Calculus on Carnot groups

In this subsection we study the Carnot groups, following [62].

Definition 1.127 (Carnot group). A Carnot group is a Lie group, nilpotent and simply connected, whose Lie algebra g admits a stratification, i.e. there exist V₁, ...V_k vector spaces such that

g = V₁⊕ ... ⊕ V_k.

Moreover, g is endowed with the C-C metric, built starting from V₁.

A Lie group, as in Definition 1.127, is also called homogeneous group or stratified group. Moreover, we say that V1 generates the Lie algebra g, while the vector space Vi, for i ≥ 2, are called slices of g.

In this subsection, we indicate by dC the C-C distance associated to a Carnot group G.

Following remark gives an algebraic version of Chow’s Theorem 1.31.

Remark 1.128. If the Carnot group satisfies the H¨ormander condition, then we know (by Chow’s Theorem) that there exists an horizontal curve joining two given points. Therefore the C-C distance produces a left-invariant metric on G (see [62], Theorem 2.15).

Now we introduce the dilations. They are very important because they show that a Carnot group is self-similar, almost in the same way as the Eu-clidean space is. In fact, we can define a family of self-similar automorphisms on the group, starting from a family of dilations defined on the Lie algebra.

Definition 1.129. A family of dilations on g is a family of smooth maps δ_λ : g → g, defined, for λ > 0, as dilations

δ_λ(X) = λⁱX, whenever X ∈ V_i. (1.62) Applying the exponential map, we can rewritten the dilations (1.62) as automorphisms of G, setting eδ_λ = exp⁻¹δ_λexp. For sake of simplicity, we

still indicate eδ_λ, like the starting dilation δ_λ.

Since a C-C distance is defined, minimizing the length-functional over all the V₁-horizontal curves (Definition 1.14), it follows immediately that

d_C(δ_λ(x), δ_λ(y)) = λd_C(x, y), (1.63) for any x, y ∈ G and λ > 0.

Let G a Carnot group and d_C the associated C-C distance. Using the family of dilations δ_λ, we can define a norm on G, setting

kxk_C := dC(0, x).

Moreover, it is immediate to verify that:

kxk_C = 0 ⇐⇒ x = 0,

kλxk_C = d_C(0, δ_λ(x)) = d_C(δ_λ(0), δ_λ(x)) = λd_C(0, x) = λ kxk_C.

It is possible to introduce a new norm on G, equivalent to the C-C norm k k_C but easier to compute, using the stratification-property characterizing a Carnot group. In fact, we can write, any point x ∈ G, as x = (x1, ..., xk), with xi ∈ Vi.

Definition 1.130. Let G a Carnot group with stratification V₁, ..., V_k, the homogeneous norm is defined as

kxk :=

k

X

i=1

|xi|^2k!i

!_2k!¹

, (1.64)

where |x_i| is the usual r-dimensional Euclidean norm defined on the space vector V_i, with r = dimV_i.

Example 1.131 (Homogeneous norm of the Heisenberg group). Since the n-dimensional Heisenberg group Hⁿ is a Carnot group with k = 2, by (1.64) we get

k(z, t)k = |z|⁴+ t^2
1₄

, z ∈ R²ⁿ, t ∈ R. (1.65)

Example 1.132 (Homogeneous norm of the Gruˇsin plane). The homoge-neous norm depends only on the step of the stratification, then the Gruˇsin plane has exactly the same homogeneous norm than the Heisenberg group, that is k(x, y)k = (|x|⁴+ |y|²)¹⁴, for (x, y) ∈ R².

Following results ensure that homogeneous norm has good properties.

In fact

kδ_λ(x)k = λ kxk , (1.66)

and

x⁻¹

= kxk , (1.67)

where by x⁻¹ we indicate the usual inverse-element inside G.

Using the homogeneous norm, we can define a left-invariant distance on G, simply setting, for any x, y ∈ G,

d(x, y) = x⁻¹y

. (1.68)

The function (1.68) is a distance on G and, moreover, it is left-invariant.

In fact, for any h ∈ G, d(hx, hy) =

(hx)⁻¹hy =

x⁻¹h⁻¹hy =

x⁻¹y

= d(x, y).

W.r.t. the dilations, the new distance d, defined in (1.68), has the same behavior as the original C-C distance d_C. In fact, for any x, y ∈ G,

d(δ_λ(x), δ_λ(y)) = λd(x, y). (1.69) We conclude this subsection, remarking that all the distances defined on a Carnot group and satisfying (1.69) are equivalent, so that we can use one of these indifferently. The advantage of using the homogeneous distance d instead of the original one, is that d is simpler to calculate than d_C. The problem is that, sometimes, d doesn’t satisfy the triangle inequality and so it doesn’t induce a metric space on G. To get over this problem, we can define an equivalent distance starting from a new homogeneous norm

| k k |, satisfying as the dilation-property (1.69) as something like a triangle inequality, i.e.

| x⁻¹y

| ≤ C | kxk | + | kyk |
, for some constant C > 0 independent of the points x and y.

Remark 1.133. Looking at the particular case of the Heisenberg group, we want to note that the homogeneous norm (1.65) satisfying the usual triangle inequality and so we don’t need to introduce a new homogeneous norm.

Remark 1.134 (Non-Euclidean nature of Carnot groups). There is a very famous result by Stephen Semmes ([102]), where it is proved that a bi-Lipschitz embedding of the Heisenberg group into an Euclidean space doesn’t exist. The Euclidean behavior of the Carnot spaces is linked to the non-commutativity of the associated Lie algebras. In fact, if we assume that a bi-Lipschitz emending exists, then the corresponding differential is an injective application between the two corresponding tangent spaces. By the identifica-tion of a tangent space with the Lie algebra and using the exponential map, we can build an isomorphism between a non-Abelian group and an Abelian one, but this is impossible.

In order to conclude this subsection, we want briefly speak about the links between Carnot groups and generic distributions (see Definition 1.65).

Recall that the Carnot groups are the main examples of generic distributions.

Moreover, in some sense, all the generic distribution has the behavior of the Carnot groups. In fact, the following result holds.

Theorem 1.135. Let H a generic distribution defined on a smooth manifold M . Then, its (sub-Riemannian) tangent cone, at a point p ∈ M , is isometric to (G, dC), where G is a nilpotent Lie group and dC is a left-invariant C-C metric defined on G.

We don’t want to give a definition of tangent cone neither a proof of this result. One can find both of them and many other information about generic distributions in [89].

We want only to remark that, using the previous result, it is possible to define a family of dilations and something like an exponential map, also starting from generic distributions.