Local Invariants and Geometry of
the sub-Laplacian on H-type foliations

Sub-Riemannian geometry and the analysis of intrinsically associated differential equations, such as the sub-Riemannian heat or sub-Riemannian wave equation, have attracted increasing interest during the last decades [1, 2, 5, 9, 13, 14, 15, 17, 24, 34, 40]. Sub-Riemannian geometry is the study of bracket generating metric distributions inside the tangent bundle of a given manifold. On the one hand, such geometries allow for the definition of a metric space structure, of tangent groups or geodesic curves. On the other hand, on a regular sub-Riemannian manifold an analytic object, namely the intrinsic hypoelliptic sub-Laplacian, can be defined in form of a second order differential operator [2, 6] and generalizes the Beltrami-Laplace operator in Riemannian geometry. The sub-Riemannian heat kernel $K_{\mathrm{sub}}$ is the fundamental solution of the corresponding heat operator and it is expected to encode specific geometric data of the sub-Riemannian structure. With varying degrees of generality the existence of the small time asymptotic expansion of $K_{\mathrm{sub}}$ or of the sub-Riemannian heat trace on compact manifolds has been obtained by analytical or stochastic methods [14, 17, 23, 24, 28, 40].

In some cases even explicit formulas for the sub-Riemannian heat kernel are available [7, 8, 11, 12, 14] and are of great use in the asymptotic analysis. However, due to a large variety of sub-Riemannian structures (e.g. see [36] for many examples) it seems that a geometric interpretation of the coefficients (heat invariants) in the small time asymptotic expansion of $K_{\mathrm{sub}}$ requires a case-by-case analysis. We mention the results for unimodular Lie groups [40], particular sub-Riemannian manifolds with symmetries [5], sub-Riemannian structures on the sphere $\mathbb{S}^{7}$ [13], and on quaternionic contact (qc) manifolds [34]. In the present paper we study the geometric interpretation of the second heat invariant for the intrinsic sub-Laplacian on $H$-type foliations. Introduced in [10], $H$-type foliations form good model spaces inside the family of sub-Riemannian manifolds with step two bracket generating distribution. An $H$-type foliation $\mathbb{M}$ carries an horizontal distribution $\mathcal{H}$ of constant rank with a totally geodesic and integrable complement $\mathcal{V}$. Moreover, $\mathbb{M}$ is the total space of a Clifford bundle adapted to the splitting $\mathcal{H}\oplus\mathcal{V}$ of the tangent space (see Section 2 for a precise definition). $H$-type foliations generalize previously studied examples such as Sasakian manifolds, Twistor spaces or 3-Sasakian manifolds and they carry a canonical connection (Bott-connection) which induces useful notions of horizontal and vertical curvature. We mention that the class of $H$-type foliations has a non-empty intersection with qc-manifolds, similar aspects of which have been recently studied by the third author in [34].

The local models (tangent groups) of an H-type foliation $\mathbb{M}$ are H(eisenberg)-type Lie groups $G$ which were introduced by A. Kaplan in the 80th in his study of fundamental solutions to hypoelliptic PDE, cf. [31]. The corresponding $H$-type Lie algebras are in one-to-one correspondence with representations of Clifford algebras [22]. Such algebras are nilpotent of step two and the complement of their center (first layer) is naturally identified with a left-invariant bracket generating distribution which we call horizontal. Equipped with an inner product, which is induced by the metric on $\mathbb{M}$, the (tangent) $H$-type Lie algebras define the first algebraic and metric invariants of $\mathbb{M}$ as a sub-Riemannian manifold (see [37]).

$H$-type groups endowed with a left-invariant metric have the structure of a sub-Riemannian manifold themselves and subsequently have become an important class of examples in PDE [20, 25, 38], geometry and analysis [18, 32] or geometric measure theory, [3, 26]; see also the survey paper [41]. In analogy to known results in Riemannian geometry or the analysis in [34] we expect that for sufficiently well-behaved examples the second heat invariant of the sub-Laplacian can be expressed using a suitable concept of curvature. We remark that the horizontal distribution on a step two nilpotent Lie group has vanishing curvature as well as vanishing second heat invariant. The class of $H$-type foliations, which we are considering in this paper, is more interesting when studying the effect of curvature terms on the heat kernel analysis of the sub-Laplacian. Roughly speaking, such manifolds may be thought of as “curved versions” of the $H$-type groups.

The task of expressing the second heat invariant of the sub-Laplacian on an $H$-type foliation in terms of curvature tensors requires the choice of a connection which is suitably adapted to the sub-Riemannian and Clifford bundle structure of $\mathbb{M}$. In this paper we choose the Bott connection in [45, Chapter 5], see also [27] which - different from the Levi-Civita connection in Riemannian geometry - has non-vanishing torsion. In case of a contact manifold or for pseudo-Hermitian or strictly pseudo-convex CR manifolds a natural connection is given by the Tanno connection [44] and the Tanaka-Webster connection [43, 46], respectively. On qc-manifolds we may as well use the Biquard connection [19, 34].

We mention that all these examples have non-empty intersections with the class of $H$-type foliations. In these intersections the Bott connection coincides with the Tanno and the Tanaka-Webster connection, respectively, and it differs from the Biquard connection by torsion properties. Different choices of a connection may lead to different representations of the second heat invariant [19].

Our heat kernel analysis is based on an appropriate choice of privileged coordinates on $\mathbb{M}$ constructed by Kunkel in [26] (see also [30]) combined with recent methods from [24]. In fact, an inspection of the proofs in [24] shows that the second heat invariant can be obtained as a convolution integral. The integrand is built from the heat kernel of the sub-Laplacian of the local nilpotent approximation (see [14] for an explicit formula) and an operator which, in particular, depends on the curvature tensor, see Eq. 5.3.

Our main result concerning the geometric interpretation of the second heat invariant for an $H$-type foliation is stated below as Theorem A and will be explained more in detail next.

Let $q\in\mathbb{M}$ and $t>0$. As is well-known the heat kernel $K_{\textup{sub}}$ of the intrinsic sub-Laplace operator $\Delta_{\textup{sub}}$ on $\mathbb{M}$ has a short time asymptotic expansion of the form

Here $Q$ denotes the Hausdorff dimension of $\mathbb{M}$ as a metric space equipped with the Carnot-Carathéodory metric. The function $c_{j}(q)$ with $j=0,1,2\ldots$ is referred to as the $(j+1)$-th heat invariant of the sub-Riemannian structure.

Let $\nabla$ denote the Bott connection on $\mathbb{M}$ with torsion $T$. Moreover, $\kappa_{\mathcal{H}}$ and $\tau$ are the horizontal scalar curvature and an invariant defined through the vertical bundle, respectively. We note that an $H$-type foliation $\mathbb{M}$ is said to have horizontally parallel torsion when $\nabla_{\mathcal{H}}T=0$.

We address two further problems in connection with $H$-type foliations which so far seemed to not have been settled in the literature. The first question concerns the asymptotic behaviour of the Popp volume vol of small balls on $\mathbb{M}$. Let $r>0$ and consider the preimage $B(q,r)\subset\mathbb{M}$ around $q\in\mathbb{M}$ of a Korányi ball in $\mathbb{R}^{n+m}$ of radius $r$ under (parabolic) privileged coordinates. In Section 4 it is shown:

Finally, we give a sufficient and necessary condition for an $H$-type foliation with horizontally parallel torsion to be locally isometric as a sub-Riemannian manifold to its tangent group ($H$-type group).

The structure of the paper is as follows: In Section 2 we recall some basic notions in sub-Riemannian geometry and we introduce $H$-type foliations $\mathbb{M}$ which form a model class of sub-Riemannian manifolds with a Clifford bundle structure, [10].

We recall the construction of privileged coordinates [33] in Section 3. Throughout the paper all local calculations will be performed within such coordinates which are well-adapted to the sub-Riemannian structure on $\mathbb{M}$. In particular, the coordinate system obeys linear scaling in horizontal directions much like Riemannian normal coordinates, but obeys parabolic scaling in vertical directions reflecting the bracket structure. An essential ingredient to our analysis is the calculation of the homogeneous parts of the associated co-frame and connection one-forms in 3.3. Finally, C (3.7) is achieved.

In Section 4 we calculate the Popp volume $\mathcal{P}$ on an $H$-type foliation which is used in the definition of the intrinsic sub-Laplacian $\Delta_{\mathrm{sub}}$. A local expression of $\Delta_{\mathrm{sub}}$ is then given in 4.4. Section 4 contains a proof of B (4.3) as its main result.

Based on our calculations in local privileged coordinates and methods from [24] we introduce a new invariant $\tau$ in Section 5 and derive a representation of the second heat invariant as a linear combination of $\tau$ and the horizontal scalar curvature $\kappa_{\mathcal{H}}$ with respect to the Bott connection. This result can be further refined when assuming horizontally parallel Clifford structure. The section concludes by proving 5.12 (A).

The paper ends with an appendix which collects various technical calculations.

In the setting of sub-Riemannian geometry, one works with a smooth manifold $\mathbb{M}$ of dimension $n+m$ equipped with a pair $(\mathcal{H},g_{\mathcal{H}})$ where $\mathcal{H}$ is a rank $n$ subbundle of $T\mathbb{M}$ and $g_{\mathcal{H}}$ is a symmetric, positive-definite (2,0)-tensor on $\mathcal{H}$. We insist that the bracket-generating condition holds, which we say is satisfied if at every point $p\in\mathbb{M}$ one can generate all of $T_{p}\mathbb{M}$ by taking sufficiently many Lie brackets of vector fields in $\Gamma(\mathcal{H})$ at $p$. Notably, this is equivalent to Hörmander’s condition in PDEs [36]. One can define the sub-Riemannian (Carnot-Carathéodory) distance $d_{cc}(p,q)$ for two points $p,q\in\mathbb{M}$ by the usual infimum formula taken over the space of smooth paths $C_{\mathcal{H},p,q}$ connecting $p$ to $q$ such that $\dot{\gamma}(t)\in\mathcal{H}_{\gamma(t)}$ at almost every point along $\gamma$. The now famous theorem of Chow [21] and Rashevsky [39] tells us that $(\mathbb{M},d_{cc})$ is a metric space when $\mathcal{H}$ is bracket-generating.

Such a triple $(\mathbb{M},\mathcal{H},g_{\mathcal{H}})$ is called a sub-Riemannian manifold. These objects arise naturally in myriad settings and have been extensively studied in recent decades with increasing interest, see [1, 36, 42] for an overview.

An established approach is to study sub-Riemannian manifolds by equipping a Riemannian penalty metric

parameterized by $\varepsilon>0$ on an appropriately chosen complementary distribution $\mathcal{V}$.

In some key examples there is a natural choice of a complement, but generically this is not the case. However, when such a choice can be made one can then attempt to recover purely sub-Riemannian results by working with Riemannian tools on $(\mathbb{M},g_{\varepsilon})$ and considering the limit $\varepsilon\rightarrow 0^{+}$, however one should understand that Riemannian definitions of curvature do not make sense in this limit.

Consider $\mathbb{R}^{n+m}=\mathbb{R}^{n}\oplus\mathbb{R}^{m}$ with coordinates $(x,z)=(x^{1},\ldots,x^{n},z^{1},\ldots,z^{m})$ and the natural dilation $\delta_{t}\colon\mathbb{R}^{n+m}\rightarrow\mathbb{R}^{n+m}$ defined for $t>0$ by

We associate the following weights $w$ for the generators $1$, $x^{\alpha}$, $z^{i}$, $\frac{\partial}{\partial x^{\alpha}}$, $\frac{\partial}{\partial z^{i}}$ of polynomial vector fields:

A formal monomial is a product of the generators, the order of which is the sum of weights of each term. A polynomial vector field on $\mathbb{R}^{n+m}$ (considered as a differential operator of first order) is said to be homogeneous if it is a sum of monomials of the same weight. For any smooth vector field, applying the Taylor expansion for the coefficients, one can rearrange the terms into a sum of homogeneous terms

where $X^{(l)}$ is a homogeneous vector field of order $l$. Written in this form, we will call this a parabolic Taylor expansion.

We say that a system of coordinates $\Psi\colon U\to\mathbb{R}^{n+m}$ centered at $q\in U\subseteq\mathbb{M}$ is privileged if for any horizontal vector field $X\in\Gamma(\mathcal{H})$ the terms of the homogeneous order $-2$ vanish, see more details in [1, Chapter 10.4] or [16].

Now we will construct a system of privileged coordinates that is convenient for the further calculations; it is reminiscent of normal Euclidean coordinates and normal coordinates for CR manifolds [30, 33]. Let $q\in\mathbb{M}$ be fixed. Consider an arbitrary tangent vector $X+Z\in\mathcal{H}_{q}\oplus\mathcal{V}_{q}=T_{q}\mathbb{M}$ and define $\gamma_{X+Z}$ to be the curve starting at $q$ with

where $D_{t}$ denotes the covariant derivative along $\gamma_{X+Z}$. According to [33] (see also [30]), there are neighborhoods $0\in O\subset T_{q}\mathbb{M}$ and $q\in U\subset\mathbb{M}$ so that the map $\varphi:O\longrightarrow U$ defined by

is a diffeomorphism, and satisfies the parabolic scaling $\varphi(tX+t^{2}Z)=\gamma_{X+Z}(t)$ whenever either side is defined. Such a curve $\gamma_{X+Z}$ will be called a parabolic geodesic.

Fix a horizontal (resp. vertical) orthonormal frame $\{X_{1}(q),\dots,X_{n}(q)\}$ (resp. $\{Z_{1}(q),\dots,Z_{m}(q)\}$) at $q$ and extend these vectors to be parallel along parabolic geodesics starting at $q$. It is known (cf. [33]) that such extensions necessarily are smooth in a neighbourhood of $q\in\mathbb{M}$. In this way, we obtain a smooth local frame $\{X_{1},\dots,X_{n},Z_{1},\dots,Z_{m}\}$ for $T\mathbb{M}=\mathcal{H}\oplus\mathcal{V}$. Note that the corresponding dual frame $\{\theta^{1},\dots,\theta^{n},\eta^{1},\dots,\eta^{m}\}$ is also parallel along parabolic geodesics. In the following we call such a frame a special frame.

We define a privileged coordinate system $(x,z)$ on $U$ by composing the inverse of $\varphi$ with the map $\lambda:T_{q}\mathbb{M}\longrightarrow\mathbb{R}^{n+m}$ defined by $\lambda(W):=(\theta^{\alpha}(W),\eta^{i}(W))$ (cf. [1, Example 10.31 (ii)]). Note that if we consider another horizontal (resp. vertical) frame at $q$, then these frames are related via an orthogonal transformation. Hence, the resulting coordinates are related via a block diagonal matrix with orthogonal entries, i.e. a matrix of the form

with $A\in{\bf O}(n)$ and $B\in{\bf O}(m)$, where ${\bf O}(k)$ with $k\in\mathbb{N}$ denotes the group of orthogonal $k\times k$ real matrices.

From now on we will work with this special frame and co-frame and frequently write

The infinitesimal generator $\tilde{P}$ of the $\mathbb{R}^{+}$-action $\delta_{t}$ (Eq. 3.1) on $\mathbb{R}^{n+m}$ is given in these coordinates by

We adapt the notion of homogeneity to tensor fields on $\mathbb{M}$. We will denote by $P$ the pullback of $\tilde{P}$ to $\Gamma(T\mathbb{M})$ and say that a tensor field $\Theta$ is homogeneous of order $l=\mathrm{ord}(\Theta)\in\mathbb{Z}$ with respect to the above dilations if

We now prove a lemma crucial for our asymptotic analysis on H-type foliations, (cf. [30, 33]):

As mentioned in [30], if $\nu$ is a differential form, then the homogeneous part $\nu^{(l)}$ of degree $l>0$ of its parabolic Taylor expansion can be computed by the formula

Furthermore, using the structure equations (see 7.1 in the Appendix) of the Riemannian foliation $\mathbb{M}$ (w.r.t. the special frame and co-frame):

We then obtain exact expressions for the homogeneous parts of the special co-frame and connection $1$-forms.

Applying the proposition, we recover the particular expressions.

The following properties will be useful in the sequel.

Now we compute the low order homogeneous parts in the parabolic Taylor series of the special frame $\{X_{1},\ldots,X_{n},Z_{1},\ldots,Z_{m}\}$. For this, we first define the following linearly independent vector fields on $\mathbb{R}^{n+m}$: