*j=1*

*X**1j**f X**1j**.*

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 groups**G1,G2*with dilations δ*_{s}^{1} *and δ*_{s}^{2}, a map
*L :* G1 *→ G*2 *is a horizontal linear map if L is a group homomorphism which*
*respects the dilations: L(δ*^{1}_{s}*x) = δ*^{2}_{s}*L(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**

**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]→ R*^{3}*with endpoints γ(0) = x and γ(1) = y, so that*

*γ*^{}*(t) = a(t)X*_{1}*|**γ(t)**+ b(t)X*_{2}*|**γ(t)* (2.8)
and

*a(t)*^{2}*+ b(t)*^{2}*≤ δ*^{2} (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*^{}*) = γ*_{3}^{}*−*1

2*(γ*1*γ*_{2}^{}*− γ*2*γ*_{1}* ^{}*) = 0 (2.10)

*a.e., where ω is the contact form on*R

^{3}

*given in (2.5) and γ = (γ*

_{1}

*, γ*

_{2}

*, γ*

_{3}).

*Remark 2.3. Let π :* *H → C denote the projection π(x) = x*1+ * i x*2. 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 γ*

_{1}

*(t) = α*

_{1}

*(t), γ*

_{2}

*(t) = α*

_{2}

*(t) and*

*γ*_{3}*(t) = h +*1
2

*t*
0

*(γ*_{1}*γ*_{2}^{}*− γ*2*γ*_{1}^{}*)(σ) dσ.*

*It is easy to see that for any choice of x = (x*_{1}*, x*_{2}*, x*_{3}*) and y = (y*_{1}*, y*_{2}*, y*_{3}),
*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 X*_{1}direction; as we begin at the origin, this is simply
*travel along the x*_{1} *axis. From the point (1, 0, 0), we travel in the X*_{2} direction to
the point

*1, 1,*^{1}_{2}

. We then travel from this point in the *−X*1 direction to the
*point (0, 1, 1). Finally, we travel in the* *−X*2 direction, arriving at the terminus
*(0, 0, 1). The smooth curve illustrated in Figure 2.2 which winds around and up*
*the x*_{3} 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 ] → R*^{3}*, γ(0) = x, γ(T ) = y,*
*and γ*^{}*= aX*1*|**γ**+ bX*2*|**γ* *with a*^{2}*+ b*^{2}*≤ 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 X*1 *and X*2are left invariant,
left translates of horizontal curves are still horizontal and it is easy to verify that
*d(x, y) = d(y*^{−1}*x, 0).*

*Note that if γ is a horizontal curve, then so is its dilation δ**s**γ. In fact, if*
*γ*^{}*(t) =*

2
*i=1*

*γ*_{i}^{}*(t)X*_{i}*|**γ(t)*

then

*(δ**s**γ)** ^{}*=

*sγ*_{1}^{}*, sγ*^{}_{2}*,*1

2*s*^{2}*(γ*1*γ*_{2}^{}*− γ*2*γ*_{1}* ^{}*)

=

2
*i=1*

*sγ*_{i}^{}*X**i**|**δ*_{s}*γ**.*

*Moreover, if γ* *∈ C(δ) then δ**s**γ* *∈ C(sδ) (observe that the endpoints must be*
dilated as well). Consequently

*d(δ*_{s}*(x), δ*_{s}*(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

*d*_{H}*(x, y) =||y*^{−1}*x||*H

and Kor´anyi gauge

*||x||*^{4}_{H}*= (x*^{2}_{1}*+ x*^{2}_{2})^{2}*+ 16x*^{2}_{3}*.* (2.11)
*To verify that d*_{H} is a metric, one needs to prove the triangle inequality:

*d*_{H}*(x, y)≤ d*_{H}*(x, z) + d*_{H}*(z, y).* (2.12)
This can be done by direct computation as we now recall.

*Proof of (2.12). By replacing z*^{−1}*x with x and y*^{−1}*z 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, x*_{3}*) and y = (w, y*_{3}) and using the group law (2.2), we ﬁnd

*||xy||*^{4}_{H}=*|z + w|*^{4}*+ 16(x*_{3}*+ y*_{3}*−*1

2*Im(z ¯w))*^{2}

=

*|z + w|*^{2}+ 4* i (x*3

*+ y*3

*−*1

2*Im(z ¯w)*

^{2}

=*|z|*^{2}+ 4* i x*3

*+ 2zw +|w|*

^{2}+ 4

*3*

**i y**^{2}

*≤*

*||x||*^{2}_{H}+ 2*|z||w| + ||y||*^{2}_{H}2

*≤ (||x||*H+*||y||*H)^{4}*.*

2.2. Carnot–Carath´eodory distance 19

*The lack of isotropy of the distance d*_{H} follows precisely the lack of isotropy
*of the CC metric d. In particular, both behave like the Euclidean distance in*
*horizontal directions (X*_{1}*and X*_{2}), and behave like the square root of the Euclidean
*distance in the missing direction (X*_{3}*). Clearly, d*_{H}is homogeneous of order 1 with
*respect to the dilations (δ** _{s}*):

*||δ*

*s*

*x||*H

*= s||x||*H. Consequently, there exist constants

*C*

_{1}

*, C*

_{2}

*> 0 so that*

*C*_{1}*||x||*H*≤ d(x, 0) ≤ C*2*||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 R* ^{n}*. For

*x∈ H \ {o}, let*

*prop-erty j*

_{H}

*(δ*

*s*

*x) = δ*

_{1/s}*x is also self-evident. Less obvious is the following Heisenberg*analog of a classical Euclidean inversion relation:

*d*_{H}*(j*_{H}*(x), j*_{H}*(y)) =* *d*_{H}*(x, y)*

*||x||*H*||y||*H*.* (2.15)
*Proof of (2.15). As in the proof of (2.12), we write x = (z, x*_{3}*) and y = (w, y*_{3})
and use the group law (2.2) to compute

*d*_{H}*(j*_{H}*(x), j*_{H}*(y))*^{4}
The relation between the Kor´anyi gauge *|| · ||*H and Kor´*anyi inversion j*_{H}
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**

**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 X*_{1} *and X*_{2} 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 X*_{1}*, X*_{2}*and X*_{3}form an orthonormal system. We denote this
*extended inner product by g*_{1} or*·, ·*1 as dictation by speciﬁc situations.

*Accordingly, we deﬁne the horizontal length of γ to be*

Length_{H,CC}*(γ) =*

1 0

*γ*^{}*(t), X*_{1}*|**γ(t)*^{2}1+*γ*^{}*(t), X*_{2}*|**γ(t)*^{2}1*dt* (2.16)

and claim that

*d(x, y) = inf*

*γ* Length_{H,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) = x*1+ * i x*2

*, dπ :*

*h → C denotes its*diﬀerential, and Length

*(*

_{C,Eucl}*·) denotes Euclidean length in the plane, then*

Length_{H,CC}*(γ) = Length*_{C,Eucl}*(π(γ)).* (2.18)
*To prove (2.17) we ﬁx x, y∈ H and let ¯d = inf** _{γ}*Length

_{H,CC}*(γ). For any δ > d(x, y)*

*we consider a curve γ*

*∈ C(δ) and note that*

*d*¯*≤ Length*_{C,Eucl}*(π(γ))≤*

1 0

*a*^{2}*+ b*^{2}*dt≤ 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 Length*_{C,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 C^{1}*curve and t*_{i}*= i/n, i = 1, . . . , n, is a partition*
*of [0, 1], then*

lim sup

*n**→∞*

*n*
*i=1*

*d*_{H}*(γ(t*_{i}*), γ(t*_{i}* _{−1}*)) =

Length_{H,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**

**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 group*G with graded Lie algebra g = V*1*⊕ · · · ⊕ V**r*, homogeneous

Then the Carnot–Carath´eodory distance onG is deﬁned to be
*d(x, y) = inf Length*_{G,CC}*(γ),*

*where the inﬁmum is taken over all horizontal paths connecting x to y. Clearly,*
*d is left invariant, moreover, the maps δ**s* are indeed a family of dilations with
respect to this metric:

*d(δ*_{s}*x, δ*_{s}*y) = sd(x, y).* (2.20)
These properties of the CC metric imply analogous properties for the resulting
Hausdorﬀ measures*H*^{α}*, α > 0. Recall that the α-dimensional Hausdorﬀ measure*
*H*^{α}*on 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 B**i* with
*diameter diam B*_{i}*< δ. The standard implication*

*H*^{α}*(S) <∞ ⇒ H*^{α}^{}*(S) = 0 for all α*^{}*> α*

*ensures the existence of a unique value α*0 *= α*0*(S)∈ [0, ∞] with the property*
that *H*^{α}*(S) = 0 for α > α*0 and *H*^{α}*(S) = +∞ for 0 ≤ α < α*0*. The value α*0 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):*

*H*^{α}*(L**y**E) =H*^{α}*(E),*
*H*^{α}*(δ*_{s}*E) = s*^{α}*H*^{α}*(E),*

*for all s, α > 0, y* *∈ G, and E ⊂ G. In particular, for each α there exists c(α) ∈*
*[0,∞] so that*

*H*^{α}*(B(x, r)) = c(α)r*^{α}

*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) =H*^{Q}*(B(o, 1))∈ (0, +∞).*

Thus the Hausdorﬀ dimension of (*G, d) is Q; indeed (G, d) is an Ahlfors Q-regular*
*space andH** ^{Q}*agrees (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*

*m*_{i}

*j=1*

*|x**ij**|*^{2r!}^{i}*,* *x = (x*_{11}*, . . . , x*_{rm}* _{r}*)

*∈ 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 d*_{G}*(x, y) =||y*^{−1}*x||*Gand the Carnot–

Carath´*eodory metric d are comparable.*