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Distance estimates and privileged coordinates

Nel documento Progress in Mathematics (pagine 41-45)

Corollary 2.6. Suppose Chow's condition holds. The topology defined by the sub-Riemannian distance d is the original topology of M

4. Privileged coordinates

4.2. Distance estimates and privileged coordinates

Now, fix a point p in M, regular or singular. We set ns = ns(p) = dimU(p), s = 0,1, ... ,r.

Consider a system of coordinates centered at p, such that the differen-tials dYl, ... ,dYn form a basis of T; M adapted to the flag

{O} = LO(p) C Ll(p) C ... C L8(p) C ... C LT(p) = TpM. (24) Such a coordinate system will be said linearly adapted at p.

The estimates we have proved for the sub-Riemannian distance in the Grusin and Heisenberg examples can be generalized, as local estimates, to all cases where r = 2. Using linearly adapted coordinates, and setting nl = dim Ll (p), one can prove without much difficulty that

d (0, (Yl,"" Yn)) ~

IYll + ... + IYnll + IYnl+11

1/ 2

+ ... + IYnI

1/ 2 , (25)

for Y near P = (0, ... ,0). Coordinates Yl, ... , Yn, are said to be of weight 1, and coordinates Yn,

+1, ...

,Yn are said to be of weight 2. We shall not give a proof of (25) now, and we will content ourselves with the examples in §3, since (25) will be superseded by more general statements. (See Theorem 7.34, taking in account the remark following Theorem 4.15.)

To define the notion of weight in the general case, observe that the structure of a flag such that

{O} = Vo C VI C ... C VT = V

may be described by two non-decreasing sequences of integers. The first one is the sequence

of dimensions of subspaces which form the flag. The second one is the sequence

WI ::; W2 ::; ••. ::; Wn

which is best understood by using a basis VI, •.. ,Vn adapted to the flag.

One sets Wj

=

s if Vj belongs to Vs and do not belong to Vs-l'

For the flag (24) we define WI ::; W2 ::; ..• ::; Wn by the same recipe, just replacing Vs by U(p). Notice we always have WI

=

1. Otherwise all the

4.2 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY 33

Xi would vanish at p, which would imply that all their brackets vanish at p, in contradiction with Chow's condition. Moreover Wn = r, the degree of nonholonomy at p.

Thus the j-th vector of an adapted basis of TpM can be written as a linear combination of (values at p of) brackets of order Wj of Xl"'" Xm ,

but it cannot be obtained from lesser order brackets. If Yl, ... , Yn form a system of linearly adapted coordinates at p, their differentials dYl, ... , dYn form the dual basis of such an adapted basis of TpM. The integer Wj

can be characterized by the fact that dYj vanish on Lwrl(p) and does not vanish identically on LWj (p).

Definition 4.10. We shall say that Wj is the weight of coordinate Yj.

With this definition, the proper generalization of (25) would be d (0, (Yl, ... , Yn)) :::::: IYll l /W1

+ ... +

IYnl l /Wn • (26)

It turns out that this estimate is generically false as soon as r ?: 3 for linearly adapted coordinates. This is the motivation for introducing priv-ileged coordinates.

A simple counter-example is given by the system

(27) on ~3. We have

so that Yl = X, Y2

=

Y, Y3

=

z are linearly adapted coordinates at 0 and have weight 1, 1 and 3. In this case, estimate (26) cannot be true. Indeed, this would imply

d(O, (x, y, z)) ?: const(lxl

+

IYI

+

Izll/3), whence

Iz(etX2(0))

I ::;

const Itl 3 (since d(O, etX2 (0)) ::; It!), but this is impossible since

ddt 22 z(etX2(0))

I = (X~z)(O) =

(X2 (X2 +y))(O)

=

1.

t=O

34 ANDRE BELLAICHE §4

However a slight nonlinear change of coordinates allows for (26) to hold.

It is sufficient to replace Y!' Y2, Y3 by Zl

=

X, Z2

=

y, Z3

=

Z - y2

/2.

In the above example, the point under consideration is singular, but one can give similar examples with regular p in dimension ~ 4.

To formulate conditions on coordinate systems under which estimates like (26) may hold, we introduce some definitions.

Definition 4.11. Call XII, ... , Xmf the nonholonomic partial deriva-tives of order 1 of

f

(with respect to the system (Xl, ... , Xm)).

If the manifold under study was M

=

]Rn with its Euclidean metric, one would have m

=

n, and one could take X I

=

OX1' ... ' Xn

=

oXn •

The nonholonomic derivations will thus play a role analogous to that of OX1' ...

,ox

n on ]Rn

Call further XiXjf, XiXjXkf, ... , the nonholonomic derivatives of order 2, 3, ... of

f.

Proposition 4.10. Let s be a non-negative integer. For a smooth func-tion

f

defined near p, the following conditions are equivalent:

(i) One has f(q) = 0 (d(p, q)s) for q near p;

(ii) The nonholonomic derivatives of order ~ s - 1 of

f

all vanish at p Proof. (i) =? (ii) . We have

(X. x·

~l • •• ~k f) ( ) -P - ~

ok

~ f( p e tlXil ... e tkXik)

I

.

utI· .. utk t=O

Since we have Therefore

ifk~s-l.

(ii) =? (i) . We argue by induction on s. For s = 0, there is nothing to prove. So let s

>

0, and assume that

(28)

4.2 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY 35

whenever k ::; s - 1. We have also

for any choices of i, and of it, ... , ik-l. So, applying the induction hy-pothesis to Xd (1 ::; i ::; m), we see that there exist c

>

0 and C

>

0 such that

(Xd)(q)::; Cd(p,q)s-l

if d(p, q) ::; c. Choose any q E B(p, c). Let T

=

d(p, q) and let '"'( : [0, T] --+

M a minimizing geodesic joining p to q, with velocity 1. Denote by Ui (1 ::; i ::; m) the corresponding control functions. We have

d m

d/('"Y(t))

=

L(Xd) ('"Y(t))Ui(t) a.e.,

i=O

with E~l Ui(t)2

=

1 a.e. It follows that

\:/('"Y(t))\ ::; mCd(p,'"'((t))S-1 = mCts-l.

By (28), applied for k = 0, we have 1(0) = 0, so by integrating we obtain

and mC

II(q)l::;

-d(p,qy, s

proving thus the proposition.

Definition 4.12. If Condition (i), or (ii), holds, we say that

1

is of order

~ s at p. We say that

1

is of order s at p if it is of order ~ s, and not of order ~ s

+

1.

Definition 4.13. We call system of privileged coordinates a system of local coordinates Zl, ... , Zn centered at p such that:

(i) Zl, ... , Zn are linearly adapted at Pi (ii) The order of Zj at p is exactly Wj.

If we suppose only that Zl, ..• , Zn are linearly adapted, then the order of Zj is always::; Wj: Fix j, and set s

=

Wj' For some choice of of the indices iI, i2, ... , is, we have

36 ANDRE BELLAICHE §4

Now,

is a linear combination of nonholonomic derivatives of order s of Zj. One of this derivatives must be non zero. So Zj cannot be of order ~ s

+

1,

and must be of order ~ s = Wj. But it may well happen that the order of Zj be

<

Wj: for the system (27), the order of coordinate Y3 = Z at 0 is 2, while W3 = 3.

Our goal in the sequel is to show that the estimate d (0, (Yl, ... ,Yn)) ~ IYlI1/W1

+ ... +

IYn 11/wn •

holds near p if and only if Yl, ... ,Yn form a system of privileged coordi-nates at p (see Theorem 7.34).

Nel documento Progress in Mathematics (pagine 41-45)