# Carnot–Carath´ eodory distance

Nel documento Progress in Mathematics Volume 259 (pagine 32-38)

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

for some a, b, c, d∈ R. This is easy to verify from the deﬁnition.

### 2.2.1CC 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

21γ2 − 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

1γ2 − 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

sγ)=



1, sγ2,1

2s21γ2 − γ2γ1)



=

2 i=1

iXi|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+ y31

2Im(z ¯w))2

=

|z + w|2+ 4i (x3+ y31

2Im(z ¯w)

2

=|z|2+ 4i x3+ 2zw +|w|2+ 4i y32



||x||2H+ 2|z||w| + ||y||2H2

≤ (||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 jHsx) = 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.

### 2.2.2CC 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

(t), X1|γ(t)21+(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). 

### 2.2.3CC 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), HαsE) = 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.

Nel documento Progress in Mathematics Volume 259 (pagine 32-38)