Canonical coordinates, almost Lie groups, nilpotent tangent cones and a sharp version of the ball-box theorem for

equireg-ular polarizations. Suppose we are given a frame of smooth vector fields Y1 , •. . ,Yn on V, n = dim V, where each

li

is assigned an integer deg

li

= deg i ~ 1, such that commuting fields at most add degrees, i.e.

[li,}j] = L Cijk(V)Yk , k

where Cijk

=

0 for degk

>

degi

+

degj (compare 0.5). Then for each v E V one defines the following Lie algebra Lv with a preferred basis, denoted

Yiv

where the Lie brackets are given by the formulae

[Yiv, Y;vJ =

LbijkCijk(V)Yk', k

where bijk

=

1 for degk

=

degi

+

^{degj and bijk} 0 otherwise. To comprehend the meaning of (*)V we E-scale the fields

li

according to their degrees, denote eli

=

^EdegYiYi, and express the multiplication table ( *) in terms of eli. This gives

[eli, e}j] = LEdijkCijkeYk, k

for dijk

=

degi

+

degj - degk. Now we see that (*)V equals the limit of

e ( *) at v for E ^----t 0 which shows that Lv is indeed a Lie algebra which is nilpotent of degree at most max deg Yi.

t

1.4 C-C SPACES SEEN FROM WITHIN 129

Definition. (compare [Good], [Bell]). The simply connected (nilpotent) Lie group Nv corresponding to Lv with distinguished left invariant fields corresponding to Yiv is called the nilpotent tangent cone of the frame {li}

at v.

We still denote the distinguished fields on Nv by Yi^v and we want to show that the formal limit relation "(*) ~ (*)V for c ~ 0 implies an actual convergence "li ~ Yiv in suitable local coordinates in V and Nv.

In fact, one can use for this purpose any system of coordinates in V near v which is made up in a canonical way out of the fields "li but we shall stick to the coordinates ti defined with the composition of the one parameter groups li(t), namely, with the map

defined on a certain cube

We identify the Lie algebra Lv

=

L(Nv) (which comes along with the basis YiV) with ]Rn and we denote by Eo : Lv = ]Rn ~ Nv ^{the map} defined by composing the one-parameter subgroups YiV(t) corresponding to the fields Yiv on Nv. Now we assume the "radius" p> 0 is so small that the maps Ev and Eo are diffeomorphisms of B(p) onto their respective images in V and Nv and we transport the (left invariant) fields Yiv from Nv to V (or rather to the image Ev(B(p)) C V) by (the differential of) the map

(Notice that the map Eol is defined for all p as Eo is, in fact, a diffeo-morphism of Lv onto Nv, but this is irrelevant for our local discussion.) We denote the transported fields by

Yiv

and we want to compare them with the fields li in smaller and smaller "boxes" around v in V obtained by the following c-scaling of a fixed (cubical) box B(p)

=

Ev(B(p))

c

V.

We denote by a" : ]Rn ~ ]Rn the linear operator sending the Euclidean basis { ei} to {cdeg i ei} and we apply this notation to our systems of vector fields li, Yiv, etc. For example, we write a,,(li) for "li = cdegili. Then the c-scaled box B,,(p) C V is defined as the Ev-image of a"B(p) C ]Rn.

Equivalently, fJ,,(p) can be defined as the image of B(p) under the Ev-map corresponding to the fields a"li

=

"li.

Next we observe that the operators a" on Lv

=

^]Rn are automorphisms of the Lie algebra Lv and we denote by A" : Nv ^~ Nv the corresponding

130 MIKHAEL GROMOV § 1

automorphisms (self-similarities) of Nv . Then we transport these for c ::; 1 to V via

EvEr;l

and denote the transported maps by

where the "transport" is defined by

Let us summarize what we have obtained so far. We have chosen a small "curved cube" B(p)

c

V around v, that is the Ev-image of an actual p-cube in ~n. We have on this cube two systems of fields,

Yi

and

¥iv

which are related as follows.

(1) The two systems of fields coincide at v,

¥iV(v)

=

Yi(v),

i = 1, ... ,no

(2) The multiplication table for

¥iv

has constant coefficients,

[¥iV, fjV] = L

^CijkYk'

k

where ^Cijk are the structure constants of the Lie algebra ^Lv with the distinguished basis

Yiv.

(3) The multiplication table (*)V is obtained from that for the fields

eYi

=

ae(Yi)

(see e(*)) by sending c ----+ 0 and by evaluating the limit at ^V.

(4) We have diffeomorphisms

Ae

of B(p) onto smaller box-like domains Bc(p)

c

B(p), such that (the differentials of)

Ae

on

¥iv

commute with ae , i.e.

Ae(¥iV)

=

ae(¥iV),

or equivalently,

A;l(ae(¥iV))

=

¥iv.

Now we want to understand what happens to the fields

fie

⁼

A;l(ae(Yi))

on B(p) (where the corresponding fields yi need be defined on Be(P)) for c ----+ O.

l.4.A. Convergence Proposition. If the fields yi are C1-smooth (which we assume all along) then the fields

fie

uniformly converge to

¥iv

on B(p).

1.4 C-C SPACES SEEN FROM WITHIN 131

Proof. The multiplication table for the fields

fie

^{on B(p)} is obtained from that of eY,: on Be(P) (see e(*) above) with the map Ae : B(p) --+

Be(p), Namely,

[fie, fje]

⁼

L

^C'fjk

^Y{,

k

where

C;jk(V')

=

EdijkCijk(Ac(v')), v' E B(p), for dijk

=

degi

+

degj - deg k.

It follows, by the continuity of Cijk at our point v (in the "center" of B(p)), that the oscillations of Cijk on Be(p) go to zero for E --+ 0 and so the same is true for the oscillation of CYjk on B(p). Thus we have uniform convergence on B(p),

C'fjk --+ Cijk for E --+ O.

Now we are going to prove the required convergence

fie

^--+

^fiv

^by

ob-serving that

fie

^and

^fiv

satisfy similar systems of ordinary differential equations (in the coordinates t1,"" t n ) with the coefficients cYjk and Cijk playing identical roles. To do this we lift all our objects to the cube B(p)

c

]Rn

=

Lv where we use the coordinates t1, .. . , tn and we introduce the following notations CYjk, the lifts of the functions CYjk to B(p) via the map Ev : B(p) --+ B(p).

"fie,

the lifts of the fields

fie

^{to B (p) by (Ev)} -1. (This makes sense since E~ is a diffeomorphism of B(p) onto B(p) by our assumption.)

"fiv, the lifts of the fields

Yi

^v to B (p)

c

]Rn

=

Lv via the map Eo : Lv --+

Nv . Notice that the map Ev : B (p) --+

B

(p) sends "fiv to

Yt.

First observation. The fields

"fie

are related to the fields Oi = a~i on B(p)

c

]Rn by the following identities,

Y{ 01 on B(p)

Y2^e 02 on the subspace {t1

=

O}

c

B(p),

ye

3 03 on{t1

=

0, t2

=

O}

c

B(p)

Y,; an

on the tn-line {ti = 0, i = 1, ... ,n -I} C B(p).

(A)

132 MIKHAEL GROMOV § 1

Furthermore, the fields Yiv satisfy the same system of relation on B(p).

To see that, we first observe that the lifts

Yi

to B(p) ofthe original fields Yi

= Y?

satisfy (A) as immediately follows from the definition of the map Ev : ^{]Rn ---t} V via the composition of the one-parameter subgroups Yi(t).

Similarly, the fields fiv satisfy (A) since these are the lifts of Yiv from Nv ^to Lv =]Rn via the map Eo : Lv ^-+ Nv obtained by composing YiV(t). What remains to show is that the passage from Yi = fil to fie = Ac ^{1 (} ae (Yi) ) does not change the fields

fie

on those parts of B(p) where the relations (A) apply. Namely

fh (= Yt) on all of B(p)

fh (=

yn

on the subspace {tl = O}

c

B(p) and so on. To see this we shall bring Ae from V to ^]Rn by taking

Then

fie ⁼

^Ac1

^(ae(Yi))

^{and (A) for}

fie

follows from the relations (A) for fiv, which are

YI

^V = 81 on B(p),

Y

2^{v =} 82 on {tl = O}

c

B(p),

etc., and the commutation between Ae and ae on the fields fiv, i.e.

Ac1 (ae(fiV))

=

fiv, which follows from the corresponding relation for Ae (and eventually for Ae). Q.E.D. (This extra argument for

fie

^was

needed as the fields

fie

were defined with Ev rather than with the map corresponding to fie (t).)

Second observation. The fields

fie

satisfy the following linear differ-ential equations,

(B)

1.4 C-C SPACES SEEN FROM WITHIN 133

Furthermore, the fields

¥iv

satisfy an identical system with (the constants) - ' t d f -²

Cijk IllS ea 0 cijk'

The equations (B) follow from the commutation relations for

¥ie,

^that

are

[fie, fie] = I:

^Ceijk

^Y:,

k

the identities (B) and the obvious formula [8i , Y] = 8iY = aa^Y for all

def t,

fields Y on

]Rn.

Similarly, we see the validity of these relations for

¥iv

^with

Cijk in place of c':fjk'

To understand the meaning of (A) and (B) let us read these equations from bottom to top. The last identity in (A), i.e.

Y;

^{= 8n ,} should be thought of as an initial value datum for the (last among (B)) equation 8n- IY; = L:~=I C~,n-l,kYk on the (tn-I, tn)-plane (given by the equa-tions ti

=

0, i

=

1, ... ,n - 2). Notice that this equation also involves the fields Yk for k ::; n - 1 on the (tn-I, tn)-plane but these are equal to 8k on this plane according to (A). Next, we take Y; on the (tn-I, tn)-plane ob-tained by solving our initial value problem and we also take Y;-I = 8n-1

on this plane as given by the second from the bottom relation (A). Then the pair (Y;, Y;-I) serves for initial values for the second from bottom equations in (B) on the (tn-2, tn-I, tn)-space which are

where the field

Y:

on the right hand side for k ::; n-2 equals 8k according to (A).

Conclusion. The fields

¥i

^{e on B(p)} are uniquely determined by their values at v and by the functions c':fjk via the equation (A) and (B). It follows, by an elementary theorem on dependence of solutions of linear O.D.E. upon initial conditions and coefficients, that the fields

¥i

^{e are}

continuous in c':fjk' In particular,

¥i

^{e - t Yijk for E - t 0 as c':fjk - t Cijk.}

Consequently, the fields

fie,

which are images of

¥ie

^{under Ev , converge}

to

fiv.

Q.E.D.

134 MIKHAEL GROMOV § 1

1.4.A'. Uniformity of the convergence. Recall that the objects ap-pearing in 1.4.A are constructed with the use of a distinguished point v E V. These are the "curved cube"

B

(p) =

B (

v, p), the fields

yt

coming from Nv and the fields

fiE

obtained by some rescaling and "homotopies"

Ac1

= A~-l of

Yi.

Now, we claim that the norm p~v

- ¥iell

can be bounded independently of v. More precisely, we have the following Uniform version of l.4.A. If the fields

Yi

are C1-smooth, then, for each compact subset Va C V, there exists a positive number p and a function 8(s) ^---+ 0 for s ^---+ 0, such that for each v E Va the fields

¥iv

and

¥ie

are well defined on B(p) = B(v, p) and

II¥iv - ¥iell ::;

8(s), where the norm refers to the Riemannian metric on V which makes the frame

Yi

orthonormal.

This is immediate by observing that the proof of 1.4.A is "uniform in v".

1.4.A". On C^r -convergence. If the fields

Yi

are Cr+1-smooth then

¥ie

converge to

¥iv

in the C r -topology and this convergence is uniform in v in the above sense.

Proof. If

Yi

are C^r+¹ then the coefficients Cijk are C^r and so for r 2': 1 the Lie derivatives of the functions Cijk with respect to the fields

y:

^go

to zero as fast as s, i.e.

iY:

^{C~j k} I ::; const s.

This estimate lifts to B(p) where it reads

iY:

C~j ^k I ::; const s,

and since the fields

Y:

are close to a fixed frame (namely

Y;)

this implies IO{tCijkl ::; c~tE.

Then by going through (A) and (B) one obtains a C1-bound on

"fie,

namely

IO{t"fie I ::;

const's. With this, if r 2': 2, one can pass from the bound on the Lie derivatives

Y:Y,;

^Cijk to the bound on o{tOvCijk which yields, in turn, a bound on O{tOv"fijk and so on. _

1.4 C-C SPACES SEEN FROM WITHIN 135

l.4.B. Approximation of equiregular Carnot-Caratheodory spaces by self-similar nilpotent groups. Let H

c

T(V) be an corresponding left invariant metric for which the differential of EvE01 at id E Nv is isometric. Now both, V and Nv , have C-C metrics, say dist sufficiently smooth then the difference between the metrics dist and dist~

on B* (v, c) is o( c), i.e. c ¹ (dist( VI, V2) - dist~ (VI, V2)) ---+ 0, for all pairs

E--->O

of points VI, V2 E B* (v, c).

Proof. To see the picture clearly for c --> 0, we rescale our metrics and neighbourhoods by c-¹ using the (expanding) diffeomorphism

Ac

¹ acting in V near v (see 1.4.A). To simplify the matter, we assume for the moment that our Riemannian metric in V comes from the left invariant metric in N v via EvE01. (This assumption, in fact, does not restrict the generality as our old metric agrees with the new one at v.) Then we observe that

Ac,

scales dist~ by c, i.e.

A;

^{dist~ = c dist~, wherever}

A;

^{and dist~ are}

defined. In particular, A;~l transform B*(v,c) to B*(v, 1), provided E;;I,

5 Compare [Mitl;2J.

136 MIKHAEL GROMOV § 1

and hence

A;

are defined on the ball B*(v, 1). In fact, this can be always achieved by multiplying the underlying Riemannian metric by a fixed large constant and we assume from now on that the unit ball B* (v, 1) is small enough for our game.

Now we compare c:-Idist and c:-Idist* in B*(v,c:) by bringing them to B*(v,1) by Acl. We observe that Ac1(c:-Idist*) = dist* and so we must prove the uniform convergence on the unit ball,

(Ac1 (c:- I dist) - dist*) -+ 0 for c: -+

o.

To prove this we use 1.4.A which shows that the polarization Ac1 (H) on B*(v, 1) converges to the polarization H* corresponding to dist* and the Riemannian metric (i.e. quadratic form) on this polarization transported from the original metric on H converges to the metric in H*, since the vectors YI , ... ^'Yn1 spanning H satisfy, according to 1.4.A,

where

Yiv

are certain vector fields spanning H* .

To conclude the proof we would need the following continuity of the Carnot-Caratheodory metrics defined by (H,g) where H C T(V) is a polarization and 9 is a Riemannian metric on H,

if (HE;' grJ converges to (H*, g*), then the C-C metrics also converge, i.e. distE; -+ dist*.

This is indeed so if the convergence HE; -+ H* is understood in the Cd-I_topology (where d is the bound on the degrees of the commutators of the fields in H* spanning T(V)) due to a uniform bound on the metrics distE (see below) but, in general, e.g. for CO-convergence HE; -+ H*, the functions distE; for arbitrarily small c:

>

0 may be, a priori, infinite on certain pairs of points in V and so one cannot speak of the ordinary (uniform) convergence distE; -+ dist*. However we do have the following weak convergence defined with the Hausdorff distance between subsets in V with respect to dist*.

1.4 C-C SPACES SEEN FROM WITHIN 137

Weak convergence lemma. If (He, ge) uniformly converge to (H* , g*) then every diste-ball Be (v, p) Hausdorff-converges to the corresponding dist* -ball B*(v, p) and this convergence is uniform on compact subsets in VxlR+, i.e.

for c5(c) ~ 0 for c ~ 0, where one may use a fixed function c5(c) for each compact subset of points (v,p).

Proof. Let HI and H2 be mutually close polarizations with close quadratic forms. Then, at least locally, there exist mutually close frames of orthonormal vector fields spanning HI and H2 and with these fields one establishes a correspondence between Hr and H2-horizontal curves as follows. The curves CI(t) and C2(t) parametrized by arc length (coming from the quadratic forms in HI and H2 respectively) correspond to each other if at each moment t their derivatives,

c~(t) E (Hdv l=Cl(t) and c;(t) E (H2)v2=c2(t)

have identical decompositions with respect to the frames in HI and H2 .

Clearly, corresponding curves issuing from nearby points remain close for a certain time which implies the required closeness of the C-C balls. (We suggest the reader would check that this "close" talk can be made rigorous and uniform.)

Corollary. If the metrics diste are uniformly bounded, i.e. if each diste-ball of radius p around v contains the dist* -ball around v of radius p*

for some strictly positive function p*(p), p> 0, then

I

diste: - dist*

I

^{~ 0}

uniformly on compact subsets in V.

This is obvious by the triangle inequality.

Conclusion of the proof of the approximation theorem for C2 d-2 -smooth polarizations H C T(V). If H is or -smooth, then the corresponding frame YI , ... ,Yn obtained by taking commutators of degree S dare Cr-d+I-smooth and then He converges to H* in Cr+d _

topology according to 1.4.A". In particular, the convergence He: ~ H*

is C^r+^{d •} Now we use the fact that T(V) is generated by commutators of H* -horizontal fields of degree S d which, according to the Chow con-nectivity theorem, makes dist*

<

^00. Since the Chow theorem appeals to the derivatives of H* of order S d - 1, it is stable under small C d- I_

perturbations of H* which implies a uniform bound on diste whenever

He: is sufficiently Cd-I-close to H*. •

138 MIKHAEL GROMOV § 1

The case of H being Cd-smooth. We start by giving a bound on a metric in pure "Hausdorff terms". To grasp the idea, imagine we have two metrics on V, say dist and dist*, such that every dist-ball B(v,p) is sufficiently dist* -Hausdorff close to the corresponding dist* -ball B*( v, p).

Say,

distifau (B( v, p), B* (v, p)) ~ plIO.

Then the triangle inequality implies that the metrics are close as functions on VxV. In particular (and most importantly)

B(v,p)

c

B*(v,ap), for a fixed a

>

^0,

(where one can take a = ~).

Now we turn to the proof of the approximation theorem and observe that the weak convergence lemma makes every small dist-ball around v quite close to the corresponding dist~-ball. But here, unlike the above discussion, the metric dist~ depends on v. To remedy that we exclude v by setting dis* (v, v')

=

dis~ (v, v'). Of course, dis* is not, in general, a metric, but the above argument also works for quasi-metrics, that are positive functions on V x V, vanishing exactly on the diagonal and satisfying the following approximate triangle inequality,

dis(v,v") ~ const(dis(v,v')

+

dis(v',

v")), (+)

which must be uniform on compact subsets in VxVxV (Le. for each K

c

V x V x V there exists canst, so that (+) holds for all v, v' ,v" E K).

In order to prove (+) for dis* we recall that the function dist* comes from the nilpotent group Nv with self-similarities Ae : Nv ^~ Nv defined with ae : Lv ^~ Lv on the Lie algebra Lv = L(Nv). Since Ae and ae commute with the (composed orbit) map Eo : Lv ---+ Nv, (i.e. Eoae =

AeEo) the €-balls in Nv around id E Nv are equivalent to the Eo-images of the €-boxes Be in Lv =]Rn defined by Itil ~ €degi. (We have already seen this picture for the exponential map in 0.3.e). Therefore, the inequality (+) for dis* reduces to the corresponding property of the Ev-images of the boxes Be C ]Rn = Tv(V). Namely, we need to show, that if two such images, say Ev(Be) and EV1(Bo) in V intersect, then EV1(Bo) is contained in Ev(Be/) for €' ~ const(€

+

8). (Warning: the commutation relations deg[Yi, lj) ~ degi

+

degj is crucial for this property.) To see this we recall that Ev (Be) consists of the second ends of piecewise smooth curves issuing from v which are built of n segments Cl, ... , Cn, where Cl

1.4 C-C SPACES SEEN FROM WITHIN 139

is a piece of orbit of Yl of length::; cdeg 1 (= c), C2 such a piece for Y2 of length::; cdeg 2 and so on. Thus the problem reduces to showing that if we add to such curve a new piece cn +! ofthe orbit of Yj of length (8)deg j , then there exists a curve of n pieces

ci, ... ,

c~, ... , c~ with lengths::; (c,)degi for c' ::; const(c

+

8), such that the second end of the new curve equals the free end v" of Cn+!, see Fig. 3 below.

Cn+I v"

cnllc~

~ ___ C_1 __ 7C_2~[ ^__ --_-_-~I~::::::!- ^______

¹

ci

v

Figure 3

In other words, we must compensate for changing the order of the orbits and this can be achieved with the relation (*) in 1.2. (compare [N-S-W]).

This is straightforward and we leave it to the reader.

Finally, the weak convergence lemma shows that the "Hausdorff dis-tance" between dist and dis* on B* (v, c) C V is O( c) uniformly in v (because of l.4.A'). This yields a bound on the metric dist by the above argument (but now, of course, with a constant different from 3/5) which concludes the proof of the approximation theorem for Cd-smooth H.

Remarks and corollaries

(a) It seems, the conclusion of the theorem should stand for H being Cd-I.

(b) If His

e

^{Ml ,} it is easy to show that o(c) in the approximation theorem can be replaced by O(c2 ).

(c) The ball-box theorem is an immediate corollary of the approxima-tion theorem as the balls in Nv are (obviously) box-like as we have mentioned several times.

(d) The Mitchell theorem concerning the tangent cones of C-C manifolds (see (iii) in O.3.D) immediately follows from the approximation the-orem. In fact, one only needs here the (weak) dis* -Hausdorff

conver-140 MIKHAEL GROMOV § 1

gence of distc rather than the final result on the uniform convergence (see [Mih;2]).

(e) The ball-box theorem implies (at least in the equiregular case) that small balls in V are "essentially contractible" ) i.e. each small E-ball is contractible within the concentric ball of radius CE for a fixed C 2': l.

In fact, one can squeeze a topological ball (box) between the E and the CE-balls. Furthermore, the balls in nilpotent groups with self-similarities are honestly (and obviously) contractible (i.e. the above C equals one). It follows by the approximation theorem that for all V one has C -7 1 for E -7 0 and it is likely (for sufficiently smooth C-C data) that C = 1 for small E, i.e. small balls are probably contractible.

1.4.C. Pinching and related problems for C-C metrics. The ap-proximation of a frame of fields

Yi

with almost constant coefficients in the multiplication table

[Yi,

Yj]

=

L:k CijkYk by a frame where the corre-sponding coefficients are truly constant (see 1.4.A) represents a simplest instance of the stability phenomenon for (homogeneous) geometric struc-tures. In general, one looks for a weakest possible local or infinitesimal criterion for a given geometrIc structure to be homogeneous or almost homogeneous in a suitable sense. More specifically, when dealing with a Riemannian metric, one makes up such a criterion in terms of curvature (and covariant derivatives of the curvature) sometimes by pinching the curvature between two constants. (This explains the "pinching" terminol-ogy.) In our case the structure was given by a frame of vector fields which form an almost Lie group in the terminology of [Ru] and there are several results due to Min-Oo and Ruh allowing an approximation of an almost Lie group by an actual Lie group which goes infinitely deeper than our proposition l.4.A. In fact, our argument with choosing O.D. equations (A) and (B) parallels the initial phase of the proof of Rauch's comparison and pinching theorems for Riemannian manifolds. (See [GrosAP] for an exposition of these techniques and ideas.)

1.4.C. Pinching and related problems for C-C metrics. The ap-proximation of a frame of fields

Yi

with almost constant coefficients in the multiplication table

[Yi,

Yj]

=

L:k CijkYk by a frame where the corre-sponding coefficients are truly constant (see 1.4.A) represents a simplest instance of the stability phenomenon for (homogeneous) geometric struc-tures. In general, one looks for a weakest possible local or infinitesimal criterion for a given geometrIc structure to be homogeneous or almost homogeneous in a suitable sense. More specifically, when dealing with a Riemannian metric, one makes up such a criterion in terms of curvature (and covariant derivatives of the curvature) sometimes by pinching the curvature between two constants. (This explains the "pinching" terminol-ogy.) In our case the structure was given by a frame of vector fields which form an almost Lie group in the terminology of [Ru] and there are several results due to Min-Oo and Ruh allowing an approximation of an almost Lie group by an actual Lie group which goes infinitely deeper than our proposition l.4.A. In fact, our argument with choosing O.D. equations (A) and (B) parallels the initial phase of the proof of Rauch's comparison and pinching theorems for Riemannian manifolds. (See [GrosAP] for an exposition of these techniques and ideas.)

Nel documento Progress in Mathematics (pagine 137-161)