Jonathan J Bernstein

Alison Whale

Justin Brown

Cort Johnson

Eugene Cook

Laurent Calvez

Xianghua Zhang

Steve W Martin

SOLID STATE ELECTROCHEMICAL ALKALI SOURCES FOR COLD ATOM SENSING

Alkali sources are used as getters, to coat thermionic emission electrodes and to provide alkali atoms for precision quantum sensors such as clocks, magnetometers, accelerometers and gyroscopes. Commercial alkali sources are electrically heated to 400 -700C, driving a chemical reaction which produces the alkali. These sources use high current, high power, operate at high temperatures, turn on and off slowly, and evolve impurities. We report here on solid state electrolytic sources for the production of cesium and rubidium at temperatures of 80C to 170C. The advantages of electrolytic sources over thermally driven sources are reduced current and power consumption, faster turn-on and turn-off, the ability to monitor the quantity of generated alkali by current integration, and reduced temperature of operation. Resistivity and activation energy of the solid ionic conductors was measured vs. temperature. Alkali vapor pressure was detected by optical absorption in a custom vacuum system. Strong alkali signals (comparable to commercial sources) were measured when current was passed through the samples. Reversing the polarity of the applied voltage results in ambient alkali absorption. These Solid State Ambipolar Alkali Sources (SSAAS) show potential as high purity, precision alkali sources for cold atom sensors where size, weight and power must be reduced.

INTRODUCTION

Alkali sources have been used for decades to coat thermionic emission electrodes and as vacuum getters. Recently, laserinterrogated atomic sensors such as highly accurate clocks [START_REF] Meekhof | Accuracy Evaluation of a Cesium Fountain Primary Frequency Standard at NIST[END_REF], magnetometers [START_REF] Budker | Optical Magnetometry[END_REF] and inertial sensors [START_REF] Gilowski | High resolution rotation sensor based on cold Rubidium atoms[END_REF] have been developed. The most accurate of these sensors operate in a high vacuum, using alkali atoms trapped by magneto-optical traps (MOT's) prior to performing an optical or microwave based measurement of time, magnetic field or acceleration. For these sensors, clouds of ~ 10 7 atoms are needed. The alkali partial-pressure must be maintained at an acceptable level (~10 -8 to < 10 -9 Torr) over a potentially wide environmental temperature range. The vapor pressure of the pure metals are too high for a solid Cs or Rb source to be used. Maintaining the correct background pressure of alkali is further complicated by the kinetics of absorption, adsorption and desorption from the vacuum chamber walls, particularly with glass which can absorb substantial quantities of alkali [START_REF] Ma | Modification of glass cell walls by rubidium vapor[END_REF]. An ideal source would be low-power, fast, well-controlled and could also act as a sink to regulate alkali vapor pressure.

Commercial alkali sources from SAES [5] and Alvatec [6] use thermally driven chemical reactions to liberate alkali, and may produce contaminant gasses as well. Typically 4-8 amps is used to resistively heat the device to its operating temperature of 400-700C [START_REF] Succi | Atomic absorption evaporation flow rate measurements of alkali metal dispensers[END_REF], producing unwanted light, heat and magnetic fields which can interfere with sensor operation.

It has long been known that alkali can be produced by solid state electrolysis of glass [8,[START_REF] Klemperer | Vacuum preparation of pure cesium by electrolysis of glass[END_REF]. Gong et al. [START_REF] Gong | Electrolytic fabrication of atomic clock cells[END_REF] demonstrated the formation of Cs vapor cells by building a CsBO3 glass piece into a vapor cell wall, and driving Cs into the cell using a molten salt anode at 325C. Later, Cs generation was demonstrated with CsBO3 glass as an independent source (not part of a vapor cell wall) and using a solid metal anode at 170C [START_REF] Bernstein | All solid state ion-conducting cesium source for atomic clocks[END_REF].

Chalcogenide glasses based on S, Se or Te doped with alkalihalides have been extensively developed for infra-red optics, and are known to have higher ionic mobility than most oxide glasses [START_REF] Pradel | Ionic conductivity of chalcogenide glasses[END_REF]. Chalcogenide glass is also less hygroscopic than oxide glasses with high alkali content. Na-''-alumina is a ceramic phase of Al2O3 and Na with high ionic mobility, which can be substituted for other alkali through a high temperature diffusion process [START_REF] Radzilowski | Dielectric Loss of Beta Alumina and of Ion-Exchanged Beta Alumina[END_REF].

We report here for the first time chalcogenide (sulfide and selenide) glasses doped with Cs and Rb and Rb--alumina which produce high purity alkali by electrolysis when small currents (µA-mA) are passed through them.

Figure 1 shows a cartoon of an electrochemical source, with Rb ions drifting from anode to cathode under a bias voltage. We have also demonstrated for the first time absorption of alkali by reversing the bias voltage of a Rb -alumina source, allowing both emission and absorption of alkali with the same device. A silver anode provides charge compensating ions for the migrating Rb ions.

GLASS FABRICATION

Selenide glass (GeSe2 + Ga2Se3) samples containing 30% or 40% CsI, and sulfide glass containing CsCl were produced at Iowa State University (ISU). Selenide glass with 25% or 30% RbI was produced at the University of Rennes. Rb-substituted -alumina was obtained from Ionotec.

The glass samples from ISU were prepared from appropriate amounts of 99.99%+ purity constituent elements (Ge, Ga, Se, CsCl or CsI) then sealed in evacuated silica ampules. The silica glass ampules were placed in a rocking furnace and heated at 1C/min to 850-950C and then held for 12 hours. Different soaking temperatures were used to determine the optimum temperature. It was found that 950C provided a high enough temperature to ensure complete reaction among the elements and yet not so high that vapor pressure would break the sealed glass tube. After soaking for 12 hours, the ampule was immediately transferred to a furnace that was preheated to the glass annealing temperature of 250 to 270C for four hours.

ELECTRODING Anode:

The back anode needs to inject mobile ions into the glass to avoid building up space charge as the alkali ions migrate up towards the cathode. This has proved a problem in previous work. In [START_REF] Gong | Electrolytic fabrication of atomic clock cells[END_REF] a molten NaNO3 salt anode was used as a source of compensating ions. Developers of high temperature alkali ion guns [START_REF] Kim | A New Solid-State Cesium Ion Source[END_REF] using ion-exchanged Cs-mordenite electrolyte found that current flow was limited by anodic processes causing oxidation of the host lattice. Space charge formation led to a rise in voltage necessary to drive current through their device.

It is well known that metals such as Ag and Cu form nonblocking electrodes which inject ions into glass and have high ionic mobility [START_REF] Bychkov | Superionic and ion-conducting chalcogenide glasses: Transport regimes and structural features[END_REF]. For this reason continuous Ag electrodes were chosen for our anodes. The use of solid ion-injecting electrodes prevents development of space charge regions or oxidation of the host lattice anions. Cathode: Cathodes were applied by evaporation of Al through an etched Invar shadow mask, held in place by magnets. The role of the cathode is to supply electrons to neutralize the alkali ions migrating to the surface. To allow the alkali atoms to evaporate without being blocked by the metal fingers, it is desirable to make these fingers narrow and to cover a small fraction of the front surface. Figures 2,3 and 5 shows samples after application of the cathode finger array. These samples are 10-14 mm diameter and 1-3 mm thick. The shadow mask has finger widths resulting in 160 m wide metal lines with 400 um spacing, or about 60% open area.

EXPERIMENTAL METHODS

A vacuum chamber was constructed from a 6 port stainless steel cube for the measurement of alkali generation. A mechanically backed turbo-pump was used for evacuation to the low 10 -7 Torr range after bake-out. This pump could be valved off and an ion pump used to maintain vacuum.

A multi-pin electrical feedthrough to the chamber has connections for a small ceramic heater with built-in thermocouple, high voltage and ground to drive the solid state source, and driving a commercial source from SAES or Alvatec, used for demonstrating alkali absorption. A lock-in amplifier referenced to the 10 kHz modulation provides the change in amplitude due to the absorption. The sensitivity of the measurement is currently limited by residual amplitude modulation of the EOM to Cs vapor densities of ~1×10 8 /cm 3 . The detected alkali density is calibrated using standard alkali vapor cells (e.g. Thorlabs GC25075-CS).

CONDUCTIVITY MEASUREMENTS

The alkali ion conductivity vs. temperature was measured at ISU by small signal AC impedance techniques extrapolated to DC, after applying and baking silver paint electrodes on both sides. Results for 6 samples (5 material compositions) are shown in Figure 8. The '' alumina has the highest conductivity. The Rb and Cs doped selenide glasses have a conductivity which depends strongly on the concentration of alkali [START_REF] Pradel | Ionic conductivity of chalcogenide glasses[END_REF], although activation energy for the various glass samples is similar at ~ 0.77 eV. High conductivity is an advantage as it allows operation at lower voltages and temperatures.

DC current-voltage vs. temperature measurements were made using evaporated Al finger electrodes on the front, and continuous silver electrodes on the back. Due to various effects including current crowding, space charge and electrode interfacial reactions, the ionic conductivity measured this way is typically 100X lower than the small signal AC measurements. A plot of current-voltage vs. temperature for a CsCl-sulfide glass is shown in Figure 9.

ALKALI EMISSION

Samples were placed on the ceramic heater in the vacuum system, held in place by a Be-Cu spring which also connects electrically to the top finger array. The resonant laser was directed over the top surface of the glass sample and its absorption was measured to determine the alkali vapor concentration. Alkali density was calibrated by comparison with the absorption of standard Cs or Rb vapor cells maintained at a known temperature.

CsCl-Sulfide Glass:

A sample of CsCl-GeSe2-Ga2Se3 glass from ISU was metallized with Ag on the back and Al fingers on the front. Current-voltage curves with temperature as a parameter were measured, shown in Figure 9. The current roughly doubles for every 10C increase in temperature.

Cs vapor emission from this sulfide glass as measured by laser absorption is shown in Figure 10, held at 130C with a bias voltage of 1200 V and current of 5 A, cycled on and off twice. Cs losses are primarily to the chamber walls and (much less) to the pumps. Kinetics of alkali absorption and desorption from the chamber walls largely determine the concentrations and rate of change of the measured alkali in vacuum. Curve fitting the decay time of the Cs concentration shown in (e.g. Figure 11) gave a value of 1.25 s regardless of whether the chamber is sealed or actively pumped, because at low pressure the chamber walls dominate the alkali partial pressure kinetics. Due to chamber wall effects, we do not yet know how fast the solid state sources can be turned on and off, but it is likely much faster than the present generation of thermally driven sources [START_REF] Fortagh | Fast loading of a magneto-optical trap from a pulsed thermal source[END_REF].

RbI GeSe2-Ga2Se3 Glass:

A RbI doped glass supplied by the University of Rennes was electroded in a similar manner to the previous samples (Al cathode fingers and continuous Ag back anode). Strong Rb absorption signals were seen when the glass was forward biased at 170C and 1 kV, (Figure 12). Current through the sample increased from 80 µA to 250 µA at constant voltage, which is most likely due to sample self-heating. A constant current drive would provide improved stability compared to constant voltage. '' Alumina Ceramic: Figure 13 shows Rb emission from a Rb '' alumina sample biased at 50 volts and a current of 0.7 mA at 120°C. Calibration of the atom density was performed by comparison with a standard Rb filled ampule. The strength of the Rb signal from this source was comparable to that from a SAES Rb source driven at 5 amps. The higher mobility of Rb in the '' alumina allows this source to operate at reduced temperature and voltage. Alkali Absorption: By reverse biasing the device, a surface region depleted of alkali ions is created. This ion-depletion region is capable of absorbing incident alkali atoms, ionizing them and driving ions into the bulk. To demonstrate absorption, a Rb SAES source was mounted in the chamber and run at 4.88 amps for several minutes to establish a constant background Rb pressure. A Rb doped '' alumina source was then reverse biased at -130 V (-0.21 mA) and 130C, creating a Rb depleted surface while the SAES source was left on. The measured Rb concentration was reduced by 30-50%, as shown in Figure 14. Considering the small size of the SSAAS device (~1.1cm 2 ) compared to the chamber walls (> 280 cm 2 ), this demonstrates remarkably efficient absorption.

The measured Rb concentration is an integral over the path of the laser through the vacuum system. At each point in the path, the vapor pressure is proportional to an average over the entire 4 steradians solid angle of the incident flux of alkali atoms emitted from the surrounding surfaces. The laser passes close to the surface of the SSAAS device, and for that part of the integral, the SSAAS device is almost half the solid angle.

CONCLUSIONS

We have demonstrated solid state electrochemical alkali sources which, when compared to commercial thermally driven chemical sources, require greatly reduced current and power, operate at lower temperatures (70-170C, compared to 400-700C), exhibit faster turn-on and turn-off, and can both source and sink alkali using direct current control. The output of alkali was demonstrated to be comparable in magnitude to commercial SAES alkali sources consuming 5 amps.

Cold atom clouds formed in Magneto-Optical Traps (MOT's) used in high precision atomic clocks, magnetometers and inertial sensors require only about 10 6 to10 8 atoms in a typical measurement. Solid state electrochemical sources will reduce the power, current and temperature required to produce alkali atoms by orders of magnitude. As cold-atom sensors enter the marketplace they will benefit from the high purity, small size and low power offered by these novel solid state atom sources.

Figure 1 .

 1 Figure 1. Cross-sectional diagram showing Rb ion migration through source from anode to cathode.

Figure 2 .

 2 Figure 2. CsI-GeSe2-Ga2Se3 glass source with Al front electrodes.

Figure 3 .

 3 Figure 3. RbI-GeSe2-Ga2Se3 glass source and spring contact.

Figure 4 .

 4 Figure 4. CsCl-sulfide glass without electrodes. Ruler markings are mm.

Figure 5 .

 5 Figure 5. Rb '' alumina source and spring contact.

Figure 6 .

 6 Figure 6. 25 mm X 25 mm heater, CsCl-sulfide glass sample and Be-Cu spring clip.Alkali Measurement: A diode laser tuned to an optical transition of the alkali species of interest is passed through the chamber just above the sample to measure the concentration of each alkali [Cs] or [Rb]. To detect small absorption signals we used a lock-in detection method, the block diagram of which is shown in Figure7. Phase modulation is applied using an electro-optic modulator (EOM) to put 2 GHz sidebands on laser light locked to an atomic transition. The EOM is driven almost to carrier suppression such that most of the optical power is in the sidebands. These sidebands are sufficiently detuned from the Doppler broadened atomic transition (~1 GHz) to pass through the vapor with minimal absorption. An RF switch driven with a 32 kHz square wave gates between light on-resonance and (modulated) light off-resonance.

Figure 7 .

 7 Figure 7. Electro-optical detection block diagram.

Figure 8 .

 8 Figure 8. Log10 conductivity (AC impedance technique) vs. 1000/T(K) for 5 materials.

Figure 9 .

 9 Figure 9. DC current-voltage-temperature plot for CsCl-Ge-Gasulfide glass shown in Figure 4.

Figure 10 .

 10 Figure 10. Optical absorption signal for Cs emission from sulfide glass sample at 130C. Current through sample is 5 A. CsI-GeSe2-Ga2Se3 Glass: A 40% CsI doped selenide glass supplied by ISU was electroded with Al fingers (cathode) and continuous Ag or Cu (anode). Strong Cs generation signals were measured after biasing to 1 kV at 170C and 44-50 µA (Figure 11).

Figure 11 .

 11 Figure 11. Cs generation from CsI-GeSe2-Ga2Se3 glass.

Figure 12 .

 12 Figure 12. Rb generation signal from 30% RbI doped selenide glass.

Figure 13 .

 13 Figure 13. Rb generation from Rb doped '' alumina.

Figure 14 .

 14 Figure 14. SSAAS device reverse biased as absorber. Background level set by SAES source at 4.88 A. SSAAS absorber turned on and off several times is able to absorb 50% of the Rb produced by the SAES source.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge funding from the DARPA Cold Atom MicroSystems (CAMS) Solid State Ambipolar Alkali Source (SSAAS) program (contract # HR0011-14-C-0115). We also acknowledge Christopher Roper for bringing Rb-'' alumina to our attention. Mitchel Hansberry is thanked for mechanical design and Joseph Louis for metallization engineering. The views expressed


Erlend Grong

Pierre Pansu
email: pierre.pansu@math.u-psud.fr

ASYMPTOTIC EXPANSION OF HOLONOMY

Keywords: Subject Classification. 53C29, 41A99, 53C17 asymptotic expansion of holonomy, radial gauge, sub-Riemannian geometry, horizontal holonomy

Given a principal bundle with a connection, we look for an asymptotic expansion of the holonomy of a loop in terms of its length. This length is defined relative to some Riemannian or sub-Riemannian structure. We are able to give an asymptotic formula that is independent of choice of gauge.

Introduction

Let G → P → M be a principal bundle with some connection ω and let Ω be its curvature form. The realization of curvature as the infinitesimal generator of holonomy of ω appears in the now classical Ambrose and Singer theorem, [START_REF] Ambrose | A theorem on holonomy[END_REF]. Further elaborated by Ozeki [START_REF] Ozeki | Infinitesimal holonomy groups of bundle connections[END_REF], this result explains how the holonomy group of ω is infinitesimally generated by Ω and its covariant derivatives 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 theorem shows that for any p ∈ P x , the corresponding holonomy Hol ω p (γ) equals exp(disk(γ) σ * Ω) where σ denotes an arbitrary gauge satisfying σ(x) = p and disk(γ) denotes the integral over an arbitrary filling disk. Using ideas from [START_REF] Pansu | Differential forms and connections adapted to a contact structure, after M. Rumin[END_REF], we are able to show that this result holds approximately for short loops relative to some Riemannian or even sub-Riemannian metric on M . Such expressions have real world applications, see [START_REF] Murray | Nonholonomic motion planning: steering using sinusoids[END_REF][START_REF] Hatton | Optimizing coordinate choice for locomoting systems[END_REF]. A question in these applications if often to look for the best choice of gauge. In the case of M = R n , we obtain the following expansion using the radial gauge.

Theorem 1.1. For every x ∈ R n , define m Ω (x) = 3 + k Ω and all its covariant derivatives of order ≤ k vanish at x, 2 otherwise.

Let σ : R n → P be a gauge in which ω is parallel in radial directions emanating from x. Then for p = σ(x) ∈ P x , there is an ε > 0 and C ≥ 0 such that

(1.1) exp -1 Hol ω p (γ) + disk(γ) σ * Ω ≤ Cℓ(γ) 2mΩ(x) ,
for every loop γ based at x with ℓ(γ) < ε. In the above formula, disk(γ) denotes the integral over a radial filling disk.

The above result also holds true on Carnot groups, which also have dilations and hence the possibility of defining a radial gauge. The details are found in Section 2.

For this reason, the radial gauge is convenient, but may not always be 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 R n with coordinates (z 1 , . . . , z n ), we have the following result.

Theorem 1.2. Write ∂ k = ∂ ∂z k and let x = (x 1 , . . . , x n ) be any point. Let σ be an arbitrary gauge such that σ(x) = p. Use the notation ω σ k = (σ * ω)(∂ k ) and Ω σ ij = (σ * Ω)(∂ i , ∂ j ) and define

F p (γ) = 1 2 n i,j=1 Ω σ ij (x) γ (z i -x i )dz j (1.2) + 1 3 n i,j,k=1 (∂ k Ω σ ij )(x) + [ω σ k (x), Ω σ ij (x)] γ (z i -x i )(z k -x k )dz j ,
for any loop based at x. Then F p does not depend on the choice of gauge. Furthermore, there exist constants ε > 0 and C ≥ 0, such that exp -1 Hol ω p (γ) + F p (γ) ≤ Cℓ(γ) 4 for every γ with ℓ(γ) < ε.

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 choice is made, the above statement remains true with respect to the dilations in this coordinate system. We give the details of this in Section 4.

It was observed in [START_REF] Pansu | Differential forms and connections adapted to a contact structure, after M. Rumin[END_REF] that integrating the curvature over a radial filling disk in the sub-Riemannian Heisenberg group will actually give a better approximation of the holonomy than in the Euclidean case. This result was found using ideas from [START_REF] Falbel | Holonomy of sub-Riemannian manifolds[END_REF][START_REF] Rumin | Formes différentielles sur les variétés de contact[END_REF]. By applying the new theory of horizontal holonomy found in [START_REF] Chitour | Horizontal holonomy and foliated manifolds[END_REF], 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 1 . Assume that the sections of D and their 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 1 generators. Then there is a"modified curvature" Ω such that exp -1 Hol ω p (γ) + disk(γ)

σ * Ω ≤ Cℓ(γ) 2k+2 ,
The result of Theorem 1.2 follows from computations on Euclidean space found in Section 6. In that Section, we also look at the examples of Riemannian manifolds, the Heisenberg group and the sub-Riemannian Hopf fibration.

Euclidean space and Carnot groups

2.1. Sub-Riemannian manifolds. Throughout our paper, any manifold M will be connected. In this section we will revisit some basic facts about sub-Riemannian manifolds. For more details, we refer to [START_REF] Montgomery | A tour of subriemannian geometries, their geodesics and applications[END_REF].

On a manifold M , a sub-Riemannian structure is a pair (D, g), where D is a subbundle of the tangent bundle T M and g ∈ Γ(Sym 2 D * ) is a positive definite, smooth, symmetric tensor defined only on D. We will refer to the subbundle D as the horizontal bundle. For any v, w ∈ D x , we write v, w g = g(v, w) and

|v| g = v, v 1/2
g . We say that a continuous curve γ : [a, b] → M is horizontal if is absolutely continuous and satisfies γ(t) ∈ D γ(t) for almost every t ∈ [a, b]. For such a curve, we define its length ℓ(γ) by

ℓ(γ) = b a | γ(t)| g dt.
We have the corresponding Carnot-Carathéodory metric of the sub-Riemannian structure,

(2.1) d(x, y) = inf ℓ(γ) : γ : [0, 1] → M horizontal γ(0) = x, γ(1) = y .
The distance in (2.1) is in general not finite for two arbitrary points x and y.

For a given horizontal bundle D, define D 1 = Γ(D) and for every j = 1, 2, . . . ,

D j+1 = span X, [X, Y ] : X ∈ D j , Y ∈ Γ(D) .
For any point x ∈ M , we define the growth vector n(x) = (n j (x)) ∞ j=1 of D at x by

n j (x) = rank X(x) : X ∈ D j .
We say that λ is the step of D at x if

λ = min {k ∈ N : n j (x) = n k (x) for any j ≥ k} .
We often give the growth vector at x only by n(x) = (n 1 (x), . . . , n λ (x)), indicating that the remaining numbers equal n λ (x). We say that D is bracket-generating if for every x there exists some j such that n j (x) = dim M . In other words, D is called bracket-generating if the sections of D and their iterated brackets span the entire tangent bundle. If D is bracketgenerating, 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. [START_REF] Jean | Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning[END_REF]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 T M ,

D 1 = D ⊆ D 2 ⊆ • • • ⊆ D λ , such that D j = Γ(D j ).
Remark 2.1. A sub-Riemannian metric can alternatively be described as a possibly degenerate cometric g * on T * M such that the image D of the map ♯ : T * M → T M , β → β, • g * is a subbundle. These two points of views are related by β, α g * = ♯β, ♯α g . Remark 2.2. We could have considered an even more general setting where we have a Finsler metric rather than a Riemannian metric on D, thus considering sub-Finsler geometry. The techniques and results of this paper are still valid in this sub-Finsler setting.

Carnot groups.

Let n be a nilpotent Lie algebra. Assume that this Lie algebra is stratified, meaning that we can write n

= n 1 × • • • × n λ with bracket relations satisfying [n 1 , n j ] = n j+1 if 1 ≤ j ≤ λ -1, 0 if j = λ.
Let N be the connected, simply connected Lie group corresponding to n. Choose an inner product on n 1 and define a sub-Riemannian structure (D, g) on N by left translation of n 1 and this inner product. The sub-Riemannian manifold (N, D, g) 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 s as the Lie group automorphism δ s : N → N uniquely determined by

(2.2) (δ s ) * A ∈ n j → s j A.
Dilation δ s is homothetic: it multiplies the Carnot-Carathéodory distance by s. Therefore, as a metric space, (N, D, g) is geodesic, locally compact, it has a transitive group of isometries and homotheties. Conversely, a theorem of Le Donne [START_REF] Donne | A metric characterization of Carnot groups[END_REF] asserts that metric spaces with such properties are Finsler Carnot groups.

Write n 0 = 0 and define n j as the dimension of

n 1 × • • • × n j for any 1 ≤ j ≤ λ. Choose a basis Z 1 , . . . , Z n of n such that (2.3) span{Z nj-1+1 , . . . , Z nj } = n j .
Since N is simply connected and nilpotent, the exponential map is a global diffeomorphism. We introduce exponential coordinates (z 1 , . . . , z n ) corresponding to the point exp(z

1 Z 1 + • • • + z n Z n ).
In these coordinates, we have δ s : (z 1 , . . . , z i , . . . , z n ) → (sz 1 , . . . , s wi z i , . . . , s λ z n ), where w i is the number such that Z i ∈ n wi . We call w i the i-th weight. We define the radial vector field S by S(x) = d dt δ 1+t (x) t=0 . It follows that if t → e tS is the flow of S, then δ s = e (log s)S for any s > 0. If Z 1 , . . . , Z n is a basis of n satisfying (2.3) and (z 1 , . . . , z n ) are the corresponding exponential coordinates, then

S = n i=1 w i z i ∂ ∂z i .
Furthermore, since δ s is a group homomorphism, it commutes with the left action, meaning that we also have

(2.4) S = n i=1 w i z i Z i ,
Here, we have used the same symbols for elements in n and their corresponding left invariant vector fields. As all Carnot groups are contractible spaces, we know that any principal bundle π : P → M can be trivialized. Furthermore, we have the following stronger statement by using our dilations δ s . Let π : P → N be a principal bundle with a connection ω. Let hS denote the horizontal lift of S with respect to ω and let t → e thS be its flow. Define δ ω s : P → P by formula δ ω s (p) = e (log s)hS (p).

Lemma 2.3. (a)

There exists a unique principal bundle isomorphism ζ : P → N × P 0 such that (i) ζ(p 0 ) = (0, p 0 ) for any p 0 ∈ P 0 , (ii) if ζ(p) = (x, p 0 ), then ζ(δ ω s (p)) = (δ s (x), p 0 ). (b) For every p 0 ∈ P 0 , there is a unique gauge σ : N → P such that σ(0) = p 0 and (σ * ω)(S) = 0. Furthermore, this satisfies (σ * ω)(v) = 0 for every v ∈ T 0 N .

Proof. (a) For a fixed x, define γ(t) = δ 1-t (x). Let γ p (t) denote this curve's horizontal lift to p ∈ P x . Define then

ζ(p) = (x, γ p (1)).
By smooth dependence on initial conditions of ODEs, this is a well defined smooth map. Furthermore, from properties of horizontal lifts,

ζ(p • a) = ζ(p) • a.
Hence, ζ is a principal bundle isomorphism. (b) Define σ by the relation ζ(σ(x)) = (x, p 0 ). We then have σ * S = hS • σ which is obviously in the kernel of ω. The last statement follows from the fact that there exist points x 1 , . . . , x n ∈ N such that tangent vectors at 0 ∈ N of the form d ds δ s (x)| s=0 form of basis of T 0 N .

We call any gauge such as in Lemma 2.3 (b) radial, centered at 0. We can define radial gauges centered at an arbitrary point x ∈ N similarly, replacing dilations δ s with δ x s := l x • δ s • l x -1 , with l x denoting left translations by x.

2.3.

Weighting of functions, forms and vector fields. For any q-form α on N with q ≥ 0, we define δ * s α as the pull-back with respect to δ s . We say that a q-form α is homogeneous of weight m ∈ Z if δ * s α = s m α. We say that α is of weight ≥ m if lim s↓0 s -k δ * s α = 0 for any k < m. We remark that if α j is homogeneous of weight m j (resp. of weight ≥ m j ) for j = 1, 2, then α 1 ∧ α 2 is homogeneous of weight m 1 + m 2 (resp. of weight ≥ m 1 + m 2 ) and α 1 + α 2 is of weight ≥ min{m 1 , m 2 }.

Write d 0 (x) := d(0, x). Since {δ s } s>0 is a dilation, any function f of weight ≥ m satisfies f = O(d m 0 ) as x → 0. For a general smooth function f , we have a weighted Taylor polynomial at 0, Tay 0 (f ; k) = k m=0 f (m) , with each f (m) being homogeneous of weight m. Each f (m) is defined by

(2.5) f (m) = 1 m! d m ds m δ * s f s=0 .
We can also give a similar Taylor expansion for forms. The one-forms dz 1 , . . . , dz n are a global basis of T * N and dz i is homogeneous of weight w i . It follows that every one-form α is of weight ≥ 1 and has a weighted Taylor expansion Tay 0 (α; k) = k m=1 α (m) as well. Here

α (m) = n i=1 f (m-wi) i dz i , whenever α = n i=1 f i dz i ,
with the convention that f (m) = 0 whenever m < 0. Similarly, any q-form α has weight ≥ q and Taylor expansion Tay

0 (α; k) by defining (α ∧ β) (m) = m k=1 α (k) ∧ α (m-k) .
For a vector field X on N , we define δ * s X such that for any function f ,

(δ * s X)(δ * s f ) = δ * s (Xf ).
In other words, δ * s X(x) = (δ 1/s ) * X(δ s (x)). 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 f X is of weight ≥ m as well. Furthermore, since

(2.6) [δ * s X 1 , δ * s X 2 ] = δ * s [X 1 , X 2 ], X 1 , X 2 ∈ Γ(T M ). it follows that if X j is homogeneous of weight m j (resp. of weight ≥ m j ), then [X 1 , X 2 ] is homogeneous of weight m 1 + m 2 (resp. of weight ≥ m 1 + m 2 ). Consider Z 1 , .
. . , Z n as the basis of n satisfying (2.3). Use the same symbols for the corresponding left invariant vector fields. By definition, Z i is a homogeneous vector field of weight -w i for 1 ≤ i ≤ n. As a consequence, any vector field with values in D has weight ≥ -1 and all vector fields are of weight ≥ -λ.

Observe that any homogeneous function, vector field or form is uniquely determined by their values in a neighborhood U of 0. Hence, we call a function on such a neighborhood U homogeneous if it is the restriction of a homogeneous function on N . Similarly, for any neighborhood U of 0, there exists a neighborhood 0

∈ Ũ ⊆ U such that δ s ( Ũ ) ⊆ Ũ for s ≤ 1. We say that a function f defined on U is of weight ≥ m if lim s↓0 s k δ * s f | Ũ = 0 for any k < m.
We use analogous definitions for forms and vector fields.

Remark 2.4. Notice that since the flow of S commutes with δ s , the vector field S is homogeneous of weight 0.

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 = ℓ d : L (x, M ) → [0, ∞] be the length relative to d, ℓ(γ) = sup    k j=1 d γ(t j-1 ), γ(t j ) : k ≥ 1, 0 = t 0 < t 1 < • • • < t k = 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, • ). 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

F (γ) ≤ Cf (ℓ(γ)).
Note that the definition of O(f (ℓ)) does not change if we replace the norm on E 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 manifold and d is defined as in (2.1). The following observations are important to note.

(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 manifold (M, g) can be parametrized by arc length, meaning that we can find a reparametrization γ : [0, 1] → M satisfying | γ(t)| g = ℓ(γ) whenever γ(t) exists, see e.g. [START_REF] Petersen | Riemannian geometry[END_REF]Chapter 5.3]. By a similar argument, we can reparametrize any horizontal curve in a sub-Riemannian manifold (M, D, g) to have constant 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) = δ x s (γ(t)
). Theorem 2.6. Let p ∈ P x , x ∈ N be an arbitrary point. Let σ be a radial gauge centered at x and satisfying σ(x) = p. Define F p : L (x, M ) → g by

F p (γ) = disk(γ) σ * Ω. Assume that σ * Ω is of weight ≥ m Ω . Then Hol ω p = exp -F p + O(ℓ 2mΩ
) . The proof is given in Section 2.5.3 2.5. Theory and proof for Theorem 2.6.

2.5.1.

Metric spaces with dilations. Let (M, d) be a metric space with possibly some points being of infinite distance. We call a collection of maps {δ s :

M → M } s>0 dilations if δ 1 = id M , δ s1 • δ s2 = δ s1s2 . and if d(δ s (x), δ s (y)) = s d(x, y).
Assume that M is a smooth manifold and define L (M ) = x∈M L (x, M ). For any curve γ ∈ L (M ), define δ s γ as the loop δ s γ(t) = δ s (γ(t)). Let d be a metric on M and assume that this metric has dilations {δ s } s>0 that are also diffeomorphisms.

Lemma 2.7. Let F : L (M ) → E be a function with values in a normed vector space (E, • ). Let x 0 ∈ M be an arbitrary point. Assume that there are constants ε ∈ (0, 1] and C > 0 such that for every y ∈ {δ s (x 0 )} s≥1 and γ ∈ L (y, M ) with

ℓ(γ) ≤ ε, F (δ s γ) ≤ Cf (εs), whenever 0 < s ≤ 1. Then F = O(f (ℓ)) at x 0 . Proof. For any γ ∈ L (x 0 ) with ℓ(γ) < ε, define γ = δ ε/ℓ(γ) γ. Then by our assumptions F (γ) = F (δ ℓ(γ)/ε γ) ≤ Cf (ℓ(γ)).
Notice that F in Lemma 2.7 needs only be defined for any γ ∈ L (y, M ), y ∈ {δ s (x 0 )} s≥1 with ℓ(γ) ≤ ε. If x 0 is a fixed point for every δ s , then we only need to consider y = x 0 in Lemma 2.7. If d only has finite values and δ s has a fixed point x 0 , then this point is unique since lim s↓0 δ s (x) = x 0 for any x ∈ M . For a Carnot group, the maps δ s defined in (2.2) are dilations in the above sense with fixed point 0. In general, for every x ∈ N , the maps δ x s = l x • δ s • l x -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 s . We want to consider functions on loop space L (x, N ) related to 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) Φ α : L (x, N ) → g, Φ α (γ) = disk(γ)
α.

Then Φ α = O(ℓ m ).
Proof. It is sufficient to consider g = R and x = 0. Choose an arbitrary loop γ ∈ L(0, U ) and assume that it is parametrized to have constant speed, i.e. | γ(t)| g = ℓ(γ) at all points where the derivative exists. Write ι S α = n j=1 f j dz j and define (2.8)

β s = n j=1 s wj 1 0 r wj -1 (δ * sr f j )dr dz j .
We compute

disk(δsγ) α = 1 0 1 0 1 r (δ * sr α)(S, γ(t)) dr dt = 1 0 β s ( γ(t)) dt,
and so

Φ α (γ) ≤ ℓ(γ) 1 0 |β s | g * (γ(t)) dt.
Since ι S α is of weight ≥ m, each f j is of weight ≥ mw j . It follows that for sufficiently small ε > 0, there are constants C j , 1 ≤ j ≤ n 1 such that f j (x) ≤ C j d(x) m-1 whenever d 0 (x) < ε. Combining these bounds, we obtain,

|β s | g * (x) ≤ s m m n1 i=1 C j d 0 (x) m-1 .
Defining C 0 = max 1≤j≤n1 C j and using that d 0 (γ(t)) ≤ ℓ(γ) 2 , gives us the inequality

Φ α (γ) ≤ n 1 ℓ(γ) m C 0 2 m m s m < n 1 ε m C 0 m s m ,
whenever ℓ(γ) < 2ε. The result follows from Lemma 2.7.

2.5.3. Proof of Theorem 2.6. Choose an embedding of g into gl(q, C) and put a Banach algebra norm • on gl(q, C). We will show that there exists an ε > 0 and C such that (i) exp -1 Hol ω σ(0) (γ) is well defined whenever ℓ(γ) < ε, (ii) For every 0 < s ≤ 1 and γ with ℓ(γ) < ε,

exp -1 Hol(δ s γ) + Φ σ * Ω (δ s γ) ≤ Cs 2mΩ .
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)| g = ℓ(γ) 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 s = δ s γ. In these coordinates, the horizontal lift (γ s (t), a s (t)) of γ s (t) to σ(0) is given by

ȧs (t) • a s (t) -1 = -σ * ω( γs (t)) =: A s (t), a s (0) = 1,
and a s (1) = Hol ω σ(0) (γ s ). By Lemma A.1, there is an ε 0 < 1 such that whenever

A s L ∞ = max t∈[0,1] A s (t) L ∞ < ε 0 , then exp -1 a s (t) is well defined and exp -1 a s (1) - 1 0 A s (t) dt ≤ C 2 A s 2 L ∞ , for some constant C 2 . Since (σ * ω)(S) = 0, -A s (t) = s 0 ∂ ∂r (σ * ω)( γr ) dr = s 0 d(σ * ω) ∂ ∂r γ r , γr dr = s 0 1 r (σ * Ω) (S(γ r ), γr ) dr. Since ι S σ * Ω is of weight ≥ m Ω , there is a constant C 3 such that A s L ∞ ≤ C 3 s m0 ℓ(γ) m0 .
Hence, if we choose

C 3 ε m0 < ε 0 , exp -1 a s (1) - 1 0 A s (t) dt = exp -1 Hol ω p (δ s γ) -F p (δ s γ) ≤ C 2 C 2 3 ε 2m0 s 2mΩ .
The result now follows.

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 introduce a vector bundle Ad(P ) → M called the adjoint bundle by considering the 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 considered as a section of Ad(P ) through the identification ϕ

(x) = [p, ϕ(p)], p ∈ P x . Similarly, a g-valued form α on P is called G-equivariant if α(v 1 • a, . . . , v q • a) = Ad(a -1 )α(v 1 , . . . , v q ).
Any G-equivariant form that vanishes on ker π * 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 G-equivariant. We denote the corresponding section of Ad(P ) by ∇ ω X ϕ, giving us an affine connection ∇ ω on Ad(P ). Remark that for any gauge (σ, U ), we have

(3.1) σ * ∇ ω X ϕ = Xσ * (ϕ) + [σ * ω(X), σ * ϕ].
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 n by

(3.2) J n (0) = {∅}, J n (j) = {1, . . . , n} j , J n = ∞ j=0 J n (j).
and define functions j, w,

|•| : J n → N by (3.3) j(µ) = j, w(µ) = j i=1 w µi , |µ| = j i=1 µ i , for any µ ∈ J n (j).
We use the convention that w(∅) = 0 and |∅| = 0. By rearranging the terms of the usual Taylor expansion, we have

(3.4) f (m) (z) = µ∈Jn w(µ)=m 1 j(µ)! ∂f ∂z µ (x 0 )z µ ,
where

z µ = z µ1 • • • z µj and ∂ ∂zµ = ∂ ∂zµ 1 • • • ∂ ∂zµ j
for any µ ∈ J n (j). We use the conventions z ∅ = 1 and ∂ ∂z ∅ f = f . On the other hand, for any element Z ∈ n, we have the Taylor expansion f

(exp tZ) = k j=0 1 j! t j (Z j f )(0) + O(t k+1
). Since z 1 , . . . , z n are exponential coordinates, we obtain

f (z 1 , . . . z n ) = µ∈Jn j(µ)≤k z µ j(α)! (Z µ f )(0) + O(|z| k+1 ).
with Z µ defined analogously to ∂ ∂zµ . Collecting all the homogeneous components of this sum with k sufficiently large, we obtain

(3.5) f (m) (z) = µ∈Jn w(µ)=m z µ j(µ)! (Z µ f )(0).

3.3.

Weighted components of equivariant forms. In order to introduce approximations 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 δ ω s : P → P and ζ : P → N × P 0 as in Section 2.2. For any ϕ ∈ Γ(Ad(P )) and p ∈ P 0 , introduce Tay ω p (ϕ; k) = k m=0 ϕ (m) where ϕ (m) : N → g is defined by

ϕ (m) (x) = d m ds m ϕ δ ω s (ζ -1 (x, p)) s=0 .
We emphasize that Tay ω p (ϕ; k) is a function on N , not on P . 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 ∇∂ j = 0. Denote the induced connection by ∇ and ∇ ω on Ad(P )-valued forms by the symbol ∇ ω as well. Next, for any j ≥ 1, define ∇ ω,1 X ϕ = ∇ ω X ϕ and iteratively ∇ ω,j+1 X0,X1,...,Xj ϕ = ∇ ω X0 ∇ ω,j X1,...,Xj ϕ -∇ ω,j ∇X 0 X1,...,Xj ϕ -• • • -∇ ω,j X1,...,∇X 0 Xj ϕ.

Finally, for any µ ∈ J n (j), we write

∇ ω µ = ∇ ω,j ∂µ 1 ,...,∂µ j .
Then we have the expansion

(3.6) ϕ (m) (z) = µ∈Jn w(µ)=m z µ j(µ)! (∇ ω µ ϕ)(p).
Alternatively, we can use (3.5). Define a connection ∇ by the rule ∇Z j = 0. Define ∇ ω,j analogously to the definition of ∇ ω,j and furthermore

∇ ω µ = ∇ ω,j Zµ 1 ,...,Zµ j , giving us
ϕ (m) (z) = µ∈Jn w(µ)=m z µ j(µ)! (∇ ω µ ϕ)(p)
as well.

We extend the definition of Tay ω p (α; k) = k m=q α (m) to a general q-form α with values in Ad(P ) as in Section 2.3. Remark that Tay ω p•a (α; k) = Ad(a -1 ) Tay ω p (α; k), and, in particular, the number 0 ≤ m α ≤ ∞ defined by (3.7) m α = sup m ∈ Z : α (m) (0) = 0 , is independent of p ∈ P 0 .

3.4. Gauge-free Taylor expansion. The curvature for Ω is an Ad(P )-valued valued 2-form on M . We will define an expansion of holonomy using the Taylor expansion of Ω that does not depend on gauge.

Theorem 3.1. Let π : P → N be a principal bundle with connection ω. Let p ∈ P 0 be an arbitrary element. Define

F k p (γ) = disk(γ)
Tay ω p (Ω; k).

Then for any k < 2m Ω ,

Hol ω p = exp -F k p + O(ℓ k+1 ) .
Remark 3.2. F k p (γ) is a linear combination of terms involving iterated covariant derivatives of Ω at 0, whose coefficients are moments (integrals of polynomial 1forms along γ). It is gauge free in the sense that ∇ ω µ Ω(p) does not depend on a choice of local gauge. This will be more apparent in the computation of examples in section 6.

Proof of Theorem 3.1. By Theorem 2.6

Hol ω p = exp -Φ σ * Ω + O(ℓ 2m0 ) .

We write

Φ σ * Ω = F k p + Φ α such that α = σ * Ω -Tay p (σ * Ω; k).
Since α is of order ≥ k + 1, the result follows from Lemma 2.8.

4.

Privileged coordinates and general Riemannian and sub-Riemannian spaces 4.1. Privileged coordinates. Let us now consider a sub-Riemannian manifold (M, D, g) with D being bracket-generating, but with no further assumptions. Such manifolds do not have dilations in general. However, we can locally find coordinate systems that play a role similar to the exponential coordinates of Section 2.2. Let x 0 be a regular point of M . Let U be a neighborhood of x 0 where the growth vector n(•) is constant. All points in this neighborhood will be regular and of the same step λ. Hence, by replacing M with U , we may assume that the horizontal bundle D is equiregular with constant growth vector n = (n 1 , n 2 , . . . ). Relative to this growth vector, for any 1 ≤ i ≤ n, define the i-th weight w i as the integer uniquely determined by n wi-1 < i ≤ n wi , with the convention that n 0 = 0. Let z : U ⊆ M → R n be a local coordinate system such that z(U ) = R n and z(x 0 ) = 0 for some x 0 ∈ U .

(a) The coordinate system (z, U ) is called linearly adapted at x 0 if dz i (D j x0 ) = 0 whenever n j < i. (b) A function f defined in some neighborhood of x 0 is said to have order ≥ m at x 0 if for any j = 0, 1, . . . , m -1 and any collection of vector fields X 1 , . . . , X j ∈ Γ(D), we have

(X 1 • • • X j f )(x 0 ) = 0.
Furthermore, f is said to have order m if it is of order ≥ m but not of order ≥ m + 1. (c) The coordinate system (z, U ) is called privileged if z i has order w i at x 0 . Any such coordinate system will be linearly adapted at x 0 .

For any sub-Riemannian manifold (M, D, g) and any x 0 ∈ M , there exist privileged coordinate systems centered at x 0 . For a construction, see [START_REF] Bellaïche | The tangent space in sub-Riemannian geometry[END_REF]Section 4.3]. Note that the definition of a privileged coordinate system at x 0 for a sub-Riemannian manifold (M, D, g) only depends on D, but not on g. The following characterization of the order of functions is found in [2, Proposition 4.10]. Let (z, U ) be a privileged coordinate system and define δ s : U → U such that δ s : (z 1 , z 2 , . . . , z i , . . . , z n ) → (s w1 z 1 , s w2 z 2 , . . . , s wi z i , . . . , s wn z n ).

Relative to this privileged coordinate system, define functions, forms and vector fields of weight m or ≥ m as in Section 2.3. We warn the reader not to confuse notions of weight and order. In particular, for an arbitrary function f, being of order m at x 0 is an intrinsic property of f and D while being homogenous of weight m is something that depends on the chosen privileged coordinate system. These notions are however related in the following way, see [2, Section 5.1].

Lemma 4.3. A function f is of order ≥ m at x 0 if and only if it has weight ≥ m relative to any privileged coordinate system centered at x 0 . In particular, f has order m at x 0 if and only if it has weight ≥ m, but not weight ≥ m + 1. Lemma 4.3 also gives us the following corollary.

Corollary 4.4. Relative to a privileged coordinate system (z, U ) centered at x 0 , if a function, differential form or vector field has weight ≥ m (resp. not weight ≥ m), then the same holds true relative to any other privileged coordinate system.

Proof. The statement is true for functions by Lemma 4.3. In particular, the function z i must have weight ≥ w i but not weight ≥ w i + 1 with respect to any privileged coordinate system. The same must then hold true for the form dz i . This gives us the desired result for forms since, near x 0 , all such forms can be written as a sum of elements f dz i1 ∧ • • • ∧ dz iq for some function f . Finally, any vector field X is of weight ≥ m if and only if for any one-form α of weight ≥ m 2 we have that α(X) is of weight ≥ m + m 2 , which shows independence of privileged coordinate system for vector fields as well.

Relative to a privileged coordinate system, we can define a radial gauge as in Lemma 2.3. Theorem 4.5. Let π : P → M be a principal bundle over a sub-Riemannian manifold (M, D, g), equipped with a connection ω. Let x 0 ∈ M be a regular point of (M, D, g). Let (z, U ) be any privileged coordinate system around x 0 . For any p ∈ P x0 , let σ : U → P | U be the corresponding radial gauge satisfying σ(x 0 ) = p. Define

F p (γ) = disk(γ) σ * Ω.
Assume that σ * Ω has weight ≥ m Ω at x 0 . Then

Hol ω p = exp -F p + O(ℓ 2mΩ
) . We will give the proof in Section 4.3.

4.2.

Tangent cone of a sub-Riemannian space at a regular point. Let x 0 be a given regular point (M, D, g). By choosing an appropriate neighborhood U of x 0 , we can make the following assumptions.

(i) All points in U are regular. Hence, D|U is equiregular with growth vector (n 1 , . . . , n λ ). (ii) D| U has a global orthonormal basis X 1 , . . . , X n1 . (iii) There is a privileged coordinate system z : U → R n centered at x 0 .

For now on, we will replace M with U and consider the sub-Riemannian manifold (U, D, g). We will use the privileged coordinates (z 1 , . . . , z n ) to identify U with R n . In particular, we identify x 0 with 0.

Define Z 1 , . . . , Z n1 as the -1 homogeneous component of our orthonormal basis, meaning that

Z j (z) := lim s↓0 s(δ * s X j )(z), z ∈ U.
Let n = Lie{Z 1 , . . . , Z n1 } be the Lie algebra generated by these vector fields. Since these vector fields are homogeneous of weight -1 and since there are no nonzero homogeneous vector fields of weight < -λ, the Lie algebra n must be nilpotent of step at most λ by (2.6). Actually, from [START_REF] Bellaïche | The tangent space in sub-Riemannian geometry[END_REF], we can conclude the following.

Theorem 4.6. (a) The vector fields Z 1 , . . . , Z n1 are linearly independent at every point. (b) Define a sub-Riemannian structure on ( D, ĝ) such that Z 1 , . . . , Z n1 is an orthonormal basis. Then D and D have the same growth vector. Furthermore, (U, D, ĝ) is a sub-Riemannian Carnot group of step λ, n is its Lie algebra and (z 1 , . . . , z n ) are exponential coordinates with respect to some basis of n. (c) If d is the Carnot-Carathéodory metric of ( D, ĝ), then (U, d) is the metric tangent cone of (M, d) at x 0 in the sense of Gromov-Hausdorff convergence. (d) For any ρ > 0 there exists an ε > 0 such that

(1 -ρ) d(0, x) ≤ d(0, x) ≤ (1 + ρ) d(0, x).
whenever d(0, x) < ε.

4.3.

Dilation of extended space and proof of Theorem 4.5. We continue with the notation and assumptions of Section 4.2. Introduce a space Ũ = R >0 × U with coordinates (s, z) = (s, z 1 , . . . z n ). Define a projection ν : Ũ → U by (s, z) → z. Consider any vector field X on U with local flow e tX as a vector field on Ũ by

(Xf )(s, z) = d dt f (s, e tX (z)) t=0 .
Introduce a map δs : Ũ → Ũ by δs (s 0 , z 0 ) = (ss 0 , δ s (z)).

Then ν • δs = δ s • ν, and so δ * s ν * α = ν * δ * s α for any form on U . Relative to the global orthonormal basis X 1 , . . . , X n1 of D, define vector fields X1 , . . . , Xn1 by Xj (s, z) := s -1 ( δ * 1/s X j )(s, z). Define a sub-Riemannian structure ( D, g) on Ũ such that the vector fields X1 , . . . Xn1 form an orthonormal basis. The subbundle D is then not bracket-generating, but we will still consider the Carnot-Carathéodory metric d of ( D, g). By definition, any D-horizontal curve is on the form γ(t) = (s, δ s γ(t)) where γ(t) is a D-horizontal curve in U and s > 0. Furthermore, if γ(t) has finite length, then

ℓ(γ) := 1 0 γ(t) g dt = sℓ(γ). As a consequence, d (s 1 , z 1 ), (s 2 , z 2 ) = ∞ if s 1 = s 2 , s 1 d(z 1 , z 2 ) if s 1 = s 2 .
The manifold U is isometric to the submanifold {(1, z) ∈ Ũ : z ∈ U } and we will identify these two manifolds from now on. Lemma 4.7. Let α be a two-form on U that is of weight ≥ m taking values in some finite dimensional vector space g. Consider the map

Φ α : L (0, U ) → g, γ → disk(γ)
α.

Then Φ α = O(ℓ m ).
Proof. It is sufficient to consider the case g = R. We consider L (0, U ) as a subset L ( Ũ ). Define Φ :

L ( Ũ ) → R by Φ(γ) = Φ α (ν(γ)). Consider any loop γ(t) = (s 0 , δ s0 γ(t))
where γ is a horizontal loop in U of length ℓ(γ) and s 0 ≥ 1. Assume that ℓ(γ) ≤ ε so that s 0 ℓ(γ) ≤ ε. Furthermore, choose ε sufficiently small such that

1 2 d(0, x) ≤ d(0, x) ≤ 3 2 d(0, x) whenever d(0, x) < 2ε. Write ι S α = n i=1 f j dz i and define β s as β s = n j=1 s wj 1 0 r wj -1 (δ * sr f j )dr dz j .
By possibly choosing ε smaller, we can assume that

|f j |(x) ≤ C j d(0, x) m-1 ≤ ( 3 2 ) m-1 C j d(0, x) m-1 , whenever x < ε. Since d(0, δ s0 γ(t)) < ε, we have (i) Φ( δs γ) = Φ α (δ s0s γ) = 1 0 β s0s ( γ(t))dt, (ii) |Φ α (δ s0s γ)| ≤ ℓ(γ) 1 0 |β s0s | g * (γ(t)) dt, (iii) If C 0 = max 1≤j≤n1 {C j }, then for 0 < s ≤ 1, |β s0s | g * (x) ≤ 3 m-1 n 1 s m 0 s m 2 m-1 C 0 1 0 r m-1 dr d(0, x) m-1 ≤ 3 m-1 n 1 s m 0 s m m C 0 d(0, x) m-1 .
As a consequence, Φ α (δ s0s γ) ≤ 3n1C0ε m 2 m-1 m s m . Hence, Φ(δ s γ) ≤ 3n1C0ε m 2 m-1 m s m and since Ũ a metric space with dilations, Φ = O(ℓ m ) by Lemma 2.7 . As a consequence, we finally have Φ = O(ℓ m ).

Proof of Theorem 4.5. Define U and Ũ as above. Consider the pull-back bundle ν * π : ν * P → Ũ and define pr P : ν * P → P by (s, z, p) → p for (s, z) ∈ Ũ and p ∈ P . This gives us a pull-back connection ω = pr * P ω on ν * π, which has curvature pr * P Ω. We may now modify the proof of Theorem 2.6 as in the proof of Lemma 4.7 for the result.

Remark 4.8. The gauge free expression of F k p found in Section 3 can be used in Theorem 4.5 as well.

Remark 4.9. Assume that (z, U ) is a coordinate system satisfying all conditions of a privileged coordinate system centered at x 0 except that z(U ) = V ⊆ R n where V is some neighborhood of 0. Define δ s : R n → R n by δ s (z 1 , . . . , z n ) = (sz 1 , . . . , s wi z i , . . . , s λ z n ). Consider the norm, z = n i=1 |z i | 1/wi , and define

V r = {z ∈ R n : z ≤ 1}.
Let ε > 0 be such that V ε ⊆ V and define U ∨ = z -1 (V ε ). We can then for s ≤ 1, define dilations δ ∨ s : U ∨ → U ∨ . Finally, taking a diffeomorphism φ : V ε → R n such that φ|V ε/2 = id, it follows that we can define a proper privileged coordinate system z = φ • z|U ∨ whose dilations equals δ ∨ s for points close to x 0 . Hence, if we have a coordinate system (z, U ) which in privileged in all ways apart from being surjective, we can still define Taylor expansions as in Section 3 with these coordinates and Theorem 4.5 is still valid. (i

) For 0 ≤ j ≤ r -1, χ(D j+1 ) ⊆ 2 D j . (ii) For 1 ≤ j ≤ r -1 and any α ∈ Γ(Ann(D j )), v → α(v) + dα(χ(v)) vanishes on D j+1 .
Relative to such a selector χ, the corresponding contraction operator

ι χ : 2 T * M → T * M is defined by (ι χ η)(v) = η(χ(v)), v ∈ T x M , η ∈ 2 T * x M , x ∈ M .
It was shown in [START_REF] Chitour | Horizontal holonomy and foliated manifolds[END_REF] that any equiregular subbundle D has at least one selector. Actually, from the construction in [3, Lemma 2.7], we know that D has a selector such that χ(D j ) ⊆ j-1 i=1 D i ⊕ D j-i for any j = 2, . . . , λ. From the same reference, we also have the following relation between holonomy along D-horizontal curves and selectors. Theorem 5.2. Let χ be a selector of D ⊆ T M . Let ω be an arbitrary connection on a principal bundle P → M . Then there exists a unique connection ω ∈ [ω] D such that its curvature Ω satisfies

ι χ Ω = 0.
Furthermore, for any x ∈ M and p ∈ P x , this connection satisfies Hol ω p (γ) :

γ is D-horizontal γ(0) = γ(1) = x = Hol ω p (γ) : γ ∈ L (x) .
Finally, if Ω is the curvature of ω, then

ω = ω + ι χ λ-1 j=1 λ -1 j (L ω ι χ ) j-1 Ω, (5.1) Ω = (id +L ω ι χ ) λ-1 Ω, (5.2)
where λ is the step of D and L ω : Γ(T * M ⊗ Ad(P )) → Γ( 2 T * M ⊗ Ad(P )) is given by

(5.3) L ω α = d ∇ ω α + 1 2 [α, α].
We see that a choice of selector gives us a unique way of choosing a connection in a D-equivalence class, which is "optimal", in the sense that the holonomy of any loop γ is also the holonomy of some D-horizontal curve. Recall that the exterior covariant derivative d ∇ ω of Ad(P )-valued j-forms is determined by the following rules, (i) For j = 0, any d ∇ ω ϕ = ∇ ω ϕ for any ϕ ∈ Γ(Ad(P )), (ii) If α is an Ad(P )-valued j-form and β a real-valued form

d ∇ ω (α ∧ β) = d ∇ ω α ∧ β + (-1) j α ∧ dβ.
If the increase in the growth vector is maximal, then we can improve our expansion by changing our connection to a D-equivalent one. Given n 1 = rank(D), the maximal increase in the growth vector (n 1 , . . . , n λ ) is given by the free nilpotent Lie algebra. The free nilpotent Lie algebra f[n 1 ; k] of step k with n 1 generators is defined as the free Lie algebra on n 1 generators divided by the ideal generated by brackets of length greater than k. where F j p (γ) = disk(γ) Tay ω p ( Ω; j), ω is defined as in (5.1) and Tay ω p ( Ω; j) is defined relative to some privileged coordinate system centered at

If we define ν[n 1 ; k] = rank(f[n 1 ; k]), then n k ≤ ν[n 1 ; λ]. Remark in particular that n 2 ≤ ν[n 1 ; 2] = 1 2 n 1 (n 1 -1). Corollary 5.
x ∈ M . (b) Assume that n k = ν[n 1 , k]. Let χ be a selector satisfying χ(D j ) ⊆ j-1 i=1 D i ⊕ D j-i , j =
Ω vanish on ⊕ k-1 i=1 D i ⊕ D k-i , it must have weight ≥ k + 1.
For the second statement, observe that n j = ν[n 1 , j] for any j = 1, . . . , k. Let hX denote the horizontal lift of a vector field on M to P with respect to ω. Write E = ker ω, E j = {w ∈ E : π * w ∈ D j } and E j = Γ(E j ). Note that rank E j = n j . If we then define F 1 = E 1 and F j+1 = F j + [F 1 , F j ], we claim that E j = F j for any i = 1, . . . , k.

We will first show that E j = F j for any j = 1, . . . , λ by induction. Assume that E j = F j for some j. For any vector field Z ∈ Γ(D j+1 ) with χ(Z) = l i=1 X i ∧ Y i and for any α, we have α

(Z - l i=1 [X i , Y i ]) = α(Z) + dα(χ(Z)) = 0. This means that Z = Z 0 + l i=1 [X i , Y i ],
for some Z 0 ∈ Γ(D j ). By our assumptions X i ∧ Y i also take values in

j i=1 D i ⊕ D j+1-i . Since we know that Ω(χ(Z)) = 0 = l i=1 Ω(X i , Y i ), we have that hZ = hZ 0 + l i=1 [hX i , hY i ] ∈ F j+1 .
This completes the induction step.

As a consequence of the above statement,

E j x ⊆ F j x = {Y (x) : Y ∈ F j } ⊆ T x M, for any x ∈ M . However, since E 1 has rank n 1 , we must have rank F j x ≤ ν[n 1
, k] = rank E j for j = 1, . . . , k and so E j = F j . This means that for any vector field

X ∈ Γ(D i ) and Y ∈ Γ(D k-i ), we have [hX, hY ] ∈ Γ(E k ),
which is equivalent to the curvature vanishing on k i=1 D i ⊕ D k-i .

6. Examples 6.1. Euclidean space. Consider the special case when N is an inner product space. Let x 0 ∈ N be an arbitrary point. We may choose coordinates (z 1 , . . . , z n ) so that we can identify N with R n with the standard inner product and x 0 = 0. Write ∂ k = ∂ ∂z k . Let π : P → R n be a principal bundle with connection ω and curvature Ω. Since any Ω has m Ω ≥ 2, we have Hol ω p = exp -F 3 p + O(ℓ 4 ) , with F k p as in Theorem 3.1 and p ∈ P 0

. Write Ω = 1 2 n i,j=1 Ω ij dz i ∧ dz j with Ω ij ∈ Γ(Ad(P )), Ω ij = -Ω ji . Then F 3 p (γ) = 1 2 n i,j=1 disk γ Ω ij (p) + n k=1 z k ∇ ω ∂ k Ω ij (p) dz i ∧ dz j = 1 2 n i,j=1 Ω ij (p) γ z i dz j + 1 3 n i,j,k=1 ∇ ω ∂ k Ω ij (p) γ z i z k dz j .
If (σ, U ) is any gauge around 0 such that σ(0) = p, and we write (σ

* ω)(∂ k ) = ω σ k and σ * Ω ij = Ω σ ij , then F 3 p (γ) = 1 2 n i,j=1 Ω σ ij (0) γ z i dz j (6.1) + 1 3 n i,j,k=1 (∂ k Ω σ ij )(0) + [ω σ k (0), Ω σ ij (0)] γ z i z k dz j .
6.2. Riemannian manifolds. For any n-dimensional Riemannian manifold (M, g) and point x 0 ∈ M , choose a normal coordinate system (z, U ) centered at x 0 . This coordinate system satisfies all assumptions of privileged coordinate system except surjectivity. It follows from Remark 4.9 that for any p ∈ P x0 , Hol ω p = exp -F 3 p + O(ℓ 4 ) with F 3 p written as in (6.1) in the coordinates (z, U ). 6.3. The Heisenberg group. Consider the 3-dimensional Heisenberg group N . This is the Carnot group of step 2 with Lie algebra

n = n 1 ⊕ n 2 , n 1 = span{Z 1 , Z 2 }, n 2 = span{Z 3 }.
where n 2 is the center and [Z 1 , Z 2 ] = Z 3 . Denote the left invariant vector fields corresponding to Z 1 , Z 2 , Z 3 by the same symbol. If we give the point exp(xZ 1 + yZ 2 + zZ 3 ) coordinates (x, y, z) then

Z 1 = ∂ ∂x - 1 2 y ∂ ∂z , Z 2 = ∂ ∂y + 1 2 x ∂ ∂z , Z 3 = ∂ ∂z .
The corresponding coframe is given by dx, dy and θ = dz + 1 2 (ydxxdy). We want to compute our approximation of holonomy on N . Using left translation, it is sufficient to consider loops based at 0. We will use Theorem 5.2. The unique selector χ of D is given by

χ : Z 1 → 0, Z 2 → 0, Z 3 → Z 1 ∧ Z 2 .
The corresponding contraction operator is

ι χ : dx ∧ dy → θ, dx ∧ θ → 0, dy ∧ θ → 0. Write Ω = Ω 1 dx ∧ θ + Ω 2 dy ∧ θ + Ω 3 dx ∧ dy. Define Ω = Ω + d ∇ ω Ω 3 ∧ θ = Ω 1 + ∇ ω Z1 Ω 3 dx ∧ θ + Ω 1 + ∇ ω Z2 Ω 3 dy ∧ θ =: Ψ 1 dx ∧ θ + Ψ 2 dy ∧ θ,
and ω = ω + β, where β = ι χ Ω = Ω 3 θ. Notice that for any ϕ ∈ Γ(Ad(P )) and vector field X, we have ∇ ω X ϕ = ∇ ω X ϕ + [β(X), ϕ].

We now want to compute Tay ω p ( Ω; 5) = Tay ω p (Ψ 1 ; 2)dx ∧ θ + Tay ω p (Ψ 2 ; 2)dy ∧ θ. With ∇ ω α defined as in Section 3.3 and for j = 1, 2, Tay p (Ψ j , 2) = Ψ j (p) + z 1 ∇ ω 1 Ψ j (p) + z 2 ∇ ω 2 Ψ j (p) + z 3 ∇ ω 3 Ψ j (p) given by basis X 1 , X 2 and X 3 and relations

+ 1 2 z 2 1 ∇ ω 1,1 Ψ j (p) + z 2 2 ∇ ω 2,2 Ψ j (p) + z 1 z 2 2 ∇ ω 1,2 Ψ j (p) + z 2 2 ∇ ω 2,1 Ψ j (p) = Ψ j (p) + z 1 ∇ ω 1 Ψ j (p) + z 2 ∇ ω 2 Ψ j (p) + z 3 ∇ ω 3 Ψ j (p) + [Ω 3 , Ψ j ] + 1 2 z 2 1 ∇ ω 1,1 Ψ j (p) + z 2 2 ∇ ω 2,2 Ψ j (p) + z 1 z 2 2 ∇ ω 1,2 Ψ j (p) + z 2 2 ∇ ω 2,
[X 1 , X 2 ] = X 3 , [X 2 , X 3 ] = X 1 , [X 3 , X 1 ] = X 2 .
This can be considered as the Lie algebra of the group of matrices SU(2) of the form

a = w 1 w 2 -w2 w1 , w 1 , w 2 ∈ C, |w 1 | 2 + |w 2 | 2 = 1.
In these coordinates and with w j = u j + iv j , j = 1, 2, we can identify X 1 , X 2 and X 3 with the left invariant vector fields

2X 1 = -u 2 ∂ u1 -v 2 ∂ v1 + u 1 ∂ u2 + v 1 ∂ v2 , 2X 2 = -v 2 ∂ u1 + u 2 ∂ v1 -v 1 ∂ u2 + u 1 ∂ v2 , 2X 3 = -v 1 ∂ u1 + u 1 ∂ v1 + v 2 ∂ u2 -u 2 ∂ v2 .
Introduce a sub-Riemannian structure (D, g) on SU(2) by defining X 1 and X 2 to be an orthonormal basis.

On the neighborhood U = {a ∈ SU(2) : u 1 > 0}, define coordinates z : U → R 3 , a → (2u 2 , 2v 2 , 2v 1 ) . This is a privileged coordinate system centered at the identity (u 1 , v 1 , u 2 , w 2 ) = (1, 0, 0, 0) apart from the requirement of surjectivity. Furthermore, the dilations δ s : U → U are well defined for 0 < s ≤ 1.

The vector fields X 1 , X 2 , X 3 in the coordinate system z are given by

X 1 = z 1 Y + 1 -|z| 2 ∂ z1 + z 3 ∂ z2 -z 2 ∂ z3 , X 2 = z 2 Y -z 3 ∂ z1 + 1 -|z| 2 ∂ z2 + z 1 ∂ z3 , X 3 = z 3 Y + z 2 ∂ z1 -z 1 ∂ z2 + 1 -|z| 2 ∂ z3 . Y = 1 1 -|z| 2 (z 1 ∂ z1 + z 2 ∂ z2 + z 3 ∂ z3 )
Define α 1 , α 2 , α 3 as the coframe dual to X 1 , X 2 , X 3 . In the coordinates (z 1 , z 2 , z 3 ) this is given by

α 1 = 1 -|z| 2 dz 1 + z 3 dz 2 -z 2 dz 3 ,

Definition 4 . 1 .

 41 Let D be a bracket-generating, equiregular subbundle with corresponding flag D ⊆ D 2 ⊆ • • • ⊆ D λ = T M and growth vector n = (n 1 , n 2 , . . . , n λ ).

Proposition 4 . 2 .

 42 Let f be a function defined around x 0 . Let d be the Carnot-Carathéodory metric of (M, D, g). The following are equivalent. (a) f has order ≥ m at x 0 . (b) f (x) = O(d(x 0 , x) m ) for x → x 0 .

5 .

 5 Increase of order in the sub-Riemannian case If d is the Carnot-Carathéodory metric of a sub-Riemannian structure (D, g), any curve of finite length is D-horizontal. Hence, when considering the expansion of holonomy in terms of length, the following concept is natural to consider. If P → M is a principal bundle and D is a subbundle of T M then two connections ω and ω are called D-equivalent if ω(v) = ω(v) for any v ∈ T P with π * v ∈ D. Let [ω] D denote the equivalence class of ω. By definition, Hol ω p (γ) = Hol ω p (γ) for any D-horizontal loop γ based at x ∈ M , p ∈ P x and ω ∈ [ω] D . Assume that D is an equiregular subbundle with corresponding flag D = D 1 ⊆ D 2 ⊆ • • • ⊆ D λ . Denote by D 0 the zero section of T M . Let Ann(D j ) denote the subbundle of T * M of covectors vanishing on D j . Definition 5.1. A two-vector-valued one-form χ ∈ Γ(T * M ⊗ 2 T M ) is called a selector of D if it satisfies the following two properties.

3 .

 3 Let (M, D, g) be a sub-Riemannian manifold with D equiregular of step λ, with canonical flag D ⊆ D 2 ⊆ • • • ⊆ D λ = T M and with growth vector (n 1 , . . . , n λ ). Let ω be some connection on π : P → M . (a) Assume that Ω defined as in (5.2) vanishes on k-1 i=1 D i ⊕ D k-i . Then Hol ω p = exp -F 2k+1 p + O(ℓ 2k+2 ) ,

  2, . . . , k, and define Ω as (5.2) relative to χ. Then Ω vanish onk-1 i=1 D i ⊕ D k-i .Proof. The first statement follows by Theorem 5.2 and from the fact that if

  1 Ψ j (p) which equals Tay p (Ψ j , 2) = Ω j (p) + z 1 ∇ ω 1 Ω j (p) + z 2 ∇ ω 2 Ω j (p) + z 3 ∇ ω 3 Ω j (p) + [Ω 3 , Ω j ] Ω 3 (p) + z 1 ∇ ω 1,j Ω 3 (p) + z 2 ∇ ω 2,j Ω 3 (p) + z 3 ∇ ω 3,j Ω 3 (p) + [Ω 3 , ∇ ω j Ω 3 ] Ω 3 (p) + ∇ ω 2,1,j Ω 3 (p) .In conclusion, we obtain Hol ω p = exp -F 5 p + O(ℓ 6 ) , where F 5 p = disk(γ) Tay ω p ( Ω; 5) = γ q 1 (xθ -2zdx) + γ q 2 (yθ -2zdy), where Ω 3 (p) + ∇ ω 2,1,j Ω 3 (p) . 6.4. Sub-Riemannian Hopf fibration. Consider the Lie algebra su(2)

	+ ∇ ω j + 1 2 z 2 1 ∇ ω 1,1,j Ω 3 (p) + z 2 2 ∇ ω 2,2,j Ω 3 (p) + 1,2,j q j (z) = z 1 z 2 2 ∇ ω 1 3 Ω j (p) + z 1 4 ∇ ω 1 Ω j (p) + z 2 4 ∇ ω 2 Ω j (p) + z 3 5 ∇ ω 3 Ω j (p) + [Ω 3 , Ω j ]
		+	1 10	z 2 1 ∇ ω 1,1 Ω j (p) + z 2 2 ∇ ω 2,2 Ω j (p) +	z 1 z 2 10	∇ ω 1,2 Ω j (p) + z 2 2 ∇ ω 2,1 Ω j (p)
		+	1 3	∇ ω j Ω 3 (p) +	z 1 4	∇ ω 1,j Ω 3 (p) +	z 2 4	∇ ω 2,j Ω 3 (p) +	z 3 5	∇ ω 3,j Ω 3 (p) + [Ω 3 , ∇ ω j Ω 3 ]
		+	1 10	z 2 1 ∇ ω 1,1,j Ω 3 (p) + z 2 2 ∇ ω 2,2,j Ω 3 (p) +	z 1 z 2 10	∇ ω 1,2,j
	+	1 2	z 2 1 ∇ ω 1,1 Ω j (p) + z 2 2 ∇ ω 2,2 Ω j (p) +	z 1 z 2 2	∇ ω 1,2 Ω j (p) + z 2 2 ∇ ω 2,1 Ω j (p)

This paper is supported by the Fonds National de la Recherche Luxembourg (AFR 4736116 and OPEN Project GEOMREV) and by the Research Council of Norway (project number 249980/F20). P.P. acknowledges support from ANR, project SRGI, ANR-15-CE40-0018.

2010 Mathematics

Let P → SU(2) be a principal bundle with connection ω. We study the holonomy of horizontal loops based at 1. Let Ω be the curvature of ω, given in the coframe by

This allows us to us to use the connection ω = ω + Ω 3 α 1 and curvature

For j = 1, 2, consider the vector fields Z j = lim s↓0 sδ * s X j and define

Hence, Hol ω p = exp -F 5 p + O(ℓ 6 ) where

Tay ω p (Ξ 2 ; 2)dz 2 ∧ θ

Tay ω p (Ξ 3 ; 3)dz 1 ∧ dz 2 .

Appendix A. Approximation of ODEs in Lie groups

A.1. Lie groups and right logarithmic derivatives. Let G be a connected Lie group with Lie algebra g. For any continuous curve [0, t 1 ] → g, t → A(t), consider the initial value problem on G,

Assume that for any t ∈ [0, t 1 ] the curve Q(t) = exp -1 a(t) in g is well defined. Note that

Consider g as a subalgebra of gl(q, C) for some q > 0 and let • be a Banach algebra norm on gl(q, C). Write A L ∞ [0,t] = sup s∈[0,t] A(s) .

Lemma A.1. Define constants B 1 and B 2 such that on the disk {z ∈ C : |z| ≤ π},

Define ad = sup{ ad(A 1 )A 2 : A 1 = A 2 = 1}. Then there exists ε > 0, such that for any A : [0, T ] → g with

Proof. We want to estimate Q(t) using Picard iteration. For any t > 0, write

. Define B = max{B 1 ad /π, B 2 ad } and assume that tM t < 1/B. Then it follows that Q(t) = lim n→∞ A n (0) is the solution of (A.1) and

The result follows from choosing ε ≤ 1/B.