j=1
X1jf X1j.
At various points in this survey we will work in this general setting to emphasize the fact that certain results do not depend on the special structure ofH. However, we make no systematic attempt to present all results in the most general framework possible.
To conclude this section, we define the notion of a linear map between Carnot groups.
Definition 2.1. Given two Carnot groupsG1,G2with dilations δs1 and δs2, a map L : G1 → G2 is a horizontal linear map if L is a group homomorphism which respects the dilations: L(δ1sx) = δ2sL(x).
Example 2.2. Each horizontal linear map L : H → H takes the form L(x) = Ax, where the matrix A takes the form
⎛
⎝a b 0
c d 0
0 0 ad− bc
⎞
⎠
for some a, b, c, d∈ R. This is easy to verify from the definition.
2.2 Carnot–Carath´ eodory distance
2.2.1 CC distance I: Constrained dynamics
Let x and y be points in H. For δ > 0 we define the class C(δ) of absolutely continuous paths γ : [0, 1]→ R3with endpoints γ(0) = x and γ(1) = y, so that
γ(t) = a(t)X1|γ(t)+ b(t)X2|γ(t) (2.8) and
a(t)2+ b(t)2≤ δ2 (2.9)
2.2. Carnot–Carath´eodory distance 17
for a.e. t∈ [0, 1]. Paths satisfying (2.8) are called horizontal or Legendrian paths.
Note that (2.8) is equivalent with the statement ω(γ) = γ3 −1
2(γ1γ2 − γ2γ1) = 0 (2.10) a.e., where ω is the contact form onR3 given in (2.5) and γ = (γ1, γ2, γ3).
Remark 2.3. Let π : H → C denote the projection π(x) = x1+ i x2. Given any absolutely continuous planar curve α : [0, 1]→ C and a point x = (α(0), h) ∈ H it is possible to lift α to a Legendrian path γ : [0, 1]→ H starting at x satisfying π(γ) = α. To accomplish this we let γ1(t) = α1(t), γ2(t) = α2(t) and
γ3(t) = h +1 2
t 0
(γ1γ2 − γ2γ1)(σ) dσ.
It is easy to see that for any choice of x = (x1, x2, x3) and y = (y1, y2, y3), the set C(δ) is nonempty for sufficiently large δ.
x1 x3
x2
Figure 2.2: Horizontal paths connecting points inH.
In Figure 2.2, we illustrate this fact by connecting the origin to the point (0, 0, 1). First, we travel in the X1direction; as we begin at the origin, this is simply travel along the x1 axis. From the point (1, 0, 0), we travel in the X2 direction to the point
1, 1,12
. We then travel from this point in the −X1 direction to the point (0, 1, 1). Finally, we travel in the −X2 direction, arriving at the terminus (0, 0, 1). The smooth curve illustrated in Figure 2.2 which winds around and up the x3 axis is a smooth horizontal curve that approximates this approach.
We define the Carnot–Carath´eodory (CC) metric d(x, y) = inf{δ such that C(δ) = ∅}.
18 Chapter 2. The Heisenberg Group and Sub-Riemannian Geometry
A dual formulation is d(x, y) = inf
T : ∃γ : [0, T ] → R3, γ(0) = x, γ(T ) = y, and γ= aX1|γ+ bX2|γ with a2+ b2≤ 1 a.e.
, that is, d(x, y) is the shortest time that it takes to go from x to y, travelling at unit speed along horizontal paths. Since the vector fields X1 and X2are left invariant, left translates of horizontal curves are still horizontal and it is easy to verify that d(x, y) = d(y−1x, 0).
Note that if γ is a horizontal curve, then so is its dilation δsγ. In fact, if γ(t) =
2 i=1
γi(t)Xi|γ(t)
then
(δsγ)=
sγ1, sγ2,1
2s2(γ1γ2 − γ2γ1)
=
2 i=1
sγiXi|δsγ.
Moreover, if γ ∈ C(δ) then δsγ ∈ C(sδ) (observe that the endpoints must be dilated as well). Consequently
d(δs(x), δs(y)) = sδ(x, y), in particular, this implies continuity of x → d(x, 0).
The Kor´anyi gauge and metric. An equivalent distance on H is defined by the so-called Kor´anyi metric
dH(x, y) =||y−1x||H
and Kor´anyi gauge
||x||4H= (x21+ x22)2+ 16x23. (2.11) To verify that dH is a metric, one needs to prove the triangle inequality:
dH(x, y)≤ dH(x, z) + dH(z, y). (2.12) This can be done by direct computation as we now recall.
Proof of (2.12). By replacing z−1x with x and y−1z with y, it suffices to prove (2.12) in the case when z = o = (0, 0, 0) is the identity element, i.e., to show that
||xy||H≤ ||x||H+||y||H. (2.13) Writing x = (z, x3) and y = (w, y3) and using the group law (2.2), we find
||xy||4H=|z + w|4+ 16(x3+ y3−1
2Im(z ¯w))2
=
|z + w|2+ 4i (x3+ y3−1
2Im(z ¯w)
2
=|z|2+ 4i x3+ 2zw +|w|2+ 4i y32
≤
||x||2H+ 2|z||w| + ||y||2H2
≤ (||x||H+||y||H)4.
2.2. Carnot–Carath´eodory distance 19
The lack of isotropy of the distance dH follows precisely the lack of isotropy of the CC metric d. In particular, both behave like the Euclidean distance in horizontal directions (X1and X2), and behave like the square root of the Euclidean distance in the missing direction (X3). Clearly, dHis homogeneous of order 1 with respect to the dilations (δs):||δsx||H= s||x||H. Consequently, there exist constants C1, C2> 0 so that
C1||x||H≤ d(x, 0) ≤ C2||x||H
for any x ∈ H. This follows immediately from compactness of the Kor´anyi unit sphere{x ∈ H : ||x||H= 1} and continuity of x → d(x, 0).
The Heisenberg group admits a conformal inversion in the Kor´anyi unit sphere analogous to the classical Euclidean inversion j(x) = x/|x|2 in Rn. For x∈ H \ {o}, let prop-erty jH(δsx) = δ1/sx is also self-evident. Less obvious is the following Heisenberg analog of a classical Euclidean inversion relation:
dH(jH(x), jH(y)) = dH(x, y)
||x||H||y||H. (2.15) Proof of (2.15). As in the proof of (2.12), we write x = (z, x3) and y = (w, y3) and use the group law (2.2) to compute
dH(jH(x), jH(y))4 The relation between the Kor´anyi gauge || · ||H and Kor´anyi inversion jH will be pursued further in Sections 3.3 and 3.4, where we discuss the connections between the Heisenberg group, CR geometry, and Gromov hyperbolic geometry.
2.2.2 CC distance II: Sub-Riemannian structure
A sub-Riemannian metric onH is determined by any choice of inner product on the horizontal subbundle of the Lie algebra. Starting from this datum one may
20 Chapter 2. The Heisenberg Group and Sub-Riemannian Geometry
define the length of horizontal curves and equipH with the structure of a metric length space, which turns out to agree with the Carnot–Carath´eodory metric. Since we have already made an arbitrary choice of coordinates to present H, we may with no loss of generality assume that X1 and X2 form an orthonormal basis of each horizontal space H(x) relative to this inner product. We extend this inner product to an inner product defined on the full tangent space, i.e., a Riemannnian metric, by requiring that the two layers in the stratification of the Lie algebra are orthogonal and that X1, X2and X3form an orthonormal system. We denote this extended inner product by g1 or·, ·1 as dictation by specific situations.
Accordingly, we define the horizontal length of γ to be
LengthH,CC(γ) =
1 0
γ(t), X1|γ(t)21+γ(t), X2|γ(t)21dt (2.16)
and claim that
d(x, y) = inf
γ LengthH,CC(γ), (2.17)
where the infimum is taken over all horizontal curves joining x and y. Note that if π : H → C denotes the projection π(x) = x1+ i x2, dπ : h → C denotes its differential, and LengthC,Eucl(·) denotes Euclidean length in the plane, then
LengthH,CC(γ) = LengthC,Eucl(π(γ)). (2.18) To prove (2.17) we fix x, y∈ H and let ¯d = infγLengthH,CC(γ). For any δ > d(x, y) we consider a curve γ ∈ C(δ) and note that
d¯≤ LengthC,Eucl(π(γ))≤
1 0
a2+ b2dt≤ δ
by (2.9). Thus ¯d≤ d(x, y).
To prove the opposite inequality, let > 0 and choose a curve γ : [0, 1]→ H connecting x and y so that LengthC,Eucl(π(γ)) = ¯d + . If we reparametrize γ to have constant velocity |dπ(γ)| = ¯d + , then γ∈ C( ¯d + ). Hence ¯d + ≥ d(x, y).
Since > 0 was arbitrary, ¯d = d(x, y).
The next lemma shows that the Kor´anyi and CC metrics generate the same infinitesimal structure.
Lemma 2.4. If γ : [0, 1]→ R is a C1curve and ti= i/n, i = 1, . . . , n, is a partition of [0, 1], then
lim sup
n→∞
n i=1
dH(γ(ti), γ(ti−1)) =
LengthH,CC(γ) if γ is horizontal,
∞ otherwise.
2.2. Carnot–Carath´eodory distance 21
The proof follows immediately from this derivation together with (2.16), (2.18)
and (2.10).
2.2.3 CC distance III: Carnot groups
The definition of a Carnot–Carath´eodory distance can be extended easily to higher-dimensional Heisenberg groups and to general Carnot groups. Consider a Carnot groupG with graded Lie algebra g = V1⊕ · · · ⊕ Vr, homogeneous
Then the Carnot–Carath´eodory distance onG is defined to be d(x, y) = inf LengthG,CC(γ),
where the infimum is taken over all horizontal paths connecting x to y. Clearly, d is left invariant, moreover, the maps δs are indeed a family of dilations with respect to this metric:
d(δsx, δsy) = sd(x, y). (2.20) These properties of the CC metric imply analogous properties for the resulting Hausdorff measuresHα, α > 0. Recall that the α-dimensional Hausdorff measure Hαon a metric space (X, d) is the outer measure defined as
22 Chapter 2. The Heisenberg Group and Sub-Riemannian Geometry
where the infimum is taken over all coverings B of the set S by balls Bi with diameter diam Bi< δ. The standard implication
Hα(S) <∞ ⇒ Hα(S) = 0 for all α > α
ensures the existence of a unique value α0 = α0(S)∈ [0, ∞] with the property that Hα(S) = 0 for α > α0 and Hα(S) = +∞ for 0 ≤ α < α0. The value α0 is the Hausdorff dimension of S.
From the left invariance and scaling properties of the CC metric, one easily deduces corresponding properties for the Hausdorff measures in (G, d):
Hα(LyE) =Hα(E), Hα(δsE) = sαHα(E),
for all s, α > 0, y ∈ G, and E ⊂ G. In particular, for each α there exists c(α) ∈ [0,∞] so that
Hα(B(x, r)) = c(α)rα
for all x∈ G and r > 0, where B(x, r) denotes the metric ball with center x and radius r in (G, d). When 0 < α < Q we have c(α) = +∞, while for α > Q we have c(α) = 0. In case α = Q,
c(Q) =HQ(B(o, 1))∈ (0, +∞).
Thus the Hausdorff dimension of (G, d) is Q; indeed (G, d) is an Ahlfors Q-regular space andHQagrees (up to a constant multiplicative factor) with the Haar measure onG. In particular, for any non-abelian Carnot group G the Hausdorff dimension strictly exceeds the topological dimension; this gives (G, d) fractal character (in the sense of the term fractal advocated by Mandelbrot).
The gauge norm (2.11) has several natural extensions to general Carnot groups. Here we recall one of the more computationally friendly ones:
||x||2r!G =
r i=1
mi
j=1
|xij|2r!i , x = (x11, . . . , xrmr)∈ G. (2.21)
In contrast with the Heisenberg situation, || · ||G is typically only a quasinorm rather than a true norm: the inequality ||xy||G ≤ ||x||G||y||G must be replaced by ||xy||G ≤ C||x||G||y||G for some (absolute) constant C < ∞. The latter fact easily follows from the Baker–Campbell–Hausdorff formula. As was the case in the Heisenberg group, the gauge quasimetric dG(x, y) =||y−1x||Gand the Carnot–
Carath´eodory metric d are comparable.