Table Of ContentASYMPTOTIC EXPANSION OF HOLONOMY
7 ERLENDGRONGANDPIERREPANSU
1
0
2 Abstract. Givenaprincipalbundlewithaconnection,welookforanasymp-
totic expansion of the holonomy of aloop interms of its length. Thislength
n
is defined relativeto someRiemannianor sub-Riemannianstructure. Weare
a
abletogiveanasymptoticformulathatisindependentofchoiceofgauge.
J
0
1
] 1. Introduction
G
Let G → P → M be a principal bundle with some connection ω and let Ω
D
be its curvature form. The realization of curvature as the infinitesimal generator
.
h of holonomy of ω appears in the now classical Ambrose and Singer theorem, [1].
t Further elaborated by Ozeki [10], this result explains how the holonomy group of
a
m ω is infinitesimally generated by Ω and its covariantderivatives at a point.
If we look at an individual loop γ based in x ∈ M, we can also make the
[
correspondence concrete in the case when G is abelian. An application of Stokes
1 theorem shows that for any p ∈ P , the corresponding holonomy Holω(γ) equals
x p
v
exp(− σ∗Ω) where σ denotes an arbitrary gauge satisfying σ(x) = p and
0 disk(γ)
7 denotes the integral over an arbitrary filling disk. Using ideas from [11],
disk(γ)R
5 we are able to show that this result holds approximately for shortloops relative to
2 R
someRiemannianorevensub-RiemannianmetriconM. Suchexpressionshavereal
0
world applications, see [9, 5]. A question in these applications if often to look for
.
1 thebestchoiceofgauge. Inthe caseofM =Rn,weobtainthefollowingexpansion
0
using the radial gauge.
7
1 Theorem 1.1. For every x∈Rn, define
:
v
3+k Ω and all its covariant derivatives of order ≤k vanish at x,
i m (x)=
X Ω 2 otherwise.
(cid:26)
r Let σ : Rn → P be a gauge in which ω is parallel in radial directions emanating
a
from x. Then for p=σ(x)∈P , there is an ε>0 and C ≥0 such that
x
(1.1) exp−1Holω(γ)+ σ∗Ω ≤Cℓ(γ)2mΩ(x),
p
(cid:12)(cid:12) Zdisk(γ) (cid:12)(cid:12)
(cid:12) (cid:12)
for every loop γ ba(cid:12)sed at x with ℓ(γ)<ε. In the(cid:12)above formula, denotes the
(cid:12) (cid:12) disk(γ)
integral over a radial filling disk.
R
The first author is supported by the Fonds National de la Recherche Luxembourg (AFR
4736116andOPENProjectGEOMREV)andbytheResearchCouncilofNorway(projectnumber
249980/F20). ThesecondauthoracknowledgessupportfromANR,projectSRGI,ANR-15-CE40-
0018.
2010 Mathematics Subject Classification. 53C29,41A99,53C17.
Key words and phrases. asymptotic expansion of holonomy, radial gauge, sub-Riemannian
geometry,horizontalholonomy.
1
2 E.GRONGANDP.PANSU
TheaboveresultalsoholdstrueonCarnotgroups,whichalsohavedilationsand
hence the possibility of defining a radialgauge. The details are found in Section 2.
For this reason,the radialgauge is convenient,but may notalwaysbe so simple
to compute in practice. We therefore continue in Section 3, showing that the
two-form σ∗Ω in (1.1) can be replaced by a two-form with polynomial coefficients
whose terms are independent of choice of gauge. For the special case of Rn with
coordinates (z ,...,z ), we have the following result.
1 n
Theorem 1.2. Write ∂ = ∂ and let x = (x ,...,x ) be any point. Let σ be
k ∂zk 1 n
an arbitrary gauge such that σ(x) = p. Use the notation ωσ = (σ∗ω)(∂ ) and
k k
Ωσ =(σ∗Ω)(∂ ,∂ ) and define
ij i j
n
1
(1.2) F (γ)= Ωσ(x) (z −x )dz
p 2 ij i i j
i,j=1 Zγ
X
n
1
+ (∂ Ωσ)(x)+[ωσ(x),Ωσ(x)] (z −x )(z −x )dz ,
3 k ij k ij i i k k j
i,j,k=1 Zγ
X (cid:0) (cid:1)
for any loop based at x. Then F does not depend on the choice of gauge. Further-
p
more, there exist constants ε>0 and C ≥0, such that exp−1Holω(γ)+F (γ) ≤
p p
Cℓ(γ)4 for every γ with ℓ(γ)<ε.
(cid:12) (cid:12)
(cid:12) (cid:12)
For a general Riemannian or sub-Riemannian manifold, we first need to choose
an appropriate local coordinate system of a type called privileged. However, after
this choiceis made, the abovestatement remains true with respectto the dilations
in this coordinate system. We give the details of this in Section 4.
It was observedin[11]that integratingthe curvature overa radialfilling disk in
the sub-Riemannian Heisenberg group will actually give a better approximationof
the holonomy than in the Euclidean case. This result was found using ideas from
[4, 13]. By applying the new theory of horizontal holonomy found in [3], we are
able to show the following (for more details, see Section 5).
Theorem 1.3. Let π : P → M be a principal bundle over a sub-Riemannian
manifold (M,D,g), where D has rank n . Assume that the sections of D and their
1
iterated brackets up to order k span a subbundle of rank equal to the rank of the
free nilpotent algebra of step k+1 and of n generators. Then there is a“modified
1
curvature” Ω˜ such that
exp−1Holω(γ)+ σ∗Ω˜ ≤Cℓ(γ)2k+2,
p
(cid:12)(cid:12) Zdisk(γ) (cid:12)(cid:12)
(cid:12) (cid:12)
The result of Th(cid:12)eorem 1.2 follows from compu(cid:12)tations on Euclidean space found
(cid:12) (cid:12)
inSection6. InthatSection,wealsolookattheexamplesofRiemannianmanifolds,
the Heisenberg group and the sub-Riemannian Hopf fibration.
2. Euclidean space and Carnot groups
2.1. Sub-Riemannian manifolds. Throughout our paper, any manifold M will
beconnected. Inthissectionwewillrevisitsomebasicfactsaboutsub-Riemannian
manifolds. For more details, we refer to [8].
On a manifold M, a sub-Riemannian structure is a pair (D,g), where D is a
subbundle of the tangent bundle TM and g ∈ Γ(Sym2D∗) is a positive definite,
ASYMPTOTIC EXPANSIONS OF HOLONOMY 3
smooth, symmetric tensor defined only on D. We will refer to the subbundle D
as the horizontal bundle. For any v,w ∈ D , we write hv,wi = g(v,w) and
x g
|v| = hv,vi1/2. We say that a continuous curve γ : [a,b] → M is horizontal if is
g g
absolutelycontinuousandsatisfiesγ˙(t)∈D for almosteveryt∈[a,b]. Forsuch
γ(t)
a curve, we define its length ℓ(γ) by
b
ℓ(γ)= |γ˙(t)| dt.
g
Za
We have the corresponding Carnot-Carath´eodory metric of the sub-Riemannian
structure,
γ :[0,1]→M horizontal
(2.1) d(x,y)=inf ℓ(γ) : .
γ(0)=x, γ(1)=y
(cid:26) (cid:27)
The distance in (2.1) is in general not finite for two arbitrary points x and y.
For a given horizontal bundle D, define D1 =Γ(D) and for every j =1,2,...,
Dj+1 =span X,[X,Y] : X ∈Dj,Y ∈Γ(D) .
For any point x∈M, we define(cid:8)the growth vector n(x)=(n (x(cid:9)))∞ of D at x by
j j=1
n (x)=rank X(x) : X ∈Dj .
j
We say that λ is the step of D at x if (cid:8) (cid:9)
λ=min{k∈N : n (x)=n (x) for any j ≥k}.
j k
We often give the growthvectorat x only by n(x)=(n (x),...,n (x)), indicating
1 λ
that the remaining numbers equal n (x).
λ
We say that D is bracket-generating if for every x there exists some j such that
n (x) = dimM. In other words, D is called bracket-generating if the sections
j
of D and their iterated brackets span the entire tangent bundle. If D is bracket-
generating, then the distance d defined in (2.1) is always finite and its topology
coincides with that of the manifold.
If n(·) is constant in a neighborhood of x, then x is called a regular point of D.
The point x is called singular if it is not regular. The set of singular points of D is
closed with empty interior, cf. [6, Sect. 2.1.2, p. 21]. If all points of M are regular,
D is called equiregular. If D is equiregular, the step λ is constant and there is a
flag of subbundles of TM,
D1 =D ⊆D2 ⊆···⊆Dλ,
such that Dj =Γ(Dj).
Remark 2.1. Asub-Riemannianmetriccanalternativelybedescribedasapossibly
degeneratecometricg∗ onT∗M suchthattheimageD ofthemap♯:T∗M →TM,
β 7→hβ,·ig∗ is a subbundle. These two points of views are related by
hβ,αig∗ =h♯β,♯αig.
Remark 2.2. We could have considered an even more general setting where we
haveaFinslermetricratherthanaRiemannianmetriconD,thusconsideringsub-
Finsler geometry. The techniques and results of this paper are still valid in this
sub-Finsler setting.
4 E.GRONGANDP.PANSU
2.2. Carnot groups. Let n be a nilpotent Lie algebra. Assume that this Lie
algebra is stratified, meaning that we can write n = n ×···×n with bracket
1 λ
relations satisfying
n if 1≤j ≤λ−1,
[n ,n ]= j+1
1 j 0 if j =λ.
(cid:26)
Let N be the connected, simply connected Lie group corresponding to n. Choose
an inner product on n and define a sub-Riemannian structure (D,g) on N by left
1
translation of n and this inner product. The sub-Riemannian manifold (N,D,g)
1
is then called a Carnot group of step λ. Observe that the case λ = 1 is just an
inner product space.
With slight abuse of notation, we will use 0 for the identity element of N, even
though N is abelian only if λ=1. For each s>0, define the dilation δ as the Lie
s
group automorphism δ : N →N uniquely determined by
s
(2.2) (δ ) A∈nj 7→sjA.
s ∗
Dilation δ is homothetic: it multiplies the Carnot-Carath´eodory distance by s.
s
Therefore, as a metric space, (N,D,g) is geodesic, locally compact, it has a tran-
sitive group of isometries and homotheties. Conversely,a theorem of Le Donne [7]
asserts that metric spaces with such properties are Finsler Carnot groups.
Write n =0 and define n as the dimension of n ×···×n for any 1≤j ≤λ.
0 j 1 j
Choose a basis Z ,...,Z of n such that
1 n
(2.3) span{Z ,...,Z }=n .
nj−1+1 nj j
Since N is simply connected and nilpotent, the exponential map is a global diffeo-
morphism. We introduce exponentialcoordinates(z ,...,z ) correspondingto the
1 n
point exp(z Z +···+z Z ). In these coordinates, we have
1 1 n n
δ : (z ,...,z ,...,z )7→(sz ,...,swiz ,...,sλz ),
s 1 i n 1 i n
where w is the number such that Z ∈n . We call w the i-th weight.
i i wi i
We define the radial vector field S by S(x) = dδ (x) . It follows that if
dt 1+t t=0
t7→etS is the flow ofS, then δ =e(logs)S for any s>0. If Z ,...,Z is a basis of
s (cid:12)1 n
n satisfying (2.3) and (z ,...,z ) are the corresponding ex(cid:12)ponential coordinates,
1 n
then
n
∂
S = w z .
i i
∂z
i
i=1
X
Furthermore,since δ is a group homomorphism,it commutes with the left action,
s
meaning that we also have
n
(2.4) S = w z Z ,
i i i
i=1
X
Here,wehaveusedthe samesymbolsforelementsinnandtheir correspondingleft
invariant vector fields.
As all Carnot groups are contractible spaces, we know that any principal bun-
dle π : P → M can be trivialized. Furthermore, we have the following stronger
statement by using our dilations δ . Let π : P → N be a principal bundle with
s
a connection ω. Let hS denote the horizontal lift of S with respect to ω and let
t7→ethS be its flow. Define δω :P →P by formula δω(p)=e(logs)hS(p).
s s
ASYMPTOTIC EXPANSIONS OF HOLONOMY 5
Lemma 2.3. (a) There exists a unique principal bundle isomorphism ζ : P →
N ×P such that
0
(i) ζ(p )=(0,p ) for any p ∈P ,
0 0 0 0
(ii) if ζ(p)=(x,p ), then ζ(δω(p))=(δ (x),p ).
0 s s 0
(b) For every p ∈P , there is a unique gauge σ :N →P such that σ(0)=p and
0 0 0
(σ∗ω)(S)=0. Furthermore, this satisfies (σ∗ω)(v)=0 for every v ∈T N.
0
Proof. (a) For a fixed x, define γ(t)=δ (x). Let γ (t) denote this curve’s hori-
1−t p
zontal lift to p∈P . Define then
x
ζ(p)=(x,γ (1)).
p
By smooth dependence on initial conditions of ODEs, this is a well defined
smoothmap. Furthermore,frompropertiesofhorizontallifts, ζ(p·a)=ζ(p)·a.
Hence, ζ is a principal bundle isomorphism.
(b) Define σ by the relation ζ(σ(x)) = (x,p ). We then have σ S = hS◦σ which
0 ∗
is obviously in the kernel of ω. The last statement follows from the fact that
there exist points x ,...,x ∈ N such that tangent vectors at 0 ∈ N of the
1 n
form d δ (x)| form of basis of T N.
ds s s=0 0
(cid:3)
We callanygaugesuchas inLemma2.3(b) radial, centered at 0. We candefine
radial gauges centered at an arbitrary point x∈N similarly, replacing dilations δ
s
with δsx :=lx◦δs◦lx−1, with lx denoting left translations by x.
2.3. Weighting of functions, forms and vector fields. Foranyq-formαonN
withq ≥0,wedefine δ∗αasthe pull-backwithrespecttoδ . We saythataq-form
s s
αis homogeneous of weight m∈Zifδ∗α=smα. We saythatαis ofweight ≥mif
s
lim s−kδ∗α= 0 for any k < m. We remark that if α is homogeneous of weight
s↓0 s j
m (resp. of weight ≥ m ) for j = 1,2, then α ∧α is homogeneous of weight
j j 1 2
m +m (resp. of weight ≥m +m ) and α +α is of weight ≥min{m ,m }.
1 2 1 2 1 2 1 2
Write d (x) := d(0,x). Since {δ } is a dilation, any function f of weight
0 s s>0
≥ m satisfies f = O(dm) as x → 0. For a general smooth function f, we have a
0
weighted Taylor polynomial at 0, Tay (f;k) = k f(m), with each f(m) being
0 m=0
homogeneous of weight m. Each f(m) is defined by
P
1 dm
(2.5) f(m) = δ∗f .
m!dsm s
(cid:12)s=0
(cid:12)
We can also give a similar Taylor expansion for f(cid:12)orms. The one-forms dz1,...,dzn
areaglobalbasisofT∗N anddz ishomogeneous(cid:12)ofweightw . Itfollowsthatevery
i i
one-form α is of weight ≥ 1 and has a weighted Taylor expansion Tay (α;k) =
0
k α(m) as well. Here
m=1
n n
P α(m) = f(m−wi)dz , whenever α= f dz ,
i i i i
i=1 i=1
X X
with the convention that f(m) = 0 whenever m < 0. Similarly, any q-form α has
weight ≥q and Taylor expansion Tay (α;k) by defining (α∧β)(m) = m α(k)∧
0 k=1
α(m−k).
P
For a vector field X on N, we define δ∗X such that for any function f,
s
(δ∗X)(δ∗f)=δ∗(Xf).
s s s
6 E.GRONGANDP.PANSU
In other words,
δ∗X(x)=(δ ) X(δ (x)).
s 1/s ∗ s
We define homogeneous vector fields and vector fields of weight ≥ m in analogy
with our definition on forms. If X is any vector field of weight ≥ m and f is any
function, then fX is of weight ≥m as well. Furthermore, since
(2.6) [δ∗X ,δ∗X ]=δ∗[X ,X ], X ,X ∈Γ(TM).
s 1 s 2 s 1 2 1 2
it follows that if X is homogeneous of weight m (resp. of weight ≥ m ), then
j j j
[X ,X ]is homogeneousofweightm +m (resp. ofweight≥m +m ). Consider
1 2 1 2 1 2
Z ,...,Z as the basis of n satisfying (2.3). Use the same symbols for the corre-
1 n
spondingleftinvariantvectorfields. Bydefinition,Z isahomogeneousvectorfield
i
of weight −w for 1 ≤ i ≤ n. As a consequence, any vector field with values in D
i
has weight ≥−1 and all vector fields are of weight ≥−λ.
Observe that any homogeneous function, vector field or form is uniquely deter-
mined by their values in a neighborhood U of 0. Hence, we call a function on
sucha neighborhoodU homogeneous if it is the restrictionof a homogeneousfunc-
tion on N. Similarly, for any neighborhood U of 0, there exists a neighborhood
0∈U˜ ⊆U such that δ (U˜)⊆U˜ for s≤1. We say that a function f defined on U
s
is ofweight ≥m if lim skδ∗f| =0 for any k <m. We use analogousdefinitions
s↓0 s U˜
for forms and vector fields.
Remark 2.4. Notice that since the flow of S commutes with δ , the vector field S
s
is homogeneous of weight 0.
2.4. Approximation of holonomy. Let M be a manifold. Define L(x,M) as
the space of all absolutely continuous loops γ : [0,1] → M based at x ∈ M with
finite length relative to some (and hence any) complete Riemannian metric g˜ on
M.
Definition 2.5. Let d be a metric on M. Let ℓ = ℓd : L(x,M) → [0,∞] be the
length relative to d,
k
ℓ(γ)=sup d γ(t ),γ(t ) :k ≥1, 0=t <t <···<t =1 .
j−1 j 0 1 k
Xj=1 (cid:0) (cid:1)
Let f be a real-valued function defined on a subset of R and let F be a function
defined on a subset of L(x,M) with values in a normed vector space (E,k·k). We
then say that F = O(f(ℓ)) as ℓ → 0 if there exist positive constants ε and C such
that
(i) f is defined on (−ε,ε),
(ii) F(γ) is well defined for any γ with ℓ(γ)<ε,
(iii) for any ℓ(γ)<ε, we have
kF(γ)k≤Cf(ℓ(γ)).
Note thatthe definitionof O(f(ℓ)) does notchangeif we replacethe normonE
with an equivalent norm. Hence, if E is finite dimensional, we do not need to
specify a norm in Definition 2.5, since all norms are equivalent.
Let us now consider the special case when (M,D,g) is a sub-Riemannian man-
ifold and d is defined as in (2.1). The following observations are important to
note.
ASYMPTOTIC EXPANSIONS OF HOLONOMY 7
(i) If γ is any absolutely continuous curve such that ℓ(γ) < ∞, then γ must be
horizontal.
(ii) Any absolutely continuous curve γ : [0,1] → M in a Riemannian mani-
fold (M,g˜) can be parametrized by arc length, meaning that we can find
a reparametrizationγ˜ :[0,1]→M satisfying |γ˜˙(t)| =ℓ(γ) whenever γˇ˙(t) ex-
g˜
ists, see e.g. [12, Chapter 5.3]. By a similar argument, we can reparametrize
anyhorizontalcurveinasub-Riemannianmanifold(M,D,g)tohaveconstant
speed.
Let π : P → N be a principal bundle with connection ω with curvature Ω. For
any γ ∈L(x,M), we define the corresponding radial disk disk(γ):[0,1]2 →M by
disk(γ)(s,t)=δx(γ(t)).
s
Theorem 2.6. Let p ∈ P , x ∈ N be an arbitrary point. Let σ be a radial gauge
x
centered at x and satisfying σ(x)=p. Define F :L(x,M)→g by
p
F (γ)= σ∗Ω.
p
Zdisk(γ)
Assume that σ∗Ω is of weight ≥m . Then
Ω
Holω =exp −F +O(ℓ2mΩ) .
p p
The proof is given in Section 2.5.3(cid:0) (cid:1)
2.5. Theory and proof for Theorem 2.6.
2.5.1. Metricspaceswithdilations. Let(M,d)beametricspacewithpossiblysome
points being of infinite distance. We call a collection of maps {δ : M → M}
s s>0
dilations if
δ =id , δ ◦δ =δ .
1 M s1 s2 s1s2
and if d(δ (x),δ (y))=sd(x,y).
s s
Assume that M is a smooth manifold and define L(M) = L(x,M).
x∈M
For any curve γ ∈ L(M), define δ γ as the loop δ γ(t) = δ (γ(t)). Let d be
s s s
`
a metric on M and assume that this metric has dilations {δ } that are also
s s>0
diffeomorphisms.
Lemma 2.7. Let F : L(M) → E be a function with values in a normed vector
space (E,k·k). Let x ∈M be an arbitrary point. Assume that there are constants
0
ε ∈ (0,1] and C > 0 such that for every y ∈ {δ (x )} and γ ∈ L(y,M) with
s 0 s≥1
ℓ(γ)≤ε,
kF(δ γ)k≤Cf(εs), whenever 0<s≤1.
s
Then F =O(f(ℓ)) at x .
0
Proof. For any γ ∈ L(x ) with ℓ(γ) < ε, define γ˜ = δ γ. Then by our
0 ε/ℓ(γ)
assumptions
kF(γ)k=kF(δ γ˜)k≤Cf(ℓ(γ)).
ℓ(γ)/ε
(cid:3)
Notice that F in Lemma 2.7 needs only be defined for any γ ∈ L(y,M), y ∈
{δ (x )} with ℓ(γ) ≤ ε. If x is a fixed point for every δ , then we only need
s 0 s≥1 0 s
to consider y = x in Lemma 2.7. If d only has finite values and δ has a fixed
0 s
point x , then this point is unique since lim δ (x)=x for any x∈M.
0 s↓0 s 0
8 E.GRONGANDP.PANSU
ForaCarnotgroup,the mapsδ definedin(2.2)aredilationsinthe abovesense
s
with fixed point 0. In general, for every x ∈ N, the maps δsx = lx ◦δs ◦lx−1 are
dilations with fixed point x.
2.5.2. Forms on Carnot groups and functionals. Let (N,D,g) be a Carnot group
with dilations δ . We want to consider functions on loop space L(x,N) related to
s
forms.
Lemma 2.8. Let α be a two-form on N with values in some finite dimensional
vector space g. Assume that α is of weight ≥m at x. Consider the map
(2.7) Φα :L(x,N)→g, Φα(γ)= α.
Zdisk(γ)
Then Φα =O(ℓm).
Proof. It is sufficient to consider g = R and x = 0. Choose an arbitrary loop
γ ∈L(0,U)andassumethatitisparametrizedtohaveconstantspeed,i.e. |γ˙(t)| =
g
ℓ(γ) at all points where the derivative exists. Write ι α= n f dz and define
S j=1 j j
n 1 P
(2.8) β = swj rwj−1(δ∗ f )dr dz .
s sr j j
j=1 (cid:18)Z0 (cid:19)
X
We compute
1 1 1 1
α= (δ∗ α)(S,γ˙(t))drdt= β (γ˙(t))dt,
r sr s
Zdisk(δsγ) Z0 Z0 Z0
and so
1
kΦα(γ)k≤ℓ(γ) |βs|g∗(γ(t))dt.
Z0
Since ι α is of weight ≥ m, each f is of weight ≥ m−w . It follows that for
S j j
sufficiently small ε > 0, there are constants C , 1 ≤ j ≤ n such that kf (x)k ≤
j 1 j
C d(x)m−1 whenever d (x)<ε. Combining these bounds, we obtain,
j 0
sm n1
|βs|g∗(x)≤ Cjd0(x)m−1.
m
i=1
X
DefiningC =max C andusingthatd (γ(t))≤ ℓ(γ),givesustheinequality
0 1≤j≤n1 j 0 2
n ℓ(γ)mC n εmC
kΦα(γ)k≤ 1 0sm < 1 0sm,
2mm m
whenever ℓ(γ)<2ε. The result follows from Lemma 2.7. (cid:3)
2.5.3. Proof of Theorem 2.6. Choose an embedding of g into gl(q,C) and put a
Banach algebra norm k·k on gl(q,C). We will show that there exists an ε>0 and
C such that
(i) exp−1Holω (γ) is well defined whenever ℓ(γ)<ε,
σ(0)
(ii) For every 0<s≤1 and γ with ℓ(γ)<ε,
kexp−1Hol(δ γ)+Φσ∗Ω(δ γ)k≤Cs2mΩ.
s s
ASYMPTOTIC EXPANSIONS OF HOLONOMY 9
Let γ be an arbitrary curve with length ℓ(γ) < ε. Without loss of generality,
we may assume that γ is parametrized to have constant speed, so |γ˙(t)| = ℓ(γ)
g
whenever it is defined. Let σ be a radial gauge. Give p ∈ P coordinates (x,a) ∈
N ×G if p=σ(x)·a and define γ =δ γ. In these coordinates, the horizontal lift
s s
(γ (t),a (t)) of γ (t) to σ(0) is given by
s s s
a˙ (t)·a (t)−1 =−σ∗ω(γ˙ (t))=:A (t), a (0)=1,
s s s s s
and a (1)=Holω (γ ).
s σ(0) s
By Lemma A.1, there is an ε <1 such that whenever
0
kAskL∞ = max kAs(t)kL∞ <ε0,
t∈[0,1]
then exp−1a (t) is well defined and
s
1
exp−1a (1)− A (t)dt ≤C kA k2 ,
s s 2 s L∞
(cid:13) Z0 (cid:13)
(cid:13) (cid:13)
for some constant C(cid:13)2. Since (σ∗ω)(S)=0, (cid:13)
(cid:13) (cid:13)
s ∂ s ∂
−A (t)= (σ∗ω)(γ˙ )dr = d(σ∗ω) γ ,γ˙ dr
s r r r
∂r ∂r
Z0 Z0 (cid:18) (cid:19)
s 1
= (σ∗Ω)(S(γ ),γ˙ ) dr.
r r
r
Z0
Since ι σ∗Ω is of weight ≥m , there is a constant C such that
S Ω 3
kAskL∞ ≤C3sm0ℓ(γ)m0.
Hence, if we choose C εm0 <ε ,
3 0
1
exp−1a (1)− A (t)dt = exp−1Holω(δ γ)−F (δ γ) ≤C C2ε2m0s2mΩ.
s s p s p s 2 3
(cid:13) Z0 (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
T(cid:13)he result now follows. (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13)
3. Gauge-free formulation
3.1. Equivariant forms and adjoint bundle. Let π : P → M be a principal
bundle with structure group G. Let g be the Lie algebra of G. We can then
introduceavectorbundleAd(P)→M calledthe adjoint bundle byconsideringthe
product bundle of P and M ×g divided out by the right action
(p,A)·a=(p·a,Ad(a−1)A), p∈P,A∈g,a∈G.
We write [p,A]∈Ad(P) for the equivalence class of (p,A)∈P ×g.
Sections of Ad(P) correspond to G-equivariant functions on P. A function
ϕ : P → g is called G-equivariant if f(p·a) = Ad(a−1)f(p). This function can be
consideredasasectionofAd(P)throughtheidentificationϕ(x)=[p,ϕ(p)],p∈P .
x
Similarly, a g-valued form α on P is called G-equivariant if
α(v ·a,...,v ·a)=Ad(a−1)α(v ,...,v ).
1 q 1 q
Any G-equivariant form that vanishes on kerπ can be considered as a form on M
∗
with values in Ad(P).
Let ω be a connection on π. Let hX denote the horizontal lift of a vector field
X with respect to ω. For any G-equivariant function ϕ, the function hXϕ is still
10 E.GRONGANDP.PANSU
G-equivariant. We denote the corresponding section of Ad(P) by ∇ωϕ, giving us
X
an affine connection ∇ω on Ad(P). Remark that for any gauge (σ,U), we have
(3.1) σ∗∇ωϕ=Xσ∗(ϕ)+[σ∗ω(X),σ∗ϕ].
X
3.2. Explicit formulas of weighted components. Let N be a Carnot group
with stratified Lie algebra n and with exponential coordinates z. We give two
formulas for the m-th homogeneous component of a general function f relative to
0∈N. Let n be the dimension of N and define an index set J by
n
∞
(3.2) J (0)={∅}, J (j)={1,...,n}j, J = J (j).
n n n n
j=0
[
and define functions j,w,|·|:J →N by
n
j j
(3.3) j(µ)=j, w(µ)= w , |µ|= µ , for any µ∈J (j).
µi i n
i=1 i=1
X X
We use the convention that w(∅)=0 and |∅|=0.
By rearrangingthe terms of the usual Taylor expansion, we have
1 ∂f
(3.4) f(m)(z)= (x )z ,
0 µ
j(µ)!∂z
µ
µX∈Jn
w(µ)=m
where z = z ···z and ∂ = ∂ ··· ∂ for any µ ∈ J (j). We use the
µ µ1 µj ∂zµ ∂zµ1 ∂zµj n
conventions z = 1 and ∂ f = f. On the other hand, for any element Z ∈ n,
∅ ∂z∅
we have the Taylor expansion f(exptZ) = k 1tj(Zjf)(0) + O(tk+1). Since
j=0 j!
z ,...,z are exponential coordinates, we obtain
1 n
P
z
f(z ,...z )= µ (Z f)(0)+O(|z|k+1).
1 n µ
j(α)!
µX∈Jn
j(µ)≤k
with Z defined analogously to ∂ . Collecting all the homogeneous components
µ ∂zµ
of this sum with k sufficiently large, we obtain
z
(3.5) f(m)(z)= µ (Z f)(0).
µ
j(µ)!
µX∈Jn
w(µ)=m
3.3. Weighted components of equivariant forms. In order to introduce ap-
proximations that are independent of choice of gauge, we will introduce Taylor
expansions of Ad(P)-valued forms. Let π : P → N be a principal bundle with
connection ω. Define δω : P → P and ζ : P → N ×P as in Section 2.2. For any
s 0
ϕ∈Γ(Ad(P))andp∈P ,introduceTayω(ϕ;k)= k ϕ(m) whereϕ(m) :N →g
0 p m=0
is defined by
P
dm
ϕ(m)(x)= ϕ δω(ζ−1(x,p)) .
dsm s
(cid:12)s=0
We emphasize that Tayωp(ϕ;k) is a funct(cid:0)ion on N, not(cid:1)(cid:12)(cid:12)on P.
(cid:12)
We want to use (3.4) to give an explicit representation of ϕ(m). Write ∂ = ∂
j ∂zj
and let ∇ be the connection on N determined by ∇∂ = 0. Denote the induced
j
