** 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, ... *,X**m *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), ***Y**2**(t), ... , Yp(t), where the corresponding vector nelds Y1, Y**2 , ••. , **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*=

*t*

*j ,*(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 Y***l , ... , **Y**n *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 *Y**k *for *k *

### >

rna is of the form*Xj+e[Xi, Xj]+O(e),*therefore the vectors

*Y*

*l , ... ,*

*Y*

*n*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(t*

*2 ),*

*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 e

^{2 }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*

*for*

^{2 }*.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 *t** ^{2 • }*
Hence, the Riemannian distance between

*v*=

*c(O)*and

*c(t)*is bounded by

*t** ^{2 }*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 X*

*2*

*(t)*in Diff V in terms of the I-parameter group corresponding to the Lie

*brackets of the fields Xl and X*

*2 ,*

[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) * *= *

*X**i ( **-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

### ~),

^{X}

^{2(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 E^{2}-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, X*2 ] *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 Y*1 , ... , **Y**k *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 **Y*1 , ... , **Y *ml , ••• , *Y **m2' ..• , **Y **m r , •.• , **Y **ms * be
*linearly independent vector fields on V, where for each **r *

### =

1, ... , s*the*

*fields Y*

*mr - 1*

*+*

*1 ,*

*Y*

*mr - 1*

*+*

*2 , ••• ,*

*Y*

*mr*are

*taken among commutators of Xi of*

*degree r, such that the fields Y*

*1 , ... ,*

*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

*the operator X*

^{aiXi, }### 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, ... , Y*

*mr*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

*V*

*1*C ... C

*V*

*2*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