Notes for Section 2.1. Useful surveys of aspects of analysis and geometry in the Heisenberg group or on more general Carnot groups include Semmes [241] and Heinonen [135]. For calculus on Heisenberg manifolds, we recommend Beals and Greiner [30] and Gaveau [120]. One of the standard references for analysis on the Heisenberg group is Chapters XII and XIII of Stein’s book on harmonic analy-sis [243]. Folland [100] has a detailed introduction to the Heisenberg group, its representations and applications, and among other things discusses polarized co-ordinates and the matrix model for H.
Notes for Subsection 2.1.3. The notion of sub-Riemannian geometry rests on the accessibility condition for horizontal paths. By the fundamental theorem of Chow and Rashevsky, local accessibility is equivalent to the bracket generating condition for a frame of smooth vector fields X1, . . . , Xmwhich generate the tangent bundle of an n-dimensional manifold M :
Rank(Lie[X1, . . . , Xm])(x) = n (2.46) for every x ∈ M. The analytic form of this condition first appeared in the liter-ature in 1967 in the celebrated paper of H¨ormander [149], who proved that it is a sufficient condition for hypoellipticity of second order differential operators of the form L = m
i=1Xj2. Operators of this form are known as sum of squares or sub-Laplacians.
Carnot groups arise naturally as ideal boundaries of noncompact rank 1 sym-metric spaces. For instance,Hn is isomorphic to the nilpotent part of the Iwasawa decomposition of U (1, n), the isometry group of the complex hyperbolic space of dimension n. The key role played by Carnot groups10become evident in the early 1970s with a number of important papers following a circle of ideas outlined by E.M. Stein in his address at the 1970 International Congress of Mathematicians in
10The name Carnot group is relatively recent, as it emerged in the late 1980s from the papers of Gromov, Pansu and others. In the work of Stein and his collaborators such groups are known as nilpotent homogeneous or stratified Lie groups.
2.5. Notes 35
Nice. The homogeneous structure of such groups allows the development of har-monic analysis [99] which, in turn, plays a central role in the regularity theory of general H¨ormander type operators. In their celebrated article [102] G. Folland and E.M. Stein use the Heisenberg group as an osculating space for strictly pseudo-convex CR manifolds inCn to study properties of singular integral operators and solve the ¯∂boperator. In the following chapter, we describe in some detail the role of the Heisenberg group in CR geometry and its natural occurrence in connection with the Gromov hyperbolic geometry of complex hyperbolic space.
The program of using harmonic analysis in stratified Lie groups as a model for harmonic analysis in more general sub-Riemannian spaces was developed in the pathbreaking paper [233], where Rothschild and Stein extended the Folland–
Stein approach to general sub-Riemannian spaces associated to smooth, bracket generating, sets of vector fields. A central result in the work of Rothschild–Stein is an approximation scheme that allows one to
• lift a set of bracket generating vector fields to a higher-dimensional space so that the lifted vectors are free (i.e., the only relations among the vectors and their commutators up to the step needed to span the whole tangent space, are those arising from the bracket structure and the Jacobi identity), and
• approximate the sub-Riemannian structure of the lifted vector fields with an osculating Carnot group structure, with very precise estimates on the nonlinear remainder terms.
Subanalyticity of real analytic Carnot–Carath´eodory metrics on sub-Riemannian manifolds was recently established in a significant and extremely intricate analysis by Agrachev and Gauthier [6].
Notes for Section 2.2. For more details on the lifting procedure and Legendrian paths as discussed in Remark 2.3, see [57], [22] and other references therein.
The metric dH defined by the gauge (2.11) is nowadays associated with the name of Adam Kor´anyi, who used it extensively in connection with harmonic analysis and potential theory in the Heisenberg group and more general Carnot groups of Heisenberg type [166]. The properties of the Kor´anyi gauge and inversion are mostly due to Kor´anyi and Reimann [168], [169], in particular, the elegant proof of (2.15) can be found in Section 1 of [169]. See also [75].
As noted in the chapter, while it is easy to write down a variety of homo-geneous gauges on general Carnot groups such as (2.21) which agree with the Kor´anyi gauge|| · ||H in the Heisenberg setting, the fact that dH defines a metric is a specific feature of that setting. No simple expressions of Kor´anyi type for homogeneous metrics in general Carnot groups are known.
We note that in the literature one can find several different gauges of Kor´anyi type, each designed for a particular application. See, for example, Franchi–Sera-pioni–Serra-Cassano [106] or Bieske [33]. See the notes to Chapter 9 for more information.
36 Chapter 2. The Heisenberg Group and Sub-Riemannian Geometry
Useful references for Hausdorff measure and dimension in metric spaces in-clude Mattila [196] and Falconer [94]. The study of sub-Riemannian spaces qua metric spaces, with a focus on the intrinsic metric geometry of subsets, was advo-cated by Gromov in [130].
Notes for Section 2.3. The description of the CC geodesics in the Heisenberg group can be found in numerous references, see for example [32], p. 28. The bubble sets were described by Pansu in his 1982 paper [217].
Notes for Section 2.4. The Riemannian approximation scheme appeared first in the paper of Kor´anyi [167], who used it to derive explicit expressions for the CC geodesics. His method is sketched in 2.4.4. Later, the same approach was used by several authors, see for instance [10] and [21]. Roughly speaking, in this scheme lies the main idea of our general approach: to define horizontal geometric objects as limits of horizontal restrictions of classical Riemannian analogs. The situation for general Carnot groups is rather more complicated than the step two case; the proof via convergence of geodesics which we gave for Theorem 2.11 encounters obstacles in the higher step setting due to the possibility of abnormal geodesics.
Theorem 2.12 was proved by Pansu in [218]. In even greater generality, Riemannian approximations to sub-Riemannian manifolds were studied by Roberto Monti in his Ph.D. dissertation at the Universit`a di Trento (unpublished).
Gromov’s notion of convergence of metric spaces was introduced in his groundbreaking paper on groups of polynomial growth [129], see also Chapter 3 of [131]. Proposition 2.7 can be found in [131, Chapter 3] as Propositions 3.7 and 3.13, respectively. A very readable account of the theory of Gromov–Hausdorff convergence of metric spaces can be found in Chapters 7 and 8 of [47].
The basic ingredients of Riemannian geometry which we use in Subsection 2.4.2 can be found in the standard texts. For the detailed definition and proof of the existence of the Levi-Civita connection, we refer the reader to [113, Theorem 2.51]. The Kozul identity can be found on p. 55 in [88].
Additional notes. Among the many advantages of the special structure of the Heisenberg groups Hn are the facts that the center is of dimension 1 and that the explicit solution of the sub-Laplacian operator L = 2n
i=1Xi2 is explicitly known. In [98], Folland proved that for a specific choice of a constant Cn, the function Γ(x, y) = Cn||y−1x||2H−(2n+2) satisfies the equation LΓ(x, y) = δx(y) in Hn. (We will prove this in the setting of the first Heisenberg group in Section 5.2.) In 1980, Kaplan [160] introduced a new class of groups, now called H-type groups, in which a generalization of Folland’s formula holds. In any step two Carnot group define the linear map J : V2→ End(V1) from the second layer of the Lie algebra stratification to the endomorphisms of the first layer via the identity
J(Y )X, X = [X, X], Y for all Y ∈ V2 and X, X ∈ V1. A step two Carnot groupG is called an H-type group (or group of Heisenberg type or Kaplan group) if the map J is orthogonal, i.e.,
J(Y )X, J(Y )X = X, X |Y |2.
2.5. Notes 37
H-type groups have a rich analytic and algebraic structure. The ideal boundaries of noncompact, constant negatively curved, symmetric spaces of rank 1 are one-point compactifications of H-type groups [73]. If we write g = exp(x(g) + y(g)) with x(g)∈ V1and y(g)∈ V2, then the fundamental solution for the sub-Laplacian in an H-type groupG is given in terms of a gauge metric
||g||4G=|x(g)|4+ 16|y(g)|2, and has the form
ΓG(g, g) = CG||(g)−1g||2G−Q,
where CG > 0 is a constant depending only on the group G and Q = dim V1+ 2 dim V2 is the homogenous dimension of G. See also [54], [138], [26] for further results connected with linear and nonlinear potential theory and H-type groups.
There is a rich theory of conformal geometry, including analogs of the Kor´anyi inversion jH on these groups; for further information, see [118].
Chapter 3
Applications of
Heisenberg Geometry
A very intuitive way to think of the sub-Riemannian Heisenberg group is as a medium in which motion is only possible along a given set of directions, changing from point to point. If the constraints are too tight, then it may be impossible to join any two points with an admissible trajectory, hence one needs to find conditions on the constraints implying “horizontal accessibility”.
Constrained motion as defined above is studied in depth in control theory and has numerous applications in engineering (motion of robot arms and wheeled motion) and biology (models of perceptual completion). It also arises naturally in other branches of pure mathematics. In this chapter we briefly describe occurrences of Heisenberg geometry in other areas of pure mathematics (CR geometry, Gromov hyperbolic geometry of complex hyperbolic space, and jet spaces), as well as in the engineering and neurobiological applications mentioned above.
3.1 Jet spaces
The Heisenberg group, as well as a large class of other Carnot groups, can be represented as jet spaces. The concept of jet space gives geometric structure to the classical framework of Taylor polynomials.
To define the first jet space J1(R, R) we begin by introducing an equivalence relation in C1(R): two functions f and g are equivalent at a point t ∈ R if f(t) = g(t) and f(t) = g(t). To stress the role of the basepoint t we will write f ∼tg.
We then define J1(R, R) as the disjoint union of the quotient spaces C1(R)/ ∼t: J1(R, R) =,
t∈R
C1(R, R)/ ∼t. (3.1)
40 Chapter 3. Applications of Heisenberg Geometry
Let us denote elements of J1(R, R) by jt(f ). The space J1(R, R) can be naturally identified with R3 via the global coordinates
jt(f )←→ x = (x1, x2, x3) = (f(t), t, f ). (3.2) Jet spaces are naturally equipped with sub-Riemannian structure. We illustrate this in the simplest case of the contact structure on J1(R, R). We want to define a 1-form θ which vanishes on all 1-jets t→ jt(f ) of C1functions. The basic relation is
df = f(t) dt, which, in local coordinates, reads
(dx3− x1dx2)jt(f ) = 0.
The latter suggests the choice θ = dx3− x1dx2; one immediately verifies that θ is contact: θ∧ dθ = dx1∧ dx2∧ dx3. The horizontal tangent bundle defined by θ is
H1x={v ∈ TxJ1(R, R) | θ(v) = 0} = span(X1,X2), (3.3) where X1= ∂x1 andX2= ∂x2+ x1∂x3. Note that
T J1(R, R) = H1⊕ [H1,H1]. (3.4) Next, we define a group law on J1(R, R) such that H1 and θ are left invariant:
(x1, x2, x3)· (y1, y2, y3) = (x1+ y1, x2+ y2, x3+ y3+ x1y2). (3.5) Observe that J1(R, R) with this group law is isomorphic with the Heisenberg group in its matrix model (2.1).