^{\mathbf{a}} (x)- I_t^{\mathbf{b}} (y) \|_{\ell^2} \le \sqrt{C_t \rho (\mathbf{a}, \mathbf{b})},\] where \(C_t := 2Z_M\sup_{x\in M}q^{(N)} (t,x,x) <\infty\) is a positive constant independent of \((\mathbf{a}, \mathbf{b})\). So, \(\mathrm{HD} (I_t^{\mathbf{a}_k} (M), I_t^{\mathbf{a}_\infty} (M)) \le \sqrt{C_t \rho (\mathbf{a}_k, \mathbf{a}_\infty)} \to 0\) as \(k\to\infty\). By the triangle inequality for \(\mathrm{HD}\), we have \(\mathrm{HD} (I_t^{\mathbf{a}_\infty} (M), I_t^{\mathbf{b}} (\hat{M})) = 0\). Thus, we can find \((\mathbf{a}, \mathbf{b})\) such that \(I_t^{\mathbf{a}} (M)=I_t^{\mathbf{b}} (\hat{M})\). We fix such an \((\mathbf{a}, \mathbf{b})\) from now on and write \(\mathbf{a}=\{\varphi_i\}_{i=0}^\infty\) and \(\mathbf{b}=\{\hat{\varphi}_i\}_{i=0}^\infty\). Thus, we have seen that
For every \(x\in M\), there uniquely exists \(\hat{y}_t \in \hat{M}\) such that \[\label{eq460526951} \mu (M)^{1/2} e^{-\lambda_i t/2} \varphi_i (x) = \hat{\mu} (\hat{M})^{1/2} e^{-\hat{\lambda}_i t/2} \hat{\varphi}_i (\hat{y}_t ), \qquad i\ge 1.\tag{9}\]
For every \(\hat{y}\in \hat{M}\), there uniquely exists \(x_t \in M\) such that \[\nonumber \mu (M)^{1/2} e^{-\lambda_i t/2} \varphi_i (x_t) = \hat{\mu} (\hat{M})^{1/2} e^{-\hat{\lambda}_i t/2} \hat{\varphi}_i (\hat{y}), \qquad i\ge 1.\]
Define a homeomorphism \(f_t\colon M \to \hat{M}\) by \(x \mapsto \hat{y}_t\) and another homeomorphism \(h_t\colon \hat{M}\to M\) by \(\hat{y} \mapsto x_t\). By definition, they are the inverses of each other.
Suppose that, at \(x_0 \in M\), there exists a proper subspace \(V\) of \(T_{x_0}M\) which contains \(\{ \nabla \varphi_i (x_0) \mid i\ge 1\}\). Here, \(\nabla\) denotes the gradient operator on \(M\) with respect to any Riemannian metric that tames \(g\). Pick any \(f \in C^\infty (M)\). By Lemma 1, Proposition 1 Lemma 3 and Proposition 2, the series 4 converges in \(C^1\)-topology, which implies that \[\nabla f (x_0) = \sum_{i=1}^\infty \langle f, \varphi_i\rangle \nabla \varphi_i (x_0)\in V.\] This is a contradiction since \(f\) is arbitrary.
Hence, for every fixed \(x_0\in M\), we can find \(1\le i_1 < \cdots < i_d\) such that \(\{ \nabla \varphi_i (x_0) \mid i= i_1, \ldots, i_d\}\) spans \(T_{x_0} M\) (recall that \(d =\dim M\)). Define a smooth map \(F_t\colon M\times \hat{M}\to {\mathbb{R}}^d\) by \[F_t (x, \hat{y}) =\{ \varphi_{i_k} (x) -c_{i_k}(t) \hat{\varphi}_{i_k} (\hat{y}) \}_{k=1}^d, \quad wherec_{i_k}(t):=e^{ (\lambda_{i_k} -\hat{\lambda}_{i_k}) t/2} \hat{\mu} (\hat{M})^{\frac{1}{2}}\mu (M)^{-\frac{1}{2}}.\] By way of construction, \(F_t ( h_t (\hat{y}), \hat{y})\equiv 0\). Set \(\hat{y}_0 = f_t (x_0)\), or equivalently \(x_0 = h_t (\hat{y}_0)\). If we apply \(\nabla\) to the \(x\)-variable at \((x_0, \hat{y}_0)\), we obtain a linear isomorphism from \(T_{x_0} M\) to \({\mathbb{R}}^d\). From the implicit function theorem, the unique implicit function on a neighborhood of \(\hat{y}_0\), which is necessarily smooth, must coincide with \(h_t\). This proves the smoothness of \(h_t\) since \(\hat{y}_0\) is actually arbitrary. In the same way, the smoothness of \(f_t\) can be shown.
The pushforward measure \((f_t)_* \mu\) can be written as \(J_t d\hat{\mu}\), where \(J_t\) stands for the Jacobian of \(h_t\). Integrating both sides of 9 , we have for all \(i\ge 1\) that \[\begin{align} 0 &= \mu (M)^{1/2} e^{-\lambda_i t/2} \int_M\varphi_i (x) \mu(dx) \nonumber\\ &= \hat{\mu} (\hat{M})^{1/2} e^{-\hat{\lambda}_i t/2} \int_M \hat{\varphi}_i (f_t (x)) \mu(dx) \nonumber\\ &= \hat{\mu} (\hat{M})^{1/2} e^{-\hat{\lambda}_i t/2} \int_{\hat{M}} \hat{\varphi}_i (\hat{y}) J_t (\hat{y})\hat{\mu}(dx). \nonumber \end{align}\] Since \(J_t\) is orthogonal to \(\hat{\varphi}_i\) for all \(i\ge 1\), \(J_t\) is identical to the constant \(\mu (M)/\hat{\mu} (\hat{M})\). Integrating the square of 9 gives \(\mu (M) e^{-\lambda_i t}=\hat{\mu} (\hat{M}) e^{-\hat{\lambda}_i t} J_t\), which implies the sets of eigenvalues coincide, i.e. \(\lambda_i = \hat{\lambda}_i\) for all \(i\ge 0\). Thus, we obtain \[\label{eq46same95EF} \varphi_i (x) = a \hat{\varphi}_i (\hat{y}_t ) \quad for all i\ge 1, \quad where a:= \hat{\mu} (\hat{M})^{1/2} /\mu (M)^{1/2} >0.\tag{10}\]
Next, we deduce the following intertwining relation from 10 : \[\label{eq46intertwine} \Delta \circ f_t^* = f_t^*\circ \hat{\Delta}.\tag{11}\] Pick any \(k= \sum_{i=0}^\infty \langle k, \hat{\varphi}_i\rangle \hat{\varphi}_i \in C^\infty (\hat{M})\). Then, \((\hat{\Delta}k) (\hat{y})= \sum_{i=1}^\infty \langle k, \hat{\varphi}_i\rangle \hat{\lambda}_i \hat{\varphi}_i (\hat{y})\) by Corollary 1 and therefore \[\begin{align} (f_t^*\circ \hat{\Delta} k) (x) = (\hat{\Delta}k) (\hat{y}_t) = \sum_{i=1}^\infty \langle k, \hat{\varphi}_i\rangle \hat{\lambda}_i \hat{\varphi}_i (\hat{y}_t) = \sum_{i=1}^\infty \langle k, \hat{\varphi}_i\rangle \frac{\hat{\lambda}_i}{a} \varphi_i (x), \quad x\in M. \nonumber \end{align}\] On the other hand, we can easily see that \[(f_t^* k) (x) = \sum_{i=0}^\infty \langle k, \hat{\varphi}_i\rangle \hat{\varphi}_i (\hat{y}_t) = \sum_{i=0}^\infty \langle k, \hat{\varphi}_i\rangle \frac{1}{a}\varphi_i (x).\] By a very similar argument to Corollary 1, we also have \[(\Delta \circ f_t^* k) (x) =\sum_{i=1}^\infty \langle k, \hat{\varphi}_i\rangle \frac{\lambda_i}{a}\varphi_i (x), \qquad x\in M.\] Thus, we have obtained 11 .
The intertwining relation 11 implies that \((f_t)_* g^* = \hat{g}^*\), that is, the two cometrics are preserved via the diffeomorphism \(f_t\colon M \to \hat{M}\). More precisely, for any \(\psi, \chi \in C^\infty (\hat{M})\) vanishing at \(\hat{y}_t=f_t (x)\), we apply the two operators in 11 to \(-\psi \chi\) and then evaluate it at \(x\). Then, we obtain \[g^*_{x} \langle(J_f)^{\top} d\psi (\hat{y}_t), (J_f)^{\top} d\chi (\hat{y}_t) \rangle = \hat{g}^*_{\hat{y}_t} \langle d\psi (\hat{y}_t), d\chi (\hat{y}_t)\rangle.\] Here, \((J_f)^{\top}\) stands for the transpose of the Jacobian map \(J_f\colon T_x M\to T_{\hat{y}_t} \hat{M}\) of \(f\). Since \(\psi\) and \(\chi\) are arbitrary, we have shown that \[g^*_{x_t} \langle(J_f)^{\top} \xi, (J_f)^{\top} \eta \rangle = \hat{g}^*_{\hat{y}} \langle\xi, \eta\rangle, \qquad \xi, \eta \in T^*_{\hat{y}}\hat{M}, \,\, \hat{y}\in \hat{M}.\] Therefore, \(M\) and \(\hat{M}\) are isometrically isomorphic as sub-Riemannian manifolds. (In particular, \(a=1\).) This completes the proof of our main theorem (Theorem 4). ◻
Let \((M,{\mathcal{D}},g)\) be a compact sub-Riemannian manifold of dimension \(d\) with the rank of \({\mathcal{D}}\) being \(n~(1 \le n \le d)\). (We do not assume the equiregularity in this appendix.) Let \(\mu\) be a smooth volume on \(M\). 1 The sub-Laplacian is defined by \(\Delta =(\nabla^{{\mathcal{D}}})^* \nabla^{{\mathcal{D}}}\). Here, \(\nabla^{{\mathcal{D}}}\) is the horizontal gradient in the direction of \({\mathcal{D}}\) and \((\nabla^{{\mathcal{D}}})^*\) is the adjoint of \(\nabla^{{\mathcal{D}}}\) with respect to \(\mu\). We study the heat semigroup \((e^{-t \Delta})_{t \ge 0}\) associated with \(\Delta\).
The main aim of this appendix is to prove the following \(L^\infty\)-estimate for the first-order derivatives of the heat semigroup. Our proof is probabilistic. It should be noted that the exponent \(\nu\) is independent of \(A, f, t\).
Proposition 5. Let the notation be as above. Then, there exists a constant \(\nu >0\) with the following property: For every \(A \in \Gamma (TM)\), there exists a constant \(C=C_A >0\) (which is independent of \(f\) and \(t\)) such that \[\| A e^{-t \Delta} f\|_{\infty} \le Ct^{-\nu} \| f\|_{\infty}, \qquad f\in C^\infty (M), t \in (0,1].\]
The aim of this subsection is to recall the basics of Malliavin calculus on the classical Wiener space. The reader unfamiliar with this topic is referred to [10]–[13]. We only use standard results in this paper except that our Wiener functionals take values in a manifold. For manifold-valued Malliavin calculus, see Taniguchi [14].
Let \(({\cal W}, {\cal H}, \mathbb{P})\) be the \(d\)-dimensional classical Wiener space, namely,
\({\cal W}= \{ w \colon [0,1]\to {\mathbb{R}}^n \mid w is continuous and w_0 =0\}\) is the Banach space of continuous functions from \([0,1]\) to \({\mathbb{R}}^d\) starting at \(0\), which is equipped with the usual uniform norm.
\(\mathbb{P}\) is the \(d\)-dimensional Wiener measure on \({\cal W}\).
\({\cal H}\) is the Cameron-Martin space: \[\begin{align} {\mathcal{H}} &=\{ h \in W \mid absolutely continuous and\| h\|_{{\mathcal{H}}}^2:=\int_0^1 | h^\prime_t|_{{\mathbb{R}}^n}^2 dt <\infty \}. \end{align}\]
As is well-known, \({\mathcal{H}}\) is a real separable Hilbert space and the coordinate process \((w_t)_{t \in [0,1]}\) is the standard \(d\)-dimensional Brownian motion.
Now, we recall some definitions and basic facts concerning Malliavin calculus on the classical Wiener space \(({\cal W}, {\cal H}, \mathbb{P})\). We often identify \({\mathcal{H}}={\mathcal{H}}^*\) by the Riesz isometry as usual.
We denote by \({\boldsymbol{D}}_{p,r} ({\cal K})\) the Sobolev spaces of \({\cal K}\)-valued Wiener functionals for the integrability index \(p \in (1, \infty)\) and the differentiability index \(r \in [0,\infty)\), where \({\cal K}\) is a real separable Hilbert space. We set \({\boldsymbol{D}}_{\infty} ({\cal K})= \cap_{k=1 }^{\infty} \cap_{1<p<\infty} {\boldsymbol{D}}_{p,k} ({\cal K})\), which is the space of smooth Wiener functional. When \({\cal K} ={\mathbb{R}}\), we write \({\boldsymbol{D}}_{p, r}\) and \({\boldsymbol{D}}_{\infty}\) for simplicity. We denote by \(D\) and \(D^*\) the \({\mathcal{H}}\)-derivative (i.e. the gradient operator in the sense of Malliavin calculus) and its adjoint i.e. (the minus of) the divergence operator, respectively. \(D\) is a bounded linear map from \({\boldsymbol{D}}_{p,r+1} ({\cal K})\) to \({\boldsymbol{D}}_{p,r} ({\cal H}^* \otimes {\cal K})\) and \(D^*\) is a bounded linear map from \({\boldsymbol{D}}_{p,r+1} ({\cal H}^* \otimes{\cal K})\) to \({\boldsymbol{D}}_{p,r} ({\cal K})\) for all \(p \in (1,\infty)\) and \(r \in [0, \infty)\).
For \(F =(F^1, \ldots, F^d) \in {\boldsymbol{D}}_{\infty} ({\mathbb{R}}^d)\), we denote by \(\sigma^{ij}_F (w) = \langle DF^i (w),DF^j (w)\rangle_{{\cal H}^*}\) the \((i,j)\)-component of Malliavin covariance matrix (\(1 \le i,j \le d\)). We denote by \(\gamma^{ij}_F (w)\) the \((i,j)\)-component of the inverse matrix \(\sigma^{-1}_F\) (if it exists). Recall that \(F\) is called non-degenerate in the sense of Malliavin if \((\det \sigma_F)^{-1} \in \cap_{1<p< \infty} L^p\). Note that \(\sigma^{ij}_F \in {\boldsymbol{D}}_{\infty}\) and \(D \gamma^{ij}_F = -\sum_{k,l} \gamma^{ik}_F ( D\sigma^{kl}_F ) \gamma^{lj}_F\). Hence, derivatives of \(\gamma^{ij}_F\) can be written in terms of \(\gamma^{ij}_F\)’s and the derivatives of \(\sigma^{ij}_F\)’s, which implies \(\gamma^{ij}_F \in {\boldsymbol{D}}_{\infty}\), too.
Let us recall the integration by parts formula in the sense of Malliavin calculus for a non-degenerate \(F \in {\boldsymbol{D}}_{\infty} ({\mathbb{R}}^d)\) (see [10]). This formula plays a key role in this appendix. Let \(\psi \colon {\mathbb{R}}^d\to {\mathbb{R}}\) be a bounded \(C^1\)-function with bounded first-order partial derivatives. Then, the following integration by parts formula holds for every \(G \in {\boldsymbol{D}}_{\infty}\): \[\begin{align} {\mathbb{E}} \bigl[ \partial_i \psi (F ) \cdot G \bigr] = {\mathbb{E}} \bigl[ \psi (F ) \cdot \Phi_i (\, \cdot\, ;G) \bigr], \label{ipb146eq} \end{align}\tag{12}\] where \(\partial_i\) stands for the \(i\)th partial differentiation on \({\mathbb{R}}^d\) and \(\Phi_i (w ;G) \in {\boldsymbol{D}}_{\infty}\) is defined by \[\begin{align} \Phi_i (w ;G) &= \sum_{j=1}^d D^* \left( \gamma^{ij }_F \cdot G \cdot DF^j \right) (w). \label{ipb246eq} \end{align}\tag{13}\]
The proof of 12 is rather easy. Indeed, we almost surely have \[\begin{align} \sum_{j=1}^d \langle D(\psi (F )), DF^j \rangle_{{\mathcal{H}}^*} \gamma^{ij }_F &= \sum_{j=1}^d \sum_{k=1}^d \left\langle \partial_k\psi (F ) DF^k, DF^j \right\rangle_{{\mathcal{H}}^*} \gamma^{ij }_F \nonumber\\ &= \sum_{k=1}^d \partial_k\psi (F ) \left( \sum_{j=1}^d \sigma^{kj}_F\gamma^{ij }_F \right) \nonumber\\ &= \sum_{k=1}^d \partial_k\psi (F ) \delta_{ik} \nonumber\\ &= \partial_i\psi (F ), \label{ipb346eq} \end{align}\tag{14}\] where \(\delta_{ik}\) is Kronecker’s delta. By multiplying \(G\), taking expectation and using the definition of \(D^*\), we obtain 12 .
Let us quickly review manifold-valued Malliavin calculus. Malliavin calculus for SDEs on manifolds was founded by Taniguchi [14]. Roughly speaking, under suitable assumptions, almost all of important results in the Euclidean case still hold true in the manifold case with natural modifications. 2
Let \({\cal N}\) be a compact smooth manifold of dimension \(d\). Choose a Riemannian metric \(g\) on \({\mathcal{N}}\) so that the determinant of the Malliavin covariance of \({\mathcal{N}}\)-valued functionals are well-defined. An \({\mathcal{N}}\)-valued Wiener functional \(F\colon {\mathcal{W}}\to {\mathcal{N}}\) is said to belong to \({\boldsymbol{D}}_{\infty} ({\cal N})\) if \(f (F) \in {\boldsymbol{D}}_{\infty}\) for every \(f \in C^{\infty} ({\mathcal{N}})\). If \(\iota\colon {\mathcal{N}}\hookrightarrow {\mathbb{R}}^k\) is an embedding, then \(F \in {\boldsymbol{D}}_{\infty} ({\cal N})\) if and only if \(\iota(F) \in {\boldsymbol{D}}_{\infty} ({\mathbb{R}}^k)\) since every \(f \in C^{\infty} ({\mathcal{N}})\) extends to a smooth function on \({\mathbb{R}}^k\) with compact support.
For \(F \in {\boldsymbol{D}}_{\infty} ({\mathcal{N}})\), \(D_h F (w) \in T_{F(w)} {\mathcal{N}}\). Hence, \(D F (w) \colon {\mathcal{H}}\to T_{F(w)}\) is a bounded linear map and \(\langle\alpha, D F (w)\rangle\in {\mathcal{H}}^* ={\mathcal{H}}\) for every \(\alpha\in T^*_{F(w)} {\mathcal{N}}\). From the Riemannian metric \(g_{F(w)}\) on \(T_{F(w)} {\mathcal{N}}\), we have \(g^*_{F(w)}\) on \(T^*_{F(w)} {\mathcal{N}}\). Take any ONB \(\{e_i\}_{i=1}^d\) of \(T^*_{F(w)} {\mathcal{N}}\). We set \(DF^i (w) := \langle e_i, D F (w)\rangle\) and \[\det \sigma_F (w):=\det \left\{ \langle DF^i (w),DF^j (w)\rangle_{{\cal H}^*} \right\}_{i,j=1}^d.\] Note that this definition is independent of the choice of the ONB, but it does depend on the choice of \(g\). However, even if we choose another Riemmanian metric \(\tilde{g}\), there exists a constant \(c>1\) such that \[\label{eq46det95ctimes} c^{-1} \det \tilde{\sigma}_F (w) \le \det \sigma_F (w) \le c \det \tilde{\sigma}_F (w).\tag{15}\] For this reason, any choice of \(g\) will do for our purpose. We say that \(F\) is called non-degenerate in the sense of Malliavin if \((\det \sigma_F)^{-1} \in \cap_{1<p< \infty} L^p\).
We provide an integration by parts formula for a non-degenerate manifold-valued Wiener functionals \(F \in {\boldsymbol{D}}_{\infty} ({\cal N})\). Below, we provide a “hand-made version" of this formula although a global version, which looks geometrically beautiful, is also known.
Let \(F \in {\boldsymbol{D}}_{\infty} ({\cal N})\) be non-degenerate in the sense of Malliavin and \(\chi \in C^\infty ({\mathcal{N}})\). Suppose that \({\rm supp}(\chi )\) is contained in a certain coordinate neighborhood \(U\). The coordinate of \(U\) is denoted by \((x^1, \ldots, x^d)\). Let \(\hat{\chi} \in C^\infty ({\mathcal{N}})\) be such that \(\hat{\chi} \equiv 1\) on \({\rm supp}(\chi )\) and \({\rm supp}(\tilde{\chi} )\subset U\). If we set \(h^i (x) :=\hat{\chi} (x) x^i\) and \(F^i := h^i (F)\), then \(h^i\) naturally extends to a smooth function on \({\mathcal{N}}\) and \(F^i \in {\boldsymbol{D}}_{\infty}\) (\(1\le i \le d\)). Moreover, on \(\{ w \in {\mathcal{W}}\mid F(w) \in {\rm supp}(\chi )\}\), we have \(F = (F^1, \cdots, F^d)\).
Similarly, if we set \(\hat{\psi} := \psi \hat{\chi}\) for \(\psi \in C^1 ({\mathcal{N}})\), then \(\hat{\psi} \equiv \psi\) on \({\rm supp}(\chi )\). Since \({\rm supp}(\hat{\psi}) \subset U\), it extends to a \(C^1\)-function on \({\mathbb{R}}^d\) with compact support, which will be denoted by the same symbol. Then, we have \[\psi (F) = \hat{\psi} (F^1, \cdots, F^d ) \qquad on \{F \in {\rm supp}(\chi )\}.\] Observe that, on RHS, an \({\mathbb{R}}^d\)-valued Wiener functional is substituted into a \(C^1\)-function on \({\mathbb{R}}^d\).
Now, we denote by \(\sigma^{ij}_F (w) = \langle DF^i (w),DF^j (w)\rangle_{{\cal H}^*}\) the \((i,j)\)-component of Malliavin covariance matrix (\(1 \le i,j \le d\)). Though \(\sigma_F\) can be degenerate outside \(\{F \in {\rm supp}(\chi )\}\), it is non-degenerate on \(\{F \in {\rm supp}(\chi )\}\) and satisfies that \[(\det \sigma_F)^{-1} \cdot \mathbf{1}_{\{F \in {\rm supp}(\chi )\}} \in \cap_{1<p< \infty} L^p, \qquad 1<p <\infty.\] We denote by \(\gamma^{ij}_F (w)\) the \((i,j)\)-component of the inverse matrix \(\sigma^{-1}_F\) on \(\{F \in {\rm supp}(\chi )\}\).
Remark 6. In the above definition of \(\sigma_F\), we used the standard metric on \(T^*{\mathbb{R}}^d\) through the obvious embedding \(U \hookrightarrow {\mathbb{R}}^d\), which is different from the original one \(g^*\) on \(T^* {\mathcal{N}}\) used in the definition of non-degeneracy of \(F\). However, this does not matter since we still have a relation as in 15 on \(\{F \in {\rm supp}(\chi )\}\).
Proposition 7. Let the notation and situation be as above. Then, we have \[\begin{align} {\mathbb{E}} \bigl[ \partial_i \psi (F ) \cdot \chi (F) G \bigr] = {\mathbb{E}} \bigl[ \psi (F ) \cdot \Phi_{i, \chi} (\, \cdot\, ;G) \bigr], \qquad G \in {\boldsymbol{D}}_{\infty}, \,\, 1 \le i \le d. \label{ipb546eq} \end{align}\qquad{(1)}\] Here, (i) \(\partial_i\) is short hand for \(\hat{\chi} (\partial/ \partial x^i)\) (which can be viewed as a vector field on \({\mathcal{N}}\) or on \({\mathbb{R}}^n\)) and (ii) \(\Phi_{i, \chi} (w ;G) \in {\boldsymbol{D}}_{\infty}\) is defined by \[\begin{align} \Phi_{i, \chi} (w ;G) &= \sum_{j=1}^d D^* \left( \gamma^{ij }_F \cdot \chi (F) G \cdot DF^j \right) (w). \label{ipb646eq} \end{align}\qquad{(2)}\] A precise definition of RHS on ?? above will be given in 18 below.
Proof. Concerning ?? , one should note that the left hand side (LHS) does not depend on the choice of \(\hat{\chi}\) and \(\partial_i \psi (F )\) can be replaced by \(\partial_i\hat{\psi} (F^1, \cdots, F^d)\), where \(\partial_i\) stands for the standard partial differentiation on \({\mathbb{R}}^d\).
We will do a similar calculation to 14 . In this case, however, the set \(\{ \det \sigma_F =0\}\) may be of positive measure and cause trouble. For a clean proof, we introduce the following approximation. For \(m\in{\mathbb{N}}\), set \(\sigma_F^m := \sigma_F +m^{-1} {\rm Id}_n\), where \({\rm Id}_n\) is the identity matrix of size \(n\). Clearly, \(\det \sigma_F^m \ge m^{-d}\) and therefore \(\gamma_F^m :=(\sigma_F^m)^{-1}\) exists for almost all \(w\). If \(\det \sigma_F (w)>0\), then \(\gamma_F^m (w) \to \gamma_F (w)\) as \(m\to\infty\). Since \((\det \sigma_F^m )^{-1} \le (\det \sigma_F )^{-1}\), both \[\{ (\det \sigma_F^m )^{-1}\mathbf{1}_{\{F \in {\rm supp}(\chi )\}} \}_{m\in {\mathbb{N}}} \quadand\quad \{ |\gamma_F^m| \mathbf{1}_{\{F \in {\rm supp}(\chi )\}} \}_{m\in {\mathbb{N}}}\] are bounded in \(L^p\) for all \(p\in (1,\infty)\). (For the latter, think of adugate matrices.)
In a very similar way to 14 , we can compute \[\begin{align}\nonumber\\ &= \sum_{j=1}^d {\mathbb{E}} \left[ \langle D(\hat{\psi} (F^1, \ldots, F^d )), \, DF^j \rangle_{{\mathcal{H}}^*} \gamma^{m, ij }_F \chi(F)G \right] \nonumber\\ &= \sum_{j=1}^d \sum_{k=1}^d {\mathbb{E}} \left[ \partial_k\hat{\psi} (F^1, \ldots, F^d ) \sigma^{kj }_F \gamma^{m, ij }_F \chi(F)G \right]. \label{ipb746eq} \end{align}\tag{16}\] On \(\{F \in {\rm supp}(\chi )\}\), \(\lim_{m\to\infty}\gamma^{m, ij }_F =\gamma^{ij }_F\), a.s. and the integrand of RHS of 16 is uniformly integrable, which follows from \(L^p\)-boundedness for all \(1<p<\infty\). So, RHS converges to \({\mathbb{E}} [\partial_i\hat{\psi} (F^1, \ldots, F^n ) \chi(F)G] = {\mathbb{E}} [\partial_i\psi (F) \chi(F)G]\), which equals LHS of ?? .
Next, we compute LHS of 16 . By the basic property of \(D^*\), we can easily see that \[\begin{align} D^* \left( \gamma^{m, ij }_F \cdot \chi (F) G \cdot DF^j \right) &= \gamma^{m, ij }_F \chi (F) G ( D^*DF^j ) -\gamma^{m, ij }_F \chi (F) \langle DG, DF^j \rangle_{{\mathcal{H}}^*} \nonumber\\ & -\gamma^{m, ij }_F G\langle D (\chi (F)), DF^j \rangle_{{\mathcal{H}}^*} -\chi (F)G\langle D \gamma^{m, ij }_F, DF^j \rangle_{{\mathcal{H}}^*}. \label{ipb846eq} \end{align}\tag{17}\] First, since \(D\) is a kind of stochastic Gateaux derivative, \(D (\chi (F))\) vanishes outside \(\{F \in {\rm supp}(\chi )\}\) and so does RHS of 17 . Hence, \(\hat{\psi} (F^1, \ldots, F^n )\) on LHS of 16 can be replaced by \(\psi (F)\) as one can easily expect. By the same argument as above the first three terms on RHS of 17 converges in \(L^1\) to \[\{ \gamma^{ij }_F \chi (F) G ( D^*DF^j ) -\gamma^{ij }_F \chi (F) \langle DG, DF^j \rangle_{{\mathcal{H}}^*} -\gamma^{ij }_F G\langle D (\chi (F)), DF^j \rangle_{{\mathcal{H}}^*}\} \mathbf{1}_{\{F \in {\rm supp}(\chi )\}}\] as \(m\to\infty\). To deal with the fourth term, one should recall that \[D \gamma^{ij}_F = -\sum_{k,l=1}^d \gamma^{m, ik}_F ( D\sigma^{kl}_F ) \gamma^{m,lj}_F.\] By the same reasoning as above, the fourth term converges in \(L^1\) to \[\chi (F)G \sum_{k,l=1}^d \langle\gamma^{ik}_F ( D\sigma^{kl}_F ) \gamma^{lj}_F, DF^j \rangle_{{\mathcal{H}}^*} \mathbf{1}_{\{F \in {\rm supp}(\chi )\}}\] as \(m\to\infty\). Thus, if we precisely define \(\Phi_{i, \chi} (w ;G)\) in ?? by \[\begin{align} \Phi_{i, \chi} (\,\cdot\, ;G) &:= \mathbf{1}_{\{F \in {\rm supp}(\chi )\}}\sum_{j=1}^d \{ \gamma^{ij }_F \chi (F) G ( D^*DF^j ) -\gamma^{ij }_F \chi (F) \langle DG, DF^j \rangle_{{\mathcal{H}}^*} \nonumber\\ &\quad -\gamma^{ij }_F G\langle D (\chi (F)), DF^j \rangle_{{\mathcal{H}}^*} +\chi (F)G \sum_{k,l=1}^d \langle\gamma^{ik}_F ( D\sigma^{kl}_F ) \gamma^{lj}_F, DF^j \rangle_{{\mathcal{H}}^*}\}, \label{ipb946eq} \end{align}\tag{18}\] we are done. (The right hand side of 18 should be understood to be \(0\) outside \(\{F \in {\rm supp}(\chi )\}\).) ◻
Corollary 4. Let the notation and situation be as in Proposition 7. Then, we have \[\begin{align} {\mathbb{E}} \bigl[ A \psi (F ) \cdot \chi (F) G \bigr] = {\mathbb{E}} \bigl[ \psi (F ) \cdot \Phi_{A, \chi} (\, \cdot\, ;G) \bigr], \qquad G \in {\boldsymbol{D}}_{\infty}, \, A\in \Gamma ({\mathcal{N}}). \label{ipb1046eq} \end{align}\tag{19}\] Here, \(\Phi_{A, \chi} (\, \cdot\, ;G)\) is defined as follows: First, write \(A (x) =\sum_{i=1}^d a^i (x) \partial_i\) on \(U\) and set \(\hat{a}^i = \hat{\chi} a^i\) so that \(\hat{a}^i \in C^\infty ({\mathcal{N}})\) for all \(1 \le i \le d\). Then, we set \[\begin{align} \Phi_{A, \chi} (\, \cdot\, ;G) := \sum_{i=1}^d \Phi_{i, \chi} (\, \cdot\, ;\hat{a}_i (F) G), \label{ipb1146eq} \end{align}\tag{20}\] where \(\Phi_{i, \chi}\) was defined in ?? and 18 .
Proof. Since LHS equals \(\sum_{i=1}^d {\mathbb{E}} \bigl[\hat{a}_i (F) \partial_i \psi (F ) \cdot \chi (F) G \bigr]\), this corollary follows immediately from Proposition 7. ◻
Since \(-\Delta\) cannot be written as a sum-of-squares operator, we cannot construct the associated diffusion process as a solution of stochastic differential equation (SDE) on \(M\). However, if we consider a suitable SDE on a principal bundle over \(M\), the projection of a solution is the desired diffusion process on \(M\). This method is called Eells-Elworthy’s construction. The classical case in the Riemannian setting is in [16]. A nice exposition on its sub-Riemannian version can br found in [17]. Similar computations are in [9], [18], too. Our exposition below is borrowed from [19].
Take a Riemmanian metirc \(\hat{g}\) on \(M\) which tames \(g\), that is, \(\hat{g}|_{{\mathcal{D}}\times{\mathcal{D}}}=g\). Any choice of \(\hat{g}\) will do. Denote by \({\mathcal{D}}_x^{\perp}\) the orthogonal complement of \({\mathcal{D}}_x\) in \(T_xM\) with respect to \(\hat{g}_x\) for \(x\in M\) and we set \({\mathcal{D}}^{\perp} =\sqcup_{x\in M}{\mathcal{D}}_x^{\perp}\), which is a subbundle of \(TM\).
Now we introduce a principal bundle over \(M\) with structure group \(O (n) \times O(d-n)\), which is the product of two orthogonal groups, acting on it from the right. \[\begin{align} O (M; {\mathcal{D}}\oplus {\mathcal{D}}^{\perp})_x &= \{u\colon {\mathbb{R}}^n \oplus {\mathbb{R}}^{d-n} ={\mathbb{R}}^d\to T_xM = {\mathcal{D}}_x \oplus {\mathcal{D}}^{\perp}_x \mid \\ & \qquad u\restriction_{{\mathbb{R}}^n} \colon {\mathbb{R}}^n \to {\mathcal{D}}_x and u\restriction_{{\mathbb{R}}^{d-n}} \colon {\mathbb{R}}^{d-n} \to {\mathcal{D}}^{\perp}_x are linear isometries \}, \\ O (M; {\mathcal{D}}\oplus {\mathcal{D}}^{\perp})&= \bigsqcup_{x\in {\mathcal{M}}} O (M; {\mathcal{D}}\oplus {\mathcal{D}}^{\perp})_x. \end{align}\] This is a subbundle of the orthonormal frame bundle over the Riemannian manifold \((M, \hat{g})\). For notational simplicity we will write \({\mathcal{P}}:= O (M; {\mathcal{D}}\oplus {\mathcal{D}}^{\perp})\) and \(G:= O (n) \times O(d-n)\). The Lie algebra of \(G\) is \({\frak o} (n) \times {\frak o} (d-n)\), which will be denoted by \({\frak g}\). Here, \({\frak o} (n)\) stands for the set of real \(n \times n\) skew-symmetric matrices. The natural projection will be denoted by \(\pi\colon {\mathcal{P}}\to M\).
If we take a suitable Ehresmann connection \(\omega\) on \({\mathcal{P}}\), which is a \({\frak g}\)-valued one-form on \({\mathcal{P}}\), the properties described below, including Relation 21 , are known to hold. (Such an \(\omega\) is concretely constructed in [19].) As usual, we define the horizontal subspace \({\mathcal{K}}_u\subset T_u {\mathcal{P}}\) by \[{\mathcal{K}}_u=\{A\in T_u{\mathcal{P}}\,|\, \omega_u(A_u)=0\}, \qquad u\in {\mathcal{P}}.\] Then, the horizontal lift \(\ell_u\colon T_{\pi(u)}M\to {\mathcal{K}}_u\), which is a linear bijection, is also defined uniquely and naturally.
Let \(\{{\mathbf{e}}_i\mid 1\le i\le n\}\) be the canonical ONB of \({\mathbb{R}}^n\). Define the canonical horizontal vector fields \(L_i\) on \({\mathcal{P}}\) by \((L_i)_u=\ell_u(u{\mathbf{e}}_i)\) for \(1 \le i \le n\). Then, there exists a unique \(V_0 \in \Gamma ({\mathcal{D}})\) such that, for all \(f \in C^\infty (M)\), \[\label{eq468964-1} \left( \sum_{i=1}^n L_i^2 +L_0 \right) (f \circ \pi ) = - (\Delta f) \circ \pi \qquad on {\mathcal{P}},\tag{21}\] where \(L_0 := \ell (V_0)\) is the horizontal lift of \(V_0\). Therefore, the law of the \((\sum_{i=1}^n L_i^2 +L_0)\)-diffusion process on \({\mathcal{P}}\) starting at \(u\) projects down to the law of the \((-\Delta)\)-diffusion process on \(M\) starting at \(\pi (u)\). Hence, for two starting points \(u\) and \(u^\prime\), the law of the projected processes coincide if \(\pi (u)= \pi (u^\prime)\). Thanks to these facts, Eells-Elworthy’s construction is available for the \((-\Delta)\)-diffusion process on \(M\).
As is well-known, we can realize the diffusion process as a solution of a stochastic differential equation (SDE). Let \((w_t)_{t \in [0,1]}\) be a standard \(n\)-dimensional Brownian motion. We consider the following SDE on \({\mathcal{P}}\): Let \((r(t, u))_{t\in [0,1]}\) be the unique solution to the following Stratonovich SDE on \({\mathcal{P}}\): \[\label{def46sde95OD} dr(t, u)=\sqrt{2}\sum_{i=1}^n L_i(r(t, u))\circ dw^i_t +L_0(r(t,u))dt, \quad r(0, u)=u\in {\mathcal{P}}.\tag{22}\] By the reason we have just stated, we have a Feynman-Kac type representation: For all \(f\in C^\infty (M)\) and \(x\in M\), we have \[\label{eq46rep95FK} e^{-t \Delta} f (x) = \mathbb{E} \left[ (f\circ \pi) ( r(t,u)) \right] \qquad for every u\in \pi^{-1}(x).\tag{23}\] Since RHS (as a function of \(u\)) depends only on \(\pi (u)\), we see that \[\label{eq46rep95FK2} Ae^{-t \Delta} f (x) = (\ell A)_u\mathbb{E} \left[ (f\circ \pi) ( r(t,u)) \right] \qquad for every u\in \pi^{-1}(x).\tag{24}\] Here, \(\ell A\) is the horizontal lift of \(A \in \Gamma (TM)\). Hence, the proof of Proposition 5 reduces to estimating the supremum of \((\ell A)\,\mathbb{E} \left[ (f\circ \pi) ( r(t,\,\cdot\,)) \right]\) over \({\mathcal{P}}\).
By the Hörmander condition on \({\mathcal{D}}\), \(\{L_1, \ldots, L_n\}\) satisfies the partial Hörmander condition at every \(u\in {\mathcal{P}}\). A precise statement of this condition is as follows: Set \[\Sigma_1 :=\{L_1, \ldots, L_n\} \quadand \quad \Sigma_k := \{ [V, L_i] \mid V\in \Sigma_{k-1}, \, 1\le i \le n\} \quadfor k \ge 2.\] Then, for every \(u \in {\mathcal{P}}\), the linear span of the subset \(\{ (\pi_*)_u W_u \mid W\in \cup_{k=1}^\infty \Sigma_k\}\) equals \(T_{\pi (u)}M\). According to [14], the \(M\)-valued Wiener function \(\pi (r(t,u))\), \(t\in (0, 1]\) and \(u\in {\mathcal{P}}\), belongs to \({\boldsymbol{D}}_{\infty} (M)\) and non-degenerate in the sense of Malliavin.
Furthermore, since \({\mathcal{P}}\) is compact, the partial Hörmander condition is automatically uniform, that is, there exists \(K\in {\mathbb{N}}\) independent of \(u\) such that \[\inf_{u\in {\mathcal{P}}} \inf \left\{ \sum_{k=1}^K \sum_{W\in \Sigma_k} \hat{g}_u ( (\pi_*)_u W_u, \eta)^2 ~\middle|~ \eta \in T_{\pi (u)} Mwith\hat{g}_u (\eta, \eta)=1\right\}>0.\] Thanks to the above condition, a Kusuoka-Stroock type estimate is known to hold as in the Euclidean case: There exist positive constants \(c_p\) and \(\nu_1\) such that \[\label{ineq46KS95est} \| \{\det \sigma_{\pi (r (t, u))} \}^{-1} \|_{L^p} \le \frac{c_p}{t^{\nu_1}}, \qquad t\in (0,1], \, u \in {\mathcal{P}}, \, 1<p<\infty.\tag{25}\] Here, \(\nu_1\) is independent of \((p, t, u)\) and \(c_p\) is independent of \((t, u)\).
Remark 8. Among some variants of the sub-Riemannian Eells-Elworthy construction (see e.g. [9], [17]–[19]), we used one in [19]. The main reason is to avoid using a partial connection. A drawback of using a partial connection is that \(A \in \Gamma (TM)\) does not in general admit a horizontal lift unless \(A \in \Gamma ({\mathcal{D}})\) and, consequently, Formula 24 breaks down.
The aim of this subsection is to prove Proposition 5 by carrying out Malliavin calculus (in particular, the integration by parts formula) for the \({\mathcal{P}}\)-valued solution \((r(t,u))_{t\in [0,1]}\). We remark that similar (or more complicated) computations were already done in the series of works [9], [18], [19].
We use a Nash embedding \(\iota\colon M \hookrightarrow {\mathbb{R}}^N\) with respect to \(\hat{g}\) for sufficiently large \(N\in{\mathbb{N}}\). (We will often identify \(M\) and \(\iota (M)\) and just write \(M \subset {\mathbb{R}}^N\).) If \(f\in C^\infty (M)\), then \(f\) extends to \(\tilde{f}\in C^\infty ({\mathbb{R}}^N)\) with compact support with the following property at every \(x\in M\): If \(\mathbf{n}_x\in T_x {\mathbb{R}}^N \cong {\mathbb{R}}^N\) is normal to \(T_xM \subset {\mathbb{R}}^N\), then \(\mathbf{n}_x \tilde{f}=0\) at \(x\). At every \(x\in M\), \(T_x {\mathbb{R}}^N\) admits a natural orthogonal decomposition into the tangent and normal subspaces. For \(V \in \Gamma (T {\mathbb{R}}^N)\), we define a vector field \(V^{{\rm tan}}\) on \(M\) by imposing \((V^{{\rm tan}})_x\) to be the tangent component to \(T_xM\) of \(V_x\). Moreover, \(V^{{\rm tan}}\) is smooth, i.e. \(V^{{\rm tan}} \in\Gamma (M)\). Then, \(V \tilde{f} \equiv (V^{{\rm tan}}) f\) on \(M\).
Next, we introduce a beautiful embedding of \({\mathcal{P}}= O (M; {\mathcal{D}}\oplus {\mathcal{D}}^{\perp})\). Let \(\{{\mathbf{e}}_i\mid 1\le i\le d\}\) be the canonical ONB of \({\mathbb{R}}^d\). For \(u\in {\mathcal{P}}\), we set \[\iota^\prime (u) :=\left( \iota (\pi (u)); \, (\iota_*)_{\pi (u)} (u{\mathbf{e}}_1 ), \ldots, (\iota_*)_{\pi (u)} (u{\mathbf{e}}_d) \right) \,\, \in {\mathbb{R}}^N \times ({\mathbb{R}}^N)^d ={\mathbb{R}}^{N(1+d)}.\] Since \(\iota\) is an isometric embedding, \(\{(\iota_*)_{\pi (u)} (u{\mathbf{e}}_i) \}_{i=1}^d\) is orthonormal in \({\mathbb{R}}^N\). Then, \(\iota^\prime\colon {\mathcal{P}}\hookrightarrow {\mathbb{R}}^{N(1+d)}\) is an embedding. Moreover, these embeddings respect the structure of projections, that is, \(\iota \circ \pi = \pi^\prime \circ \iota^\prime\). Here, \(\pi^\prime \colon {\mathbb{R}}^N \times ({\mathbb{R}}^N)^d \to {\mathbb{R}}^N\) is the projection that picks up the leftmost component of \({\mathbb{R}}^N \times ({\mathbb{R}}^N)^d ={\mathbb{R}}^{N(1+d)}\). Thanks to this fact, we need not distinguish \(\pi\) and \(\pi^\prime\) when we identify \({\mathcal{P}}\) and \(\iota^\prime({\mathcal{P}})\), which makes our computations quite simple. When there is no fear of confusion, we will identify \(\iota^\prime({\mathcal{P}}) = {\mathcal{P}}\) and write \(\pi^\prime =\pi\) for simplicity of notation. (For the orthonormal bundle over a Riemannian manifold, this kind of embedding is well-known. Our version here is a slight modification.) In coordinates, a generic element \(u \in {\mathbb{R}}^N \times ({\mathbb{R}}^N)^d\) is denoted by \(u=(u_{kl})_{0\le k \le d, 1\le l\le N}\) and \(\pi^\prime u=(u_{0l})_{1\le l\le N}\).
Any smooth vector field on \({\mathcal{P}}\) extends to one on \({\mathbb{R}}^{N(1+d)}\) with compact support, which will be denoted by the same symbol again (any such extension will do for our purpose). Then, SDE 22 can be realized in \({\mathbb{R}}^{N(1+d)}\): \[\label{def46sde95OD2} dr(t, u)=\sqrt{2}\sum_{i=1}^n L_i(r(t, u))\circ dw^i_t +L_0(r(t,u))dt, \quad r(0, u)=u\in {\mathbb{R}}^{N(1+d)}.\tag{26}\] When \(u \in \iota^\prime({\mathcal{P}})\), the solution coincides with that of 22 . For this reason, we will use the same symbol \((r(t, u))_{t\in [0,1]}\) again by slightly abusing the notation. Since \(\ell A\) extends to a vector field on the ambient space with compact support, there is a constant \(C>0\) such that \[\label{eq46starbucks} \sup_{u\in {\mathcal{P}}} |(\ell A)_u\,\mathbb{E} \left[ (f\circ \pi ) ( r(t,u)) \right] | \le C \sup_{k,l} \sup_{u\in {\mathbb{R}}^{N(1+d)}} \left| \frac{\partial}{\partial u_{kl}} \mathbb{E} \left[ (\tilde{f} \circ \pi )( r(t,u)) \right] \right|.\tag{27}\] Hence, it suffices to estimate RHS of 27 above.
Since SDE 26 is on a Euclidean space, it has been thoroughly studied. We recall some results now. Denote by \((J (t,u))_{t \in [0,1]}\) the Jacobian process of \((r (t,u))_{t \in [0,1]}\), namely, \(J (t,u)= \nabla r (t,u)\). Here, \(\nabla\) stands for the standard gradient on \({\mathbb{R}}^{N(1+d)}\) with respect to the \(u\)-variable. Firstly, for all \(p \in (1, \infty)\) and \(r \in [0,\infty)\), we have \[\label{ineq46SDE1} \sup_{u\in {\mathbb{R}}^{N(1+d)}} \sup_{t\in [0,1]} \{ \|r(t,u)\|_{{\boldsymbol{D}}_{p,r}} + \|J (t,u)\|_{{\boldsymbol{D}}_{p,r}} \}<\infty.\tag{28}\] Secondly, we also have \[\label{ineq46SDE2} \mathbb{E} \left[ \sup_{u\in {\mathbb{R}}^{N(1+d)}} |J (t,u) | \right] <\infty, \qquad 0\le t \le 1.\tag{29}\] Note that the sup is inside the expectation in 29 .
Take a partition of unity \(1 \equiv \sum_{m =1}^K \chi_m\) such that \({\rm supp} (\chi_m)\) is contained in a coordinate chart for each \(m\). By 29 and Lebesgue’s dominated convergence theorem, one can see that \[\begin{align}\nonumber\\ &= \mathbb{E} \left[\partial_{kl} \bigl\{ (\tilde{f} \circ \pi )( r(t,u)) \bigr\}\right] \nonumber\\ &= \sum_{j=1}^N \mathbb{E} \left[ (\partial_{0j} \tilde{f}) (\pi( r(t,u)) )\cdot J_{0j, kl} (t,u)\right] \nonumber\\ &= \sum_{j=1}^N \sum_{m=1}^K \mathbb{E} \left[ (\partial_{0j}^{{\rm tan}} f) (\pi( r(t,u))) \cdot \chi_m (\pi( r(t,u))) \cdot J_{0j, kl} (t,u)\right], \label{ineq46SDE3} \end{align}\tag{30}\] where \(\partial_{kl} =\partial/\partial u_{kl}\) and \(J_{0j, kl} (t,u)\) denotes the matrix component of \(J (t,u)\). To check the second equality above, just recall that \(\pi( r(t,u)) = \{ r(t,u)_{0j}\}_{1 \le j\le N}\).
Since \(\partial_{0j}^{{\rm tan}} \in \Gamma (M)\), \(J_{0j, kl} (t,u) \in {\boldsymbol{D}}_\infty\) and \(\pi( r(t,u)) \in {\boldsymbol{D}}_\infty (M)\) is non-degenerate, we can apply Corollary 4 (and Proposition 7) to RHS of 30 by setting \[F = \pi( r(t,u)),\,\, G =J_{0j, kl} (t,u), \,\, A = \partial_{0j}^{{\rm tan}}, \,\, \psi =f, \,\, \chi =\chi_m.\] In the explicit expression of the “Malliavin weight" \(\Phi_{A, \chi}\) in 18 and 20 , all but one factor are \(L^p\)-bounded in \((t, u)\) for every \(p \in (1,\infty)\), thanks to 28 . The only exception is \(\gamma_F\), the inverse of Malliavin covariance matrix. As for \(\gamma_F\), it holds that \[|\gamma_F| \le c^\prime (\det \sigma_{F})^{-1} |\sigma_{F}|^{d-1} \le c^\prime (\det \sigma_{F})^{-1} \|F\|_{{\boldsymbol{D}}_{p,1}}^{2(d-1)},\] where the constant \(c^\prime >0\) does not depend on \(F\). 3 Thanks to the Kusuoka-Stroock estimate 25 , which is uniform in \((t,u)\), we have \[\| \gamma_{\pi( r(t,u))} \|_{L^2} \le \frac{c_1}{t^{\nu_1}}, \qquad t\in (0,1], \, u \in {\mathcal{P}}.\] Here, \(c_1\) and \(\nu_1\) are certain positive constants independent of \((t, u)\). Combining all arguments above, we can see that RHS of 30 is dominated by \(c_2/t^{\nu_2}\) for all \(t\in (0,1]\) and \(u \in {\mathcal{P}}\), where \(c_2\) and \(\nu_2\) are certain positive constants independent of \((t, u)\). This completes the proof of Proposition 5.
Acknowledgement: The author is grateful to Professor Shouhei Honda for helpful comments. The author is supported by JSPS KAKENHI Grant No. 26K06846.
| Yuzuru Inahama | |
| Faculty of Mathematics, | |
| Kyushu University, | |
| 744 Motooka, Nishi-ku, Fukuoka, 819-0395, JAPAN. | |
Email: inahama@math.kyushu-u.ac.jp |
A measure on \(M\) is said to be a smooth volume if its restriction to any coordinate chart is absolutely continuous with respect to the Lebesgue measure of the chart and the density is smooth and strictly positive.↩︎
In a recent book [15] by Taniguchi himself, this topic (including the content of [14]) is explained in details. Unfortunately, this book is written in Japanese, however.↩︎
The inverse of a matrix \(Z\) equals \((\det Z)^{-1}\) times the adjunct (adugate) matrix \(\mathrm{adj} (Z)\) of \(Z\).↩︎