Construction of privileged coordinates

To prove in an effective way the existence of privileged coordinates, we first choose vector fields Y1 , ... , Yn whose values at p form a basis of TpM:

First, choose among Xl, ... , Xm a number nl of vector fields such that their values form a basis of Ll(p). Call them Yl, ... , Yn1 •

Then for each s (s = 2, ... , r) choose vector fields of the form [Xi!, [Xi2 , ... [X^{is _1 '} Xi.] ...

J]

which form a basis of LS(p) mod p-l(p), and call them YnS -1 +1, ... 'Yns .

We obtain in this way a sequence of vector fields Y1 , ... , Yn whose values at p---and at points near p---form a basis of the tangent space. At p---but not at neighbour points, if p is singular-this basis is adapted to the flag (24).

Lemma 4.11. Any vector field Y E £S(X1, ... , Xm) can be written near

pas n

Y

= 2:

^{CjYj, (29)}

j=l

where each Cj is a smooth function, of order ~ Wj - s at point p. In particular, Cj(p)

=

0 ifwj

>

s.

If p is a regular point, we have

ns

Y = 2:CjYj, j=l

that is, (29) holds with Cj identically zero for Wj

>

s.

(30)

4.3 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY 37

Note that having order 2: Wj - S at P is a restrictive condition only when Wj

>

s.

Proof. Let Y E

.c

^{S(X1 , ... ,} X'/7t). For all q in a neighbourhood of p, tangent vectors Y1(q), ... , Yn(q) form a basis of TqM, depending smoothly on q. Whence (29), with smooth Cj.

In the case of a regular p, tangent vectors Y1 (q), ... , Y_ns (q) are indepen-dent for q near p, so they form a basis of Y(q) depending smoothly on q.

On the other side, if a vector field Y belongs to

.c

^{s , we have Y(q) E Y(q)}

for all q near p. Whence (30). It can be noted that, conversely, a vector field Y such that Y(q) E Y(q) for all q near p is in

.c

^s (regular case only).

What we have left to prove for singular p amounts to the following: For all k 2: 0, the function Cj is of order 2: k at p whenever Wj 2: S

+

^k.

We use induction on k. Our assertion is trivial for k = O. Assume that it holds for 0, 1, ... , k. We have

n

[XiI' ["', [Xik' Yj·· .]](p) = L:(Xil .. . XikCj)(P)0(P)

+

n j=l ⁽³¹⁾

L:

L:(XOI ... XOtCj)(p)[X~I' ["', [X~k_t' 0j·· .j](p), j=l ^o,~

where the last sum is taken on all partitions of the sequence (iI, ... ,ik) into subsequences a = (a1,"" al') and 13 = (131,"" f3k-l') such that

O~£~k-1.

In (31), the right hand side is a tangent vector belonging to y+k(p).

Let us write Tjo~ as an abbreviation for

(X

OI " ,XOtCj)(p)[X~I' ["', [X~k_l' 0j·· .j](p).

If 1 2: Wj - s, then Tjo~ E LS+k(p). If, on the contrary, 1

<

Wj - s, we have Wj 2: S

+

¹

+

1. By the induction hypothesis, the function Cj is of order 2: 1

+

^{1 at} p, and the coefficient (XOI ... XOtCj) (p) vanishes. Therefore, in any case, we have Tjo~ E LS+k(p). We conclude that the tangent vector

n

L(XiI , "Xik Cj)(P)0(P) j=l

belongs to y+k(p). Now, tangent vectors 0(P) with Wj 2: S

+

^k

+

^{1 are}

independent mod Ls+k. We get immediately that

(XiI" ,XikCj)(p) = 0 (32) whenever Wj 2: ^S

+

k

+

1.

38 ANDRE BELLAICHE §4

Let us take j such that ^Wj 2 s

+

k

+

^1. We know from the induction hypothesis that Cj is of order 2 k at p. Since (32) holds for all choices of indices i1 , ... , ik, the function is of order 2 k

+

1, as desired. _ It will be convenient, in the following lemma and in the sequel, to introduce the notation w(a)

=

W1a1

+ ... +

wnan,

Lemma 4.12. (i) Any product X i1 X i2 ... Xis, where i1 , ... , is are integers, can be written as a linear combination of ordered monomials

'""' V·a1 'Jan

L..,.,ca1 ... an.I1 ···.I n ,

where the ca1 ... an are smooth functions, and cal ... an is of order 2 w(a)-s.

In particular, Cal ... a n (p) = 0 if W (a)

>

^s.

If p is a regular point, one may take Cal .. . an (p) identically zero for w(a)

>

^s.

(ii) A function

f

is of order> s at p if and only if

for all a

=

(a1, ... , an) such that w(a) ~ s.

Proof. First, note that by Lemma 4.11 with s = 1, we have Xi

2::7=1

^Aij 1j, where the Aij are smooth functions of order 2 ^{Wj -} 1 at p. One deduces easily that

where the sum is taken on sequences (j1, ... ,jq) such that q :=:; sand the J.tilh ... jq are smooth functions of order 2 ^Wj1

+

^Wj2

+ ... +

^{Wjg - s}

at p. Therefore, it is enough to prove that any product 1j1 .. . 1jg' with wil

+

^Wj2

+ ... +

^Wjg

=

P can be written as a linear combination

where each Ca is smooth and of order 2: w( a) - p at p.

We will argue by double induction, first on q, next on the number i of inversions in the sequence (j1, ... , jq). Let us take two indices such that jk

>

^jk+l (if they don't exist, there is nothing to prove). Using Lemma

4.3 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY

4.11, we obtain

n

=

^ljk+lljk

+ L

^{Vj lj,}

j=l

39

where the order of Vj at p is 2: Wj - Wjk - Wjk+l' Now, we replace ljk ljk+1

in the product ljl ... ljq by the right hand side in the last equality. This yields

n

ljl ... ljq = ljl .. ·ljk+lljk .. . ljq

+

^L ljl .. ·ljk-l (Vj lj )ljk+2 .. . ljq, j=l

n

L L(Y)'l '" YACVj)Y/Ll ... Y/Lk-l-£ljk+2" .ljq, j=l A,/L

where in the last sum (A, JL) runs on the set of all partitions of the sequence (j1, ... ,jk-1) into subsequences A = (A1, ... ,Ae) and JL =

(JLI, . .. , JLk-I-e).

To compute the order of Y A1 ... YA£Vj at point p, observe that Xi'P is of order 2: s - 1 at p if 'P is of order 2: s, so lj 'P is of order 2: s - Wj at p, and a factor like Y A1 ... YA£Vj is of order 2: Wj - Wjk - Wjk+l - W Al - ... - WAR'

Applying the induction hypothesis on L for the term ljl .. . ljk+lljk ...

ljq, and the induction hypothesis on q for the terms in the last sum, we see that ljl .. . ljq is a linear combination with smooth coefficients of

Yt 1 ... Yn"'n, as was to be shown, the coefficients c'" having the desired orders.

The case of a regular point is treated along the same lines, noticing only that one may write in this case

The second part of the lemma is an immediate consequence of (i), which proves the sufficiency, and of the fact that products Yt 1 ... Yn"'n are themselves noncommutative polynomials of the X:s of degree ~ W1001

+

... +

WnOO n , which proves the necessity. _

lowe the idea of the proof of Lemma 4.12 to J.-J. Risler [4].

40 ANDRE BELLAICHE §4

Choose now any system of coordinates Yl, ... ,Yn such that (Yj, Yk)

=

bjk at p.

These coordinates are linearly adapted at p.

Lemma 4.13. Let P(y) be a homogeneous polynomial of degree q. Then, we have

(Yt'l ... ynan P) (0) = (O~ll ... o~::

p)

(0). (33) if q = al

+ ... +

an. If q

>

^al

+ ... +

an, both sides are O.

Proof. The lemma will be proved if we show that

Yt'l ... ynan = O~ll ... o~::

+

^L aa{1(y)oe: ... oe::

+Q,

(34)

{(11 {1t::;al,· .. ,{1n::;aⁿ,{1#a}

where aa{1(O)

=

0 for all

f3

occurring in the sum, and Q is a differential operator of (usual) order

<

^al

+ ... +

an, without constant term. This is proved, first, by noticing that each Yi can be written as

n

Yi

=

0Yi

+

L aij(y)oYi j=1

where aij(O) = 0, i = 1, ... , n, then, by applying repeatedly the formula

n n

oylYi = O;lYi

+

Laij(Y)O;lYi

+

L(OYlaij)(Y)OYi

•

j=1 j=1

Lemma 4.14. Let

f

be a linear form in the variables of weight

>

s, that

is

f =

an8+1Yn.+l

+ ... +

anYn·

Then there exists a polynomial h in the variables Yl, ... , Yn., having only terms of order ~ 2, such that the function

g(y) = h(Yl, ... ,Yns )

+

an.+1Yn.+l

+ ... +

^anYn

has local order ~

s+

1 at point p. Moreover, the polynomial can be chosen of the form

and can be obtained by an effective procedure.

4.3 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY 41

Proof. By Lemma 4.12, the function 9 will have local order:::: s

+

^{1 at}

p if

(35) for all ^a = (al,"" an) such that ^WIQl

+ ... +

^{wnan S; s. Since Wi}

>

s when i

>

ns , we may content ourselves to ask for (35) with n replaced by ns·

We shall construct h as a sum h = hI

+

^h2

+ ... +

hs where hq is a homogeneous polynomial of degree q in the variables Yl, ... , Yn., such that for q

=

1, ...

,s,

the relation

holds for all ^a

=

(al,"" ans' 0, ... ,0) such that Wlal

+ ...

+wnsans ~ s and al

+ ... +

^{an. S; q.} We need precisely that this condition be satisfied for q

=

s.

We proceed by induction on q. We take hI

=

O. The induction hypoth-esis is then satisfied for q

=

1, since (Yjf)(p)

=

0 for any j with j S; ns.

Suppose h2, ... ,hq- l have been found. We have to find hq such that (36) holds. We know by the induction hypothesis that

(yt^{1 •••} yn~ns (J

+

^h2

+ ... +

^{hq-d) (p)}

=

0 and, by Lemma 4.13, that

if ^Wial

+ ... +

WnsQn S; sand ^al

+ ... +

^ans S; q - 1.

So, we have only to find hq such that

or

(yt^{1 •••} yn~n. hq) (p) ⁼ _(ylQl ... y~n. (J

+

h2

+ ... +

hq-l))(p) for all ^a with ^{WI al}

+ ... +

W n • an S; sand ^al

+ ... +

^{ans = q} exactly. The problem boils down to the construction of a homogeneous polynomial of degree q having some partial derivatives of degree q specified at the origin.

This construction is immediate, using Taylor's formula. _ Theorem 4.15. One can in an effective way, compute for each j (j = 1, ... , n) a polynomial Hj in the variables Yl, ... , Yv(j) , with-out linear term, nor constant term, such that the functions Zj

=

Yj

+

Hj (YI, ... , YV(j)) form a system of privileged coordinates at p.

42 ANDRE BELLAICRE §4

Proof. Apply Lemma 4.14 to

f =

^{Yj, S}

=

^{Wj -} 1, yielding 9

=

^Zj. The functions Zj obtained in this way vanish at p, and have the same linear parts as the Yj, so they form a system of coordinates around Zj. They have order 2: Wj by Lemma 4.14, and, since YjZj

=

^YjYj

=

1, they have order:::; Wj in virtue of Lemma 4.12. The theorem is therefore proved. _ Remark. The coordinates Yj having weight 1 need not be changed. In case r

=

2, no change at all is needed, the coordinates Yj (with weight 1 and 2) form already a system of privileged coordinates, as it follows immediately from the definition.

Notice that the coordinates Zl, ... , Zn supplied by the construction of Theorem 4.15 are given from original coordinates by expressions of the form

Zl

=

Yl

Z2 = Y2

+

^POI(Yl)

Zn = Yn

+

POI(Yl , ... , Yn-l)

where pol denotes a polynomial, without constant or linear term. It is easy to see that the reciprocal change of coordinates has exactly the same form.

Other ways of getting privileged coordinates are to use the mappings (Zl, ... , zn) ^f-> p exp(zlY1

+ ... +

znYn)

(Zl, ... , zn) ^f-> p exp(znYn)·· . exp(zlY1)

(compare [12,28]), (compare [19]).

Following the usage in Lie group theory, these coordinates are called canonical coordinates of the first (resp. second) kind. We shall not prove here that canonical coordinates of the first or second kind are privileged coordinates, as we will not use them in the sequel. One of the points of this paper is indeed to show that otherwise unspecified privileged co-ordinates, or privileged coordinates obtained from a simple polynomial change of coordinates, are better suited than canonical coordinates in many kinds of computations.

5.1 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY

5. The tangent nilpotent Lie algebra and the

Nel documento Progress in Mathematics (pagine 45-52)