Privileged coordinates

In this subsection we want to highlight one of the most important properties of the distributions satisfying the H¨ormander condition.

We prove that it is possible to write any tangent vector as (finite) linear combination of suitable (finite-length) brackets. Moreover, the correspond-ing local coordinates vary smoothly w.r.t. the point. This fact is the key in order to prove the existence-result for the generalized eikonal equation, given in Theorem 2.35. We try to give the main ideas without quoting all the proofs (which one can find in [20]).

So, let M a n-dimensional smooth manifold and X₁, ..., X_m smooth vector fields, satisfying the H¨ormander condition. We indicate by

L¹ = L¹(X₁, ..., X_m)

the set of all the smooth coefficients-linear combinations of X₁, ..., X_m. Then, recursively, we define

L^s+1 := L^s+ [L¹, L^s], for s ≥ 1,

so that L^s is the sub-bundle, spanned by the brackets of X₁, ..., X_m, with length less (or equal) than s.

Remark 1.60. By the Jacobi identity, we have [Lⁱ, L^j] ⊂ L^i+j, ∀ i, j ∈ N.

For any p ∈ M , we indicate by L^s(p) the vector subspace of T_pM , spanned by the brackets with length ≤ s, evaluated at the point p.

By the H¨ormander condition, we know that, for any p ∈ M , there exists k = k(p) (which we suppose the smallest) such that L^k(p) = T_pM .

Remark 1.61. Note that the natural number k(p) is equal to the step of the distribution at the point p but, following the notation of [20], we call k(p) degree of nonholonomy of the point p.

Definition 1.62. We call nonholonomic-degree-map the function k : M → [1, +∞), acting as p 7→ k(p).

Remark 1.63. The nonholonomic-degree-map is an upper semicontinuous function and hence k(q) ≤ k(p), for q near p.

We define, for any s ≥ 1, n_s(p) as the dimension of the vector space L^s(p).

Then n_s(p) is a real number such that 0 ≤ n_s(p) ≤ n.

By the following definition, we divide the points of M in two different groups, w.r.t. the behavior of n_s(p).

Definition 1.64. Let p ∈ M , we say that p is a regular point, if and only if, n_s(p) is constant near p, i.e., if and only if, there exists U , neighborhood of p, such that n_s(q) is constant for any q ∈ U . We say that p is a singular point otherwise.

The most “regular” case, among the s-R. distributions, is the following one.

Definition 1.65. A generic distribution is a distribution such that n_s(p) is constant, for any p ∈ M and for any s ≥ 1.

Remark 1.66. The generic distributions are, in particular, a rank-constant distribution (Remark 1.21).

In the generic distributions all the points are regular. Instead, the Gruˇsin plane (Example 1.24) is a s-R. geometry where one can find both of the previous types of points. In fact, all the points (0, y), for y ∈ R, are singular while the other ones are regular.

Remark 1.67. n_s(p) is a non decreasing function, for 1 ≤ s ≤ k(p).

Moreover, whenever p is a regular point, n_s(p) is strictly increasing.

Since the set of all the regular points is open and dense in M , then, in the analytic case and whenever M is a connected manifold, the sequence

0 < n₁(p) < n₂(p) < ... < n_k(p) = n is the same for all the regular points.

We need to introduce some new definitions, but we give these directly in our special case, nevertheless they can be introduced also starting from a

general chain of vector spaces.

Let p ∈ M (regular or singular point), we consider a system of local coordi-nates, centered at p, such that the differentials dy₁(p), ..., dy_n(p) form a basis of T_p^∗M (the cotangent space of the manifold M at the point p).

We say that this basis is adapted to the flag

L¹(p) ⊂ ... ⊂ L^s(p) ⊂ ... ⊂ L^k(p)(p) = T_pM, (1.31) if and only if, dy₁(p), ..., dy_i1(p) span the tangent space of the vector space L¹(p), dy_i1+1(p), ..., dy_i2(p) span the tangent space of L²(p), and so on.

Then we indicate, by Y₁, ..., Y_n, the corresponding basis for the tangent bundle T M .

Definition 1.68. The coordinates, defined as above, are called linearly adapted at the point p.

Definition 1.69. Chosen an adapted basis y₁, ..., y_n at p, we set w_i := s, if and only if, Yⁱ belongs to L^s(p)\L^s−1(p). The natural number w_i is called weight of the coordinate y_i.

Remark 1.70. The weight is called formal degree, too.

Note that w₁ = 1 and, by the H¨ormander condition, w_nk(p) = w_n = k, with k nonholonomic degree (or also step of the distribution) at p.

In order to define a system of privileged coordinates, we need to in-troduce the notion of nonholonomic partial derivatives (or X-partial derivatives) and the (nonholonomic) order of the smooth functions.

Definition 1.71. Let f : M → R, the first-order nonholonomic partial derivatives of f are the vectors X₁f, ..., X_mf .

Analogously, X_iX_jf , X_iX_jX_kf ,... are the nonholonomic derivatives of order 2,3, etc..

Definition 1.72. A function f has order ≥ s at the point p, if and only if, all the nonholonomic derivatives of order ≤ s − 1 vanish at p.

Remark 1.73. For smooth functions, the condition given in Definition 1.72 is equivalent to require that f (q) = O(d(p, q)^s), for q near p (see [20], Propo-sition 4.10).

Finally, we can give the definition of privileged coordinates.

Definition 1.74 (Privileged coordinates). A system of local coordinates z₁, ..., z_n, centered at p, is a system of privileged coordinates, if and only if,

(i) z₁, ..., z_n are linearly adapted at p,

(ii) the order of z_i at p is exactly the weight w_i.

Remark 1.75. Given a system of linearly adapted (local) coordinates z1, .., zn, the order of zi is less (or equal) than the corresponding weight wi. So, privileged coordinates are adapted coordinates with maximum order.

In [20] (Sec.7), privileged coordinates are used in order to prove some estimates of homogeneous-type for general C-C distances. Instead, here, we are interested in building privileged coordinates ([20], Sec.4.3).

Let p ∈ M , the first step consists in choosing a basis of vector fields Y₁(p), ...Y_n(p) for the tangent space at p.

So we choose n₁ brackets of X₁(p), ..., X_m(p), which are a basis for L¹(p) and we indicate these by Y₁(p), ...Y_n1(p) (note that n₁ = n₁(p) and 1 ≤ n₁ ≤ m, for any p). In the same way, for s ≥ 2, we choose n_s− n_s−1 vector fields, with the form [X_i1, [X_i2, ..., [X_is−1, X_is]...]], which form a basis of L^s(p) modulus L^s−1(p), and we indicate them by Y_ns−1+1(p), ..., Y_ns(p). In k-steps we get a basis Y₁(p), ..., Y_n(p), adapted to the flag (1.31), spanning the whole tangent space at p.

Moreover, for any point p ∈ M (as regular as singular), the vector fields Y₁, ..., Y_n are a basis for the tangent bundle T M , near p. More precisely, there exists U neighborhood of p, such that Y₁(q), ..., Y_n(q) are a basis of T_qM , for any q ∈ U . The difference between regular and singular points, is that, when p is a regular point, Y₁, ..., Y_n are adapted to the flag

L¹ ⊂ ... ⊂ L^r = T M , for any q ∈ U . Instead, when p is a singular point, they are adapted to a such flag at the point p but, in general, they are not so near p.

Chosen Y₁, ..., Y_n, it is not difficult to build privileged coordinates.

Now we recall some preliminary results.

Lemma 1.76. Any Y ∈ L^s(X₁, ..., X_m) can be written near p as

Y =

n

X

i=1

c_iY_i, (1.32)

with ci smooth functions, of order ≥ wi− s at p.

In particular ci(p) = 0, whenever wi > s.

Moreover, if p is a regular point, ci ≡ 0 for wj > s, i.e.

Y =

ns

X

i=1

c_iY_i. (1.33)

Proof. Let Y ∈ L^s(X1, ..., Xm) and U neighborhood of p such that Y1(q), ..., Yn(q) is a basis of TqM , for any q ∈ U . X1, ..., Xm are smooth vector fields, then the corresponding brackets are so. Hence, (1.32) is imme-diately with smooth coordinates.

It remains to investigate the order of the coordinates ci, for i = 1, ..., n.

If p is a regular point, we know that Y1, ..., Yn are adapted near p, too. So Y1(q), ..., Yns(q) form a basis of L^s(q), depending smoothly on q. Moreover, it is trivial to remark that, if Y ∈ L^s, then Y (q) ∈ L^s(q) near p, so that (1.33) is verified.

The case when p is a singular point is more complicate. It is possible to get it by induction on k, showing that, for any k ≥ 0, the functions ci are of order ≥ k at p, whenever wi ≥ s + k (see [20], Lemma 4.11). Nevertheless, we prefer to don’t recall the proof of this claim.

Now we quote the following lemmas without giving any proof (which one can find in [20], Lemma 4.12, Lemma 4.13 and Lemma 4.14, respectively).

For sake of simplicity, we set αⁿ = (α₁, ..., α_n) and w(αⁿ) = w₁α₁+...+w_nα_n.

Lemma 1.77. It is possible to write a product X_i1X_i2...X_is as a linear com-bination of ordered monomials:

X

αⁿ

c_α1...αnY₁^α1...Y_n^αn,

where c_α1...αn are smooth function, of order ≥ w(αⁿ) − s.

In particular c_α1...αn(p) = 0, if w(αⁿ) > s.

Moreover, if p is a regular point, one can take c_α1...αn ≡ 0, for any w(αⁿ) > s.

At last, a function f is of order > s at p, if and only if, Y₁^α1...Y_n^αnf
(p) = 0,

for any αⁿ such that w(αⁿ) ≤ s.

Chosen a system of local coordinates y₁, ..., y_n such that hy_i, Y_ji = δ_i,j, at the point p, these coordinates are linearly adapted at p.

Now we give a property for homogeneous polynomials.

Lemma 1.78. Let P (y) an homogeneous polynomial of degree d. Then Y₁^α1...Y_n^αnP
(0) = ∂_y^α_i¹...∂_y^α_nⁿP
(0),

if d = α₁+ ... + α_n.

While, if d > α₁+ ... + α_n, both of the sides are equal to 0.

The following result gives an effective procedure in order to build privi-leged coordinates.

Lemma 1.79. Let f a linear form in the variables of weight > s, i.e.

f = αns+1yns+1+ ... + αnyn.

Then, there exists a polynomial h in the variables y₁, ..., y_ns, made only by terms of order ≥ 2, such that the function, defined as

g(y) := h(y₁, ..., y_ns) + λ_ns+1y_ns+1+ ... + λ_ny_n, has (local) order ≥ s + 1 at the point p.

Moreover, the polynomial can be chosen of the form h(y₁, ..., y_ns) = X

w(α^ns)≤s

λⁿ_α^sy₁^α1...y_n^αns

s .

Finally we can write a result of existence for a system of privileged coor-dinates.

Theorem 1.80. Let j = 1, ..., n, we can compute (in an effective way) a polynomial H_j in the variables y₁, ..., y_{nwj −1} without linear or constant terms such that the function z_j := y_j+ H_j(y₁, ..., y_{nwj −1}) form a system of privileged coordinates at p.

Proof. We can apply Lemma 1.79, with f = yj and s = wj − 1, and so get g = zj. Therefore zj vanishes at p and it has the same linear part as yj. Hence, the functions zj form a system of local coordinates near p and they have order ≥ wj. Since Yjzj = Yjyj = 1, applying Lemma 1.77, we get that zj has order ≤ wj, too. So zj are adapted coordinates at p and their order is exactly the corresponding wight wj, then they are a system of privileged coordinates.

Remark 1.81. To sum up, starting from linear adapted coordinates y₁, ..., y_n near a point p ∈ M , we have built a system of privileged coordinates

z1 = y1

z₂ = y₂+ P (y₁) . . . .

z_n = y_n+ P (y₁, ..., y_n),

where P (·) are polynomials without constant or linear terms.

Moreover, the inverse-change of coordinates has exactly the same form.

In [20] is remarked that there are others ways in order to get privileged coordinates, for example using the exponential map. We want to recall that the exponential map is usually defined on the Carnot groups (see Sec.1.3, Definition 1.111). Nevertheless, it is possible to introduce something very similar in the sub-Riemannian geometries associated to some generic distri-butions (see [89]). The coordinates, built by the exponential map, are called canonical coordinates and they are, in particular, privileged coordinates, be-cause they satisfy Definition 1.74.