Proof of the Chow connectivity theorem. (compare [Herm])

Nel documento Progress in Mathematics (pagine 122-137)

o. Basic definitions, examples and problems

1. Horizontal curves and small C-C balls

1.1. Proof of the Chow connectivity theorem. (compare [Herm])

Recall that we are given vector fields Xl, ... ,Xm on a connected manifold V which Lie-generate T(V) and we want to join a given pair of points in V by a piecewise smooth curve where each piece is a segment of an integral curve of some field Xi, i = 1, ... , m.

Since the problem is local, we may assume the fields Xi integrate to one-parameter groups of diffeomorphisms of V and we must join given points by piecewise orbit curves. In other words we must prove that the group G of diffeomorphisms of V generated by these subgroups is transitive on V. We start with the following

Trivial Lemma. If G contains one-parameter subgroups, say Y1(t), Y2(t), ... , Yp(t), where the corresponding vector nelds Y1, Y2 , ••. , Yp span T(V) (without taking commutators), then G is transitive on V.

Proof. This follows from the implicit function theorem. Namely, for each v E V we consider the composed action map Ev : jRP -+ V defined by

The differential of Ev at the origin 0 E jRP sends jRP onto the span of the fields

Yi

in Tv(V) and, hence, is surjective in our case. Thus the orbit G(v) is open in V for each v E V by the implicit function theorem and,

as V is connected, G(v) = V.

Now, let X(t) be a one-parameter group (flow) on V, Y be a vector field and let us look at the transport of Y by X(t), denoted X*(t)Y. We observe that for small t -+ 0

X*(t)Y = Y

+

t[X, Yj

+

o(t)

(by the very definition of the Lie bracket) and conclude that, since the commutators of Xi span T(V), there exist vector fields Yj, j = 1, ... ,p ~

m, on V which span T(V) and such that (i)

Yi

= Xi for i = 1, ...

,m,

114 MIKHAEL GROMOV § 1

(ii) each field Yj for j

>

m equals (Xi (tj) LYjI, i.e. the transport of some Yjl for j'

<

j by the flow Xi(t) at t = tj , (where i also depends on j).

Finally we note that the one-parameter groups Yj(t) are contained in G because the transport of a field Y by X(t) corresponds to the conjugation of the one parameter group yeT), i. e.

and the proof follows by the Trivial lemma.

1.1.A. Quantitative version of the Chow theorem for deg Xi ~ 2 and a preliminary HOlder bound on the C-C metric. Let us try to estimate the length of the piecewise integral curve between given nearby points v and Vi in V provided by the above argument, where the length of an orbit is given by the time parameter. (Thus, for example, the length of the orbit for the field

t %t

in ~ between the points t = 1 and t = e E ]0, 1[

equals Ilogel). We assume that T(V) is spanned by the commutators of degree::; 2 of Xi (i.e. by Xi and [Xi, Xj]) and we want to show that v and Vi can be joined by a piecewise integral curve, such that

length of the curve

:s

(Riemannian distance between v and Vi)! .

Proof. Take vector fields Yl , ... , Yn on V such that

(i) The first mo, for certain ma

<

m, among

Yi

are some of the fields Xi say Xl,'" ,Xmo ' which are linearly independent at v.

(ii) The remaining fields Ymo+l ,"" Yn are of the form (Xi(e)LXj for some small e (specified below), such that the corresponding commu-tators [Xi, X j ] complete Xl,"" Xmo to a full n-frame at v.

Recall, that every field Yk for k

>

rna is of the form Xj+e[Xi, Xj]+O(e), therefore the vectors Yl , ... , Yn are linearly independent at v for small e, and so the differential D : ~n ~ Tv (V) of the composed orbit map

does not degenerate. Moreover, it is clear that the norm of the inverse operator is bounded by

liD-III::; const C 1 for small e ~ O.

1.1 C-C SPACES SEEN FROM WITHIN 115

It follows that the Ec:-image of the c:-ball in ]Rn around the origin contains the Riemannian ball Bv (8)

c

V for 8 2 canst' c:2 , provided c: > 0 is sufficiently small. This follows from the implicit function theorem since the second derivatives of the map Ec: are uniformly bounded in c: (as this is true for the fields

Yi

of the form (Xi (c:)

L

Xj ).

Reformulation in terms of the C-C metric. The length of piecewise integral curves defines a metric on V which is of a slightly more general type than Carnot-CaratModory since the fields Xi are not supposed to be independent. But if these fields are independent this metric essentially majorizes the Carnot-CaratModory one as our curves are (special) H-horizontal for H = Span{Xi}' In fact, the two metrics are equivalent as we shall see later and we call our present metric C-C anyway. Now we can reformulate the above proposition in terms of the following Holder bound on this C-C metric by a Riemannian one,

C-C dist;S (Riem.dist) b .

1.1.A'. Upper box bound on C-C dist for deg Xi

<

2. Let us refine (*) by taking into account the difference in the behavior of the C-C distance in different directions. Namely, we want to show that in the direction of H the C-C metric is equivalent to the Riemannian one.

Namely,

let c( t) be a smooth curve in V issuing from v E V parametrized by the length parameter t. If c(t) is tangent to H at t

=

0, then

C-C dist(v = c(o),c(t));S t.

Proof. The tangent vector of c(t) at v

=

c(o) can be written as a linear combination of Xi at v, say c'(t) =

'2::':1

aiXi. Then the value of the composed orbit map

at the point (tal,"" tam) approximates c(t) for small t as E(tal,"" tam)

=

c(t)

+

O(t2 ),

by the Taylor remainder theorem (applied to c(t) and to E with respect to some Euclidean structure in V near v). It follows, with (*), that

C-c dist(c(t), E(tal,"" tam)) ;S t

116 MIKHAEL GROMOV

and the proof of (**) is implied by the triangle inequality as C-C dist

(v,

E(tal, ... , tam)) S. t

§ 1

by the very definition of our C-C distance with piecewise integral curves.

Ll.B. Lower box bound on C-C dist for deg Xi

S

2. The above shows that every small C-C ball of radius c around v contains a box of size about e in the directions tangent to Hv and about e2 in the transversal direction (where "about e" signifies const e for const

> °

independent of e). Now we want to show that the e-C-C ball is contained in such a box.

Namely, we want to show that along a curve co(t) transversal to H the C-C distance is ~ VRiemannian distance. Here are two slightly different proofs.

First proof. Assume for the moment that the dimension of the span {Xd is constant at v and let 0 be a smooth I-form defined near v, such that

O(Xi) = 0, i = I, ... ,m and O(c~(t)) = 1, for our curve c(t) issuing from v in the direction c/(t) transversal to H.

Let c be a piecewise smooth curve tangent to H joining v with Vc: =

co(t = e) and let b denote the closed curve formed by Co [0, e] and c, see Fig. 2 below.

c v

Figure 2

This b bounds a disk D of Riemannian area about

.e

2 for

.e =

length b

=

length c

+

e,

and by the Stokes formula

AreaD

~ r

do =

r

0

~

e,

JD Jco[O,c:]

1.1 C-C SPACES SEEN FROM WITHIN 117

since 0: vanishes on c (which is tangent to H). It follows that (length c

+

e)2 ~ e and consequently

lengthc ~

Vi. •

Now, even without assuming that the span of Xi has constant rank, we have a form 0: which annihilates all Xi at v and has o:(c~(O))

=

1. Then the integral

J D

do:

= Jb

0: acquires an extra term, namely

Je

0:, which is of the order at most (length C)2 since o:(Xi ) is of order::; e in the Riemannian e-ball around v. This gives us the relation

(lengthcI

+

e)2 ~ e ± (lengthc)2, which implies length c ~

Vi

just the same.

Second Proof. Take a smooth hypersurface Va C V passing through v and tangent to Hv C Tv(V). Then every curve c issuing from v and tangent to H is also tangent to Va at v and by the Taylor remainder theorem the Riemannian distance from c(t) to Va is bounded by >:::i t2 • Hence, the Riemannian distance between v = c(O) and c(t) is bounded by

t2 on each curve Co transversal to

Yo. •

Remark. Notice that both proofs need the fields Xi to be CI-smooth and fail for continuous fields where, in fact, the quadratic HOlder bound may be invalid.

1.1.C. Corollary: Ball-box theorem for deg = 2. The small e-C-C ball in V around v is equivalent to the following exe2-box. Take a smooth rna-dimensional sub manifold VI C V through v with Tv(VI ) = Hv, take the Riemannian e-ball in VI and then the Riemannian e2-neighbourhood of this ball in V. This is our exe2-box. It is equivalent to the e-C-C ball in the sense that

where 8 and .6. are positive constants independent of e (which can be chosen continuously depending on v in a suitable sense).

118 MIKHAEL GROMOV § 1

1.2. A new proof of the Chow theorem and the HOlder bound on the C-C metric for arbitrary degree. (compare [Mitl;2] and

[N-s-

W]). Our proof of the bound

c-c

dist

-:5

(Riem.dist) ~ (+ ) for the case where T(V) is spanned by the commutators of the fields Xi of degree d ::; 2, does not easily extend for d 2: 3 but the following more straightforward approach proves out to be more forceful as it yields (+) for all d. The key ingredient is the following (well known) approximate expression for the commutator of one-parameter groups XI(t) and X2(t) in Diff V in terms of the I-parameter group corresponding to the Lie brackets of the fields Xl and X2 ,

[XI(t), X 2(tW ~f XI(t) 0 X 2(t) 0 Xll(t) 0 Xil(t)

=

[Xl, X 2](e)

+

o(t2),

(*) where the additive notation refers to some Euclidean structure in a relevant neighbourhood (and where one should note that Xi-let) =

Xi ( -t), i = 1,2).

Proof of (*). We need the following three elementary formulas

(1) (TY)(t) = Y(tT)

(2) (X +TY)(t)=X(t)oY( Tt)+O(tT)=Y( Tt)OX(t)+o(tT), for t, T --+ O.

(3) X I(t)X2(T)X

1

1(t) = (X2

+

T[XIX 2])t

+

o(tT).

Notice, that (1) is obvious, (2) follows from the Taylor remainder theorem for the composition X(t) 0 Y(Tt), and (3) is implied by (1), (2) and the definition of the Lie bracket (compare 1.1).

Now we obtain (*) in the form

XI(t) 0 X 2(t) 0 Xll(t) = [Xl, X2](t2) 0 X2(t)

+

o(e)

by applying first (3) and then (2) and (1) to the left-hand side.

Next we observe that (*) by induction implies

1.2 C-C SPACES SEEN FROM WITHIN 119

Proof of (+). We concentrate on the simplest case where dim V = 3 and T(V) is generated by Xl, X 2 and Y

=

[Xl, X2]' We denote by YO(t) the one-parameter family (not a subgroup) of diffeomorphisms defined by

° { [Xl (It I

~),

X2(ltl

~)

r for t

~

0

Y (t) =

[X2(ltl~),Xl(ltl~)r

for t:::; 0

and we observe that the composed map

EO : (tl' t2, t3) f---+ Xl(t) 0 X 2(t) 0 YO(t)(v)

sends the box B(E) C ]R3 defined by Itll :::; E, It21 :::; E, It31 :::; E2 into the E'-C-C ball in V around v for E' ~ E (in fact, for E' :::; WE). What remains to show is that the image of this box contains a Riemannian 8-ball around v for 8 ~ E2. To see that we compare EO with the composed map

for which the image of the E-box is 8-large by the implicit function theo-rem. In fact, the E-image of the E2cube defined by Itll :::; E2, i = 1,2,3, contains the required 8-ball. Then we observe with (*) that EO = E+O(E2) in the E2-cube. It follows by elementary topology (see below) that the EO-image of the E2-cube is essentially as large as the E-image. _ Elementary topology lemma. Let E and EO be continuous maps of a compact manifold with boundary into a Riemannian manifold, say B ---'> V such that

(i) distv(E, EO) = sup dist(E(b),EO(b)) :::; 80 for some 8o

>

O.

def bEE

(ii) Every two points in V within distance 8 :::; 80 can be joined by a unique geodesic segment of length 8.

(iii) The map E is a homeomorphism of B onto its image E(B) in V.

Then the image EO(B) contains every point v E E(B) for which the 8-ball in V around v is contained in E(B).

The (standard) proof of this is left to the reader.

Finally we observe that with the provisions we have made the above proof of (+) extends to the general case (of any number of fields and arbitrary d = 1,2,3, ... ) by just adjusting the notations.

120 MIKHAEL GROMOV § 1

1.2.A. Upper box bound on the C-C distance for arbitrary de-gree. The above argument does not provide decent box-shaped domains inside the C-C balls but the following simple modification of this argu-ment does just that. To see this we concentrate again on the case of three fields Xl, X2 and Y = [Xl, X2 ] spanning T(V) and we observe the following

1.2.A'. C1-Lemma. The above family YO(t) is Cl-smooth.

Proof. What we know about YO(t) is the relation

where YO(t2) and [Xl, X2](t2) are smooth (as we assume our fields Xi are COO-smooth). Hence (*)' implies that

where 'P(t) is smooth, and so

is Cl-smooth.

We observe that the differential of YO(t) at t = 0 equals that of Y(t) and so E(t) and EO(t) also have equal differentials at t = 0 (where the Cl-smoothness of EO is ensured by that of yO(t)), and so the EO-image of the box

B(E) =

{Itll ::;

E,

It21 ::;

E,

It31 ::;

E2}

is sent by EO onto a box-shaped domain in V by the implicit function theorem (which we now can apply to EO). Thus the proof of the upper box bound on distH is concluded.

1.2.B. Making "smooth" instead of "piecewise smooth" in the Chow connectivity theorem for polarizations H C T(V). In the original Chow theorem curves joining given points must necessarily con-sist of pieces of orbits of different fields and so they cannot be made smooth. But if we have a smooth polarization H, where the commutators of H-horizontal fields span T(V), we may slightly improve the result by showing that

1.3 C-C SPACES SEEN FROM WITHIN 121

Every two points in V can be joined by a smooth H -horizontal curve in V, i.e. by a smooth immersion

f :

[O,IJ -> V with f(O) = (va) and f(l) = (v) and f'(t) E H for t E [O,IJ.

Proof. Either of our two proofs of the Chow theorem provides a smooth family <1> of piecewise smooth curves issuing from vo, say 'P(t) E <1>, t E [0,1]' such that the map <1> -> V defined by 'P f-+ 'P(I) contains a given point v in its image, i.e. 'P(I) = v for a certain 'P, and this map <1> -> V is a submersion near 'P. The curve 'P consists of segments of orbits of certain H-horizontal vector fields Y1 , ... , Yk and when these fields are fixed, then 'P is uniquely determined by the lengths of the segments. In fact, these lengths serve as coordinates in <1> and so the curves in <1> close to 'P are obtained by slightly varying these lengths, called £i = £i('P), i = 1, ... , k.

Next, let us smoothly interpolate between

Xi

and

Xi+!

for all i = 1, ... , h.

Namely, we introduce a smooth family of fields yt = yt(£l, ... ,£k)' t E

[0, LkJ for Lk

=

L~=1 £i, such that

(i) yt =

Xi+!

for t E [Li

+

C, Li+l - c], for Li = £1

+ ... +

£i and small c > O.

(ii)

Ilyt II :::;

const for some const ~ 0 independent of c.

Now we define f(t) as the integral curve of the field yt issuing from Vo, i.e. f(O) = Va and f'(t) =

yt

at v

=

f(t), and observe that f -> 'P for c -> O. It easily follows (e.g. with Elementary topology lemma) that the map f f-+ f(l) contains v in its image for a sufficiently small c. • Acknowledgment. The smoothing problem in Chow's theorem was brought to my attention by Lucas Hsu.

1.3. Lower box bound on the C-C distance for deg ~ 2. We want to show that on each smooth curve ca(t) in V issuing from v in the direction c~(t) E Tv(V) transversal to the span of the commutators of given smooth fields Xi, i

=

1, ... , m, of degrees :::; s, the C-C distance satisfies for small t

>

0,

C-C dist(c(o)

=

v,c(t)) 2, tS~l.

We shall do it by adopting the second proof from 1.1.B for which we need the following

122 MIKHAEL GROMOV § 1

Definition of tangency. A smooth function

J

on V is called constant of order s at v E V with respect to given fields Xi, if for every differential operator of order r obtained by composing r :S s of our fields, denoted

XI = Xi1Xi2 ... Xir for 1= (ib ... , ir),

the function XI(J) vanishes at v, where vector fields are thought of as differential operators of first order acting on functions. Next, a subman-ifold Va C V passing through v is called s-order tangent to Xi, if every function on V vanishing on Va is constant of order s at v.

Examples

(a) If s

=

1 then this tangency means that the tangent space Tv(Vo) C Tv(V) contains the span of the fields Xi at v.

(b) If the system of fields Xi is integrable in the sense that it gives a tangent frame to a foliation of V then each leaf is tangent to Xi of infinite order and the same is true for every submanifold containing a leaf.

The following obvious lemma relates the above definition with our prob-lem.

Lemma

(a) Let J be constant of order s with respect to Xi at v and c(t) be a smooth curve issuing from v and tangent to Xi in the sense that c'(t)

= 2::::1

ai(t)Xi, for some smooth functions ai(t). (If the span H of Xi has dimension independent of v, then this tangency amounts to the inclusion c'(t) E H for all t.) Then J(c(t)) ;S tsH.

(b) Let Va be tangent to Xi at v with order s then the Riemannian distance from c( t) to Va is bounded by

Riem.dist (c(t), Va) ;S tsH.

In other words, the C-C E-neighbourhood of V near va is contained in the Riemannian ESH-neighbourhood. In particular, the C-C dis-tance on each curve Co (t) transversal to

Va

at v is ;::: (Riemannian distance) s+ 1 1 •4

4 Notice, that the only assumption on the fields Xi is C=-smoothness.

1.3 C-C SPACES SEEN FROM WITHIN 123

In order to use this lemma and exhibit actual boxes in small C-C balls, we need sufficiently many sub manifolds tangent to Xi with prescribed order. These are provided by the following

Infinitesimal lemma. Let To C Tv (V) be a linear subspace containing the commutators of Xi of orders :S sat v. Then there exists a submanifold Vo C V passing through v, having Tv (Vo) = To and tangent to Xi at v with order s.

One may prove this by a straightforward linear algebra in the space of d-th order jets of submanifolds (and functions) at v E V. But one can also make a short-cut with the following

Exponential lemma. Let Y1 , ... , Y ml , ••• , Y m2' ..• , Y m r , •.• , Y ms be linearly independent vector fields on V, where for each r

=

1, ... , s the fields Ymr - 1+1 , Ymr - 1+2 , ••• , Ymr are taken among commutators of Xi of degree r, such that the fields Y1 , ... , Y mr have the same span at v as the commutators of Xi of degree :S r. (Obviously, such lj exist.) Then the image Vo of the exponential map expv : IRms ---+ V corresponding to lj is tangent to Xi at v with order s.

Proof. To make it simple we start with the case s

=

2 and assume for the moment that the fields Xi are linearly independent and so m1 = m and lj = Xj for j = 1, ... , m. Let

f

be a smooth function vanishing on Vo (or rather on a germ of Vo at v) and observe that

(1) [Xi,Xj]f(v)

=

0, i,j = 1, ...

,m,

since Tv(Vo) contains the second degree commutators of Xi.

(2) For every field X =

2:::1

aiXi, the operator X

2

satisfies X

2

f(v)

=

O. In fact XP f(v)

=

0 for all p = 1,2,3, ... , as f vanishes on the orbit X(t)(v) (which is contained in the exponential image of IRm corresponding to Xi).

It follows that

(XiXj

+

XjXi)f(v) = ((Xi

+

X j )2 -

xl -

XJ)f(v) = 0

which implies with (1) that (XiXj)f(v)

=

0 as well. This gives us the required tangency of Vo to Xi in the case where Xi are independent and the dependent case is taken care of as follows. Every field among Xi, say X io ' can be written as a sum, Xio

=

Yi~

+

X~, where Yi~ is

124 MIKHAEL GROMOV § 1

some combination of lj, j

=

1, ... , mo, with constant coefficients, such that Yi~ equals Xio at v and so XPo = Xio - li~ vanishes at v. The relations [Xi,XjlJ(v) = 0 imply that [XPo,Xi]f(v) = 0 since XPo is a combination of Xi. Therefore the (obvious) vanishing XPoXd(v) = 0 implies XiXPof(v)

=

0, and consequently

XiXjf(v) =

(Yi' +

XP)(Yj

+

XJ)f(v) =

Y:Yjf(v)

+

li'XJf(v) +XPYjf(v) +XPXJf(v)

=

0, where the vanishing of the first term is ensured by (the argument in) the independent case.

Case 8 = 3. Now we start with the relations [Xi, [Xj,XkJ]f(v) = 0 and we use yP f(v) = 0 for every combination Y of lj with constant coefficients. Thus we get Yilj f( v)

=

0 for i

=

1, ... , ml and j

=

ml

+

1, ... , m2 as well as for i = ml

+

1, ... , m2 and j = 1, ... , ml, by using [Yi, lj]f(v) = 0 and y2 f(v)

=

0 for Y = Yi, lj and Yi

+

lj. This implies that the (second order) operators XdXj,Xk] and [Xj,Xk]Xi also vanish at f(v) where dependencies among Xi and [Xj,Xk] are taken care of as above for s = 2.

Next we show that the symmetrization of YiYjYkf(v), over all (six) per-mutations of the three indices i, j and k, vanishes for i, j, k

=

1, ... , mo.

This is done with the identity (aYi+blj+cYk)3 f(v)

=

0 for a, b, c

=

0, ±l.

This implies with the above that YiljYkf(v)

=

0 for i,j,k

=

1, ... ,mo.

What remains is to pass from Yi to Xi, where Xi may be dependent, but these dependencies are easily taken care of as earlier. _

Case 8 ~ 4. The proof of this is clear by now.

We conclude by explicitly describing a box-shaped domain in V con-taining the C-C ball around v provided by the above lemmas.

Step 1. For every r = 1,2, ... we take a submanifold

Vr

C V passing through v such that Tv (Vr) equals the span of the commutators of Xi of degrees 1, ... , r. A specific such

Vr

is provided by the exponential map corresponding to Y1, ... , Ymr from the exponential lemma.

Step 2. We take the Riemannian cr+1-neighbourhood of

Vr

for each r and intersect these over r

=

0, 1, ... (with the convention Vo = {v}). This intersection is, indeed, box-shaped if we take Vo C V1 C ... C V2 C ... , (which is possible with the exponential lemma) and it contains the c'-C-C ball around v E V for c' ~ c.

1.3 C-C SPACES SEEN FROM WITHIN 125

Matching upper and lower box bounds. Our (families of) boxes in-side (see 1.2.A) and outin-side (see above) the C-C balls are slightly different but this does not bring any confusion into the geometric picture as these families are equivalent in an obvious sense as a simple argument shows.

(See [N-S-W] for further information.)

1.3.A. Doubling and covering properties for balls; equisingular-ity and the Hausdorff dimension. The ball-box theorem (as stated in 0.5.A) immediately implies the following universal bound on the Rie-mannian volume of concentric C-C balls in a compact manifold V,

for all v E V and real p and some constant C

=

C(V). (This is one of the major applications of the ball-box theorem indicated in [N-S-W].) Consequently we obtain, as an obvious corollary, the following (purely internal) metric property of V.

Every ball Bv(2p) can be covered by at most k balls of radii p for some k depending only on V (but not on v or p).

Now, suppose H is equiregular. Then, clearly, the ball-box theorem shows that small p-balls have volumes ~ pD (see (+) in 0.6) and thus dimHau V = D with 0

<

meSD

<

00, where D is computed in terms of the commutator filtration 0 C H = Hl C H2 C ... C Hd = T(V) by D

= 'L-1=l

i rank(Hd Hi-d. Furthermore, let V' C V be a submanifold in V such that the intersections HI

=

Hi

n

T(V') have constant ranks for all i

=

1, ... , d. Then in the (proof of the) ball-box theorem one may use frames adapted to V': if some vector from the frame is tangent to V' at a

Now, suppose H is equiregular. Then, clearly, the ball-box theorem shows that small p-balls have volumes ~ pD (see (+) in 0.6) and thus dimHau V = D with 0

<

meSD

<

00, where D is computed in terms of the commutator filtration 0 C H = Hl C H2 C ... C Hd = T(V) by D

= 'L-1=l

i rank(Hd Hi-d. Furthermore, let V' C V be a submanifold in V such that the intersections HI

=

Hi

n

T(V') have constant ranks for all i

=

1, ... , d. Then in the (proof of the) ball-box theorem one may use frames adapted to V': if some vector from the frame is tangent to V' at a

Nel documento Progress in Mathematics (pagine 122-137)