Rigid Two-Step Nilpotent Lie Groups relative to Multicontact Structures

Rigid Two-Step Nilpotent Lie Groups Relative to Multicontact Structures

IRENEVENTURI

ABSTRACT- LetfXi;Ykgdenote a fixed basis of the Lie algebranof aconnected and simply connected nilpotent Lie group N of step two. Under a technical as- sumption onfXi;Ykg, we prove that the Lie algebraMof vector fieldsV onN that satisfy [V;Xi]ˆliXiis finite dimensional, a property that we refer to as rigidity. Our proof allows the explicit count ofdimM.

1. Introduction.

In a broad sense, the study of rigidity of stratified niplotent Lie groups has attractedmuchattentioninrecentyears,startingfromtheseminalworkbyN.

Tanaka ([19] and references therein) and his school (notably Yamaguchi [20], Yatsui[21]and Morimoto[15])andthen moving intomoreexplicitlygeometric settings in the important papers by KoraÂnyi and Reimann [11] and Pansu [17], where various generalised contact-type structures were considered. At a more general and formal level, the notion of multicontact structure was first formulated by Cowling, De Mari, KoraÂnyi and Reimann [4, 5] in the context of the ``boundaries'' G=P of symmetric spaces (G semisimple, P aparabolic subgroup ofG). In particular, the local nature ofG=Pas a nilpotent stratified Lie group stemming from the Iwasawa decomposition ofGˆKANand from the Bruhat decomposition, together with the root structure of the Lie algebra ofN, leads to a notion of multicontact vector field that can easily be formulated for all stratified nilpotent Lie algebras once an additional sub-stratification of theground layer is singled out. Inthe caseofanIwasawa nilpotentLie algebra nin its standard form as the sum of the (restricted) positive root spaces, the stratification is provided by the height function on the root system, whereas

(*) Indirizzo dell'A.: DIMA, UniversitaÁ di Genova, via Dodecaneso, 35, 16146 Genova.

E-mail: venturi@dima.unige.it

the sub-stratification refers to the simple roots. As explained by KoraÂnyi [10], the problem has a clear differential geometric counterpart: if one selects smooth distributionsD_ion asmooth manifoldMsuch that the vector fields spanning the distributions satisfy a HoÈrmander type condition, then a mul- ticontact mapping ofMis adiffeomorphism whose differential preserves each distributionD_i. Passing to the infinitesimal level yields the notion of multi- contact vector field, that is, a vector field V onM whose flow consists of multicontact mappings. IfVis such avector field, the requirement amounts to the condition that [V;X]2 D_ifor everyX2 D_i. Thus one may formulate a general problem as follows: given a stratified nilpotent Lie algebra nˆn₁‡ ‡n_s (parametrizing the tangent bundle of the connected and simply connected nilpotent Lie groupN with Lie(N)ˆn) together with a multicontactstructure, that is, a decomposition ofn1intothesumofsubspaces n₁ˆn⁽¹⁾₁ ‡ ‡n^(t)₁, is the Lie algebraMof vector fields onNwhich pre- serve (under bracket) each direct summandn⁽ⁱ⁾₁ finite dimensional? If yes, we say thatnis rigid and in this case it is natural to ask if we can describe the structure ofM. The remarkable answer in [20] and [4, 5] is that in most cases M 'Lie(G) ifMˆG=Pand if eachn⁽₁^j)is determined bythe structure ofPin terms of simple roots. These questions are formulated by Tanaka and his school in terms of prolongation Lie algebras and rather deep results are presented in [19, 20, 21]. Other very interesting results along these lines have been obtained by Reimann onH-type groups [18], by Warhurst on filiform groups [22], jet spaces [23] and free algebras [24], and by Ottazzi on Hes- senberg manifolds [16].

In this paper we address the following problem: ifnhas step two and we are given certain canonical basesfX_igofn₁ andfY_kgofn₂, isnrigid re- lative to the ``finest'' multicontact structure, that is, when eachn⁽ⁱ⁾₁ is just the span of the basis vectorXi? Our main theorem answers this question positively. Its proof is based on a method suggested by Eastwood [14] and combines prolongation notions together with genuine differential geo- metric constructs and provides a method for bounding explicitly the di- mension ofMonce the structure ofnis known.

2. The multicontact system.

2.1 ±Choice of bases.

Let n denote a two-step stratified nilpotent Lie algebra. This means thatnˆn1‡n2as a direct sum of vector spaces, wheren1is asubspace of

dimension n which generates the m-dimensional subspace n2 under bracket, that is

[n1;n1]ˆn2:

We select basesfX₁;. . .;X_ngandfY₁;. . .;Y_mgofn₁andn₂, respectively, and we assume that (¹)

(a)z(n)ˆn₂, that is,z(n)\n₁ˆ f0g, wherez(n) is the center ofn;

(b) for everyk2mthere existi;j2nsuch thatY_kˆ[X_i;X_j]:

(c) for everyi2nthere existj2nandk2msuch that [Xi;Xj]ˆg^k_ijYk, whereg^k_ijis anonzero real constant.

Clearly, assumption (a) does not harm generality, for otherwisenwould split as a direct sum of an abelian Lie algebra and a nilpotent Lie algebra that does satisfy (a). Further, we can always select bases such that (b) is satisfied. Indeed, once we choose a basisfX₁;. . .;X_ngofn₁, the brackets [Xi;Xj] generaten2 by assumption, so we can select a basis ofn2 among them and denote its elements byfY1;. . .;Ymg. As for (c), we donot know if all two-step nilpotent Lie algebras satisfy it or not.

Observe that, by assumption (b), which is not restrictive, for any fixed k2m there exist distinct indices i;j2n such that [X_i;X_j]ˆY_k. Hence there cannot be more directions inn2than distinct pairs (i;j)2nn. This implies thatmn(n 1)=2.

2.1.1 - Examples.

We list below two classes of two-step nilpotent Lie algebras that satisfy our hypotheses. We remark that in all these cases rigidity in our sense is essentially known (see [18] when the center has dimension at least three, and [4, 16]). Our result can thus be seen also as a unified approach.

H-type algebras.For the notation and basic properties of the Heisen- berg type algebras, introduced first in [6], we refer to [3]. Following standard notation, an H-type algebra is a direct sumv‡z, wherevˆn1

and zˆn2, that is orthogonal with respect to a positive definite inner producth;iand for which a mapJ:z!End (v) is defined that satisfies hJ_YX;X⁰i ˆ hY;[X;X⁰]i for all Y 2z and all X;X⁰2v. Evidently, J_Y is skew-symmetric with respect to h;i. The defining condition for such an algebra to be called of Heisenberg type is thatJ_Y² ˆ jYj²I(soJY is non-

(¹) Ifnis apositive integer, we writenˆ f1;. . .;ng.

singular for everyY). Among the several properties ofJ, we have [X;J_YX]ˆ jXj²Y

for everyX2vand everyY2z.

Now, for afixedY6ˆ0 withjYj ˆ1, the endomorphismJ_Yofvis skew- symmetric and non-singular. Hencevis even dimensional and we can se- lect abasisX₁;. . .;X_2prelative to which

JY ˆ 0 Ip

Ip 0

: We may obviously normalizejX_ij ˆ1 and obtain

Y ˆ[X_i;J_YX_i]ˆ [X_i;X_i‡p] if ip [Xi;Xi p] if i>p;

so that settingYˆY1, and completing to a basis ofzby choosing vectors Y2;. . .;Ymamong the various brackets [X_i;X_j], yields a basis that satisfies (b). Indeed, for everyithere exists eitherjˆi‡p(ifip) orjˆi p(if i>p) a ndkˆ1 such that [X_i;X_j]ˆ Y₁whereg¹_ijˆ 1.

Hessenberg algebras.This is a class of nilpotent algebras that includes algebras of arbitrary step. They were introduced first in [12], [13] and then studied in [16] from the point of view of rigidity, where it is shown that most of them are rigid, in the multicontact sense that we are interested in.

Given asimple Lie algebragwith Iwasawa decompositiongˆk‡a‡n and with root systemSone selects asubsetR S^‡of positive roots which satisfy the following Hessenberg-type condition: ifb2 Randa2S^‡ sa- tisfy b a2S^‡, then b a2 R. The most important property of Hes- senberg type sets is clarified by the next proposition, which is essentially contained in all of the aforementioned references.

PROPOSITION2.1. A subsetR S^‡is of Hessenberg type if and only if the vectorspacen_R⁰ˆ P

g2R⁰g_gis a (nilpotent) ideal ofnwhereR⁰ˆS^‡n R.

PROOF. Suppose R S^‡ is of Hessenberg type and let g2 R⁰ and b2 R. Ifb‡g2S^‡, then necessarilyb‡g2 R⁰, for otherwiseb‡g2 R and this implies thatg2 R, contrary to the assumption. Thereforen_R⁰is an ideal inn, obviously nilpotent.

Conversely, ifn_R⁰ is an ideal inn, then it is nilpotent and the following property holds: if g2 R⁰ and b2 R are such that b‡g2S^‡, then ne- cessarilyb‡g2 R⁰. This means that ifb‡g2 Rfor some positive rootsg andb, then bothbandgmust belong toRfor otherwise at least one of them is inR⁰and, by assumption, so is their sum. p

We stratifynby height, that is,niis the sum of all root spaces of height equal toi, where the height refers to the number of simple roots that occur in the (unique) decomposition of apositive root as asum of simple roots.

Evidently, ifR is of Hessenberg type, thenn_R⁰ is a stratified ideal and therefore the quotient n_Rˆn=n_R⁰ is again a stratified nilpotent Lie al- gebra, which we shall refer to as a Hessenberg algebra. Among the more interesting sets Rof Hessenberg type are those that include all positive roots of height smaller than or equal to a positive integerp. If we denote by R_psuch aset and letn^(p)ˆn_Rp, the sum of the root spaces with roots inR_p then it is clear thatn^(p)has step exactly equal top.

For example, if gˆsl(n;R), thenR ˆ R₂ may be identified with the set of indices that correspond to the first two superdiagonals of annn upper triangular matrix, i.e.f(i;j):1j i2g. Thusn_R⁰consists of the upper triangular matrices whose nonzero entries can only occur in the upper-right corner whose entriesa_ijsatisfyj i>2, and thereforen⁽²⁾is the quotient of the (strictly) upper triangularnnmatricesnmodulon_R⁰. It is natural to identify its first stratumn⁽²⁾₁ with the first superdiagonal and its second stratumn⁽²⁾₂ with the second superdiagonal.

Obviously,fX_iˆE_i;i‡1:1in 1g andfY_kˆE_k;k‡2:1kn 2g satisfy both (b) and (c), because

[X_i;X_j]ˆ Y_i if jˆi‡1 0 if j6ˆi‡1

[X_i;Y_k]ˆ0 mod n_R⁰ [Y_`;Y_k]ˆ0 mod n_R⁰: (1)

We remark that the dimension ofn⁽²⁾₁ is exactlyn 1, which can thus be any positive integer.

2.2 ±Multicontact vectorfields.

A vector fieldVon the connected simply connected stratified groupN with Lie algebra denoted by n is said to be a multicontact vector field relative to the basisfX₁;. . .;X_ng, if it satisfies

[V;X_i]ˆl_iX_i (2)

for alli2nand for some smooth functionsl1;. . .;lnonN. The vector field V is not assumed to be left invariant, so we write

VˆX

t2n

f_tX_t‡X

k2m

g^kY_k (3)

where theftand theg^kare smooth functions onN. Here and in the following we use subscripts to indicate components along the first stratum and su- perscripts to indicate components along the second stratum.

In preparation for what follows we recall some basic notation used for computing Lie brackets. A bracket of elements inn₁is of course inn₂. For i;j2nwe write

[Xi;Xj]ˆX

k2m

a^k_ijYk; (4)

where thea^k_ij are real constants. Evidently,a^k_ijˆ a^k_ji anda^k_ijˆ0 for all k2m if and only if [Xi;Xj]ˆ0. Moreover, if [Xi;Xj]ˆg^k_ijYk as in as- sumption (c), we havea^k_ijˆg^k_ijanda<sup>`</sup><sub>ij</sub>ˆ0 for all`6ˆk. Finally, dijstands for the Kronecker symbol.

LEMMA2.2. The vectorfield V 2X(N)as in(3)is a multicontact vector field on N if and only if forall i;t2n and all k2m

Xi(g^k)ˆX

t2n

a^k_tift

(5)

Xi(ft)ˆ ditlt: (6)

PROOF. Fori2nwe have l_iX_iˆX

t2n

f_tX_t‡X

k2m

g^kY_k;X_i

ˆX

t2n

f_t[X_t;X_i] X_i(f_t)X_t X

k2m

X_i(g^k)Y_k

ˆX

t2n

X

k2m

a^k_tiftY_k X_i(ft)Xt

X

k2m

X_i(g^k)Y_k:

Equating the coefficients ofXiandYkwe get (5) and (6). p

2.3 ±Rigidity.

In this section we prove the following

THEOREM2.3. Underthe assumption (c), the Lie algebraMof mul- ticontact vectorfields on a two-step stratified nilpotent connected and simply connected Lie group is finite dimensional.

2.3.1 - Prolongation.

The procedure of prolongation has a long history. It may be traced back to the work of Elie Cartan and was developed by many people in several ways (see e.g. [1, 7, 8, 9, 15, 19, 20] and references therein). A formulation of prolongation in the language of exterior differential systems may be found for example in [2], while from the algebraic point of view the notion was formalized by Tanaka [19]. As explained in [1] the main idea is that ``a generic overdetermined system of partial differential equations may be rewritten as a first order ``closed system'' in which all first partial deri- vatives of the dependent variables are expressed in terms of the variables themselves. To do this, we must introduce extra dependent variables for unknown derivatives until all derivatives of the original and extra vari- ables can be determined as consequences of the original equation''. In this sense, a system may or may not ``close up''. We prove in this section that our system does indeed ``close up''. This can also be expressed by saying that our system admits a finite prolongation.

The linear nature of the equations that express the various ``jet'' variables as functions of the others allows us to prove rigidity as follows.

As usual we first define a vector bundle whose fibers have as many dimensions as the number of variables that have been introduced. Then we introduce a connection on this bundle in such a way that finding a solution to the original system is equivalent to finding a covariant con- stant section of the bundle. Since the space of parallel sections cannot have dimension exceeding the dimension of the bundle, we infer the finite dimensionality of the space of solutions. Furthermore, equality is achieved in the flat case and we show that our connection is indeed flat, thereby obtaining precise dimension counts. Our approach is inspired by that of Eastwood [14].

In this section we focus on the heart of our proof, showing that taking sufficiently many derivatives yields a closed system. Since the computa- tions are slightly involved, it is perhaps helpful to keep in mind the the following picture:

This means that we consider the real valued functions onNlabeledft,g^k andlt, witht2nandk2m. We then take derivatives with respect toXi

andY_kof the functionsft,g^kandltand produce new variablesa^k_t andb^k_t. All Y-derivatives ofa^k_t and allX-derivatives ofb^k_t are zero for all choices of the indices if the functionsf_t,g^k are the coefficients of a multicontact vector field as in (3) and if thel_t satisfy (2). Hence the system associated to our multicontact problem ``closes up'', that is, the prolongation is finite. The extra variables are defined by

a^k_t ˆYk(ft); t2n; k2m;

b^k_t ˆYk(lt); t2n; k2m:

For computational reasons, we also introduce the variables c<sup>`k</sup><sub>t</sub> ˆY<sub>`</sub>(a^k_t); t2n; `;k2m;

although in fact they all vanish in the multicontact case (see Lemma 2.16).

From now on we shall use the symbolg^k_ijto mean nonzero real constants that satisfy assumption (c) without appealing explicitly to it. Evidently, we assume (5) and (6).

The following lemmadescribes Y_k(g<sup>`</sup>) for all possible pairs (k; `)2 2mm.

LEMMA2.4. Forany k; `2m and all i;j2n forwhich[Xi;Xj]ˆg^k_ijYk

we have

Y_k(g<sup>`</sup>)ˆ d<sub>k`</sub>(l_i‡l_j):

(7)

PROOF. Fixk2mand takei;j2nsuch that [X_i;X_j]ˆg^k_ijY_k. Then for any`2m, by (5) and (6)

g^k_ijY_k(g^{`</sup>)ˆ[X<sub>i</sub>;X<sub>j</sub>](g<sup>`})

ˆX_i(X_j(g^{`</sup>)) X<sub>j</sub>(X<sub>i</sub>(g<sup>`}))

ˆX_iX

t2n

a^`_tjf_t

X_jX

r2n

a^`_rif_r

ˆ X

t2n

a^`_tjditlt‡X

r2n

a^`_ridjrlr

ˆ a^{`</sup><sub>ij</sub>l<sub>i</sub>‡a<sup>`}_jil_j:

Now, [X_i;X_j]ˆg^k_ijY_k implies that a<sup>`</sup><sub>ij</sub>ˆ0 if `6ˆk, whereas if `ˆk, then

a^k_ijˆ a^k_jiˆg^k_ij6ˆ0. p

COROLLARY2.5. Letk2mandi; j2nbe such that[Xi; Xj]ˆg^k_ijYk. (i) If there exist p;q2n such thata^k_pq6ˆ0, thenl_i‡l_jˆl_p‡l_q; (ii) if there exists p2n with p6ˆj such thata^k_pi6ˆ0, thenl_jˆl_p. PROOF. By (4) and (7)

[Xp;Xq](g^k)ˆX

`2m

a<sup>`</sup><sub>pq</sub>Y`(g^k)

ˆa^k_pqY_k(g^k)‡ X

`2m;`6ˆk

a<sup>`</sup><sub>pq</sub>Y`(g^k)

ˆ a^k_pq(li‡lj):

But by (5) and (6)

[Xp;Xq](g^k)ˆXp

X

t2n

a^k_tqft

Xq

X

r2n

a^k_rpfr

ˆ X

t2n

a^k_tqd_ptl_t‡X

r2n

a^k_rpd_qrl_r

ˆ a^k_pq(lp‡lq):

Hence, ifa^k_pq6ˆ0, thenl_i‡l_j ˆlp‡lq, as claimed in (i). Now, ifa^k_pi6ˆ0 for somep6ˆj, then by part (i)l_i‡l_jˆl_i‡l_p, whencel_jˆl_p. p Next we analyse the functions a^k_i, showing that the only indices (i;k)2nmfor whicha^k_i is possibly nonzero are those for which we can findj2nsuch that [Xi;Xj]ˆg^k_ijYk.

LEMMA 2.6. Let i2n and k2m. If there is no j2n such that [X_i;X_j]ˆg^k_ijY_k, then a^k_i ˆ0.

PROOF. Fix k2m. By assumption (b), there existm;n2n such that [X_m;X_n]ˆY_k. Then for anyi2n

a^k_i ˆY_k(f_i)ˆ[X_m;X_n](f_i)ˆX_m(X_n(f_i)) X_n(X_m(f_i)):

Thus, if there does not existj2nsuch that [X_i;X_j]ˆg^k_ijY_k, theni62 fm;ng and so by (6)X_m(f_i)ˆ0ˆX_n(f_i), whencea^k_i ˆ0. p

LEMMA2.7. Let i;j;t2n with i6ˆj. Then X

`2m

a^{`</sup><sub>ij</sub>a<sup>`}_t ˆ

0 if t62 fi;jg X_j(l_i) if tˆi

X_i(l_j) if tˆj:

8>

<

>: (8)

Thus, if k2m and i;j2n are such that[Xi;Xj]ˆg^k_ijYk, then a^k_t ˆ0for t62 fi;jg, Xi(lj)ˆ g^k_ija^k_j and Xj(li)ˆg^k_ija^k_i.

PROOF. By definition ofa^`_t, for anyt2n X

`2m

a^{`</sup><sub>ij</sub>a<sup>`}_t ˆX

`2m

a<sup>`</sup><sub>ij</sub>Y<sub>`</sub>(f_t)ˆ[X_i;X_j](f_t)ˆX_i(X_j(f_t)) X_j(X_i(f_t));

hence (8) follows from (6). If [X_i;X_j]ˆg^k_ijY_k, then the sum on the left hand side reduces tog^k_ija^k_t and the conclusions follows from (6). p

LEMMA2.8. Let i2n.

(i) Forany k; `2m with k6ˆ`,P

t2na^`_tia^k_t ˆ0.

(ii) If j2n and k2m are such that[X_i;X_j]ˆg^k_ijY_k, then X_i(l_i)ˆ2g^k_ija^k_j

(9)

PROOF. Sincenis two-step, for anyi2nand anyk; `2mwe have 0ˆ[Xi;Yk](g<sup>`</sup>)ˆXi(Yk(g^{`</sup>)) Yk(Xi(g<sup>`})):

Thus, if `6ˆk, then by (7) Y<sub>k</sub>(g<sup>`</sup>)ˆ0 and so using (5) and the defi- nition of a^k_i

0ˆY_k(X_i(g^`))ˆY_kX

t2n

a^`_tif_t

ˆX

t2n

a^`_tiY_k(f_t)ˆX

t2n

a^`_tia^k_t:

Next take`ˆkand suppose thati;j2nare such that [X_i;X_j]ˆg^k_ijY_k. Then by (7) we haveY_k(g^k)ˆ (l_i‡l_j) and so by (5) and (8) we have

0ˆ[X_i;Y_k](g^k)

ˆX_i(Y_k(g^k)) Y_k(X_i(g^k))

ˆ Xi(li‡lj) Yk

X

t2n

a^k_tift

ˆ X_i(l_i)‡g^k_ija^k_j X

t2n

a^k_tia^k_t

ˆ X_i(l_i)‡g^k_ija^k_j g^k_jia^k_j X

t2nt6ˆj

a^k_tia^k_t

ˆ X_i(l_i)‡2g^k_ija^k_j X

t2nt6ˆj

a^k_tia^k_t;

where we have used thata^k_jiˆ g^k_ij. Finally, by Lemma 2.7 we havea^k_t ˆ0 for allt62 fi;jgand whentˆiwe havea^k_iiˆ0, so that all summands in the

last sum vanish. p

Observe that (8) and (9) describe X_j(l_i) for all possible pairs (i;j)2nn.

LEMMA2.9. If k2m and i;j2n are such that[X_i;X_j]ˆg^k_ijY_kand there exists p2n, p6ˆj, such thata^k_pi 6ˆ0, then a^k_i ˆ0.

PROOF. By (ii) in Corollary 2.5,ljˆlp, so thatXj(lj)ˆXj(lp) and thus, by Lemma 2.8 and Lemma 2.7, respectively, we obtain

2g^k_ija^k_i ˆX

`2m

a^{`</sup><sub>pj</sub>a<sup>`}_pˆX

`2m`6ˆk

a^{`</sup><sub>pj</sub>a<sup>`}_p

sincea^k_pˆ0 whenp62 fi;jg. Now, eithera^{`</sup><sub>pj</sub> ˆ0 for all`2mwith`6ˆk, in which case a<sup>k</sup><sub>i</sub> ˆ0, or else there exists `2m with `6ˆk such that a<sup>`}_pj6ˆ0. We show now that for all such indices`we havea<sup>`</sup>_pˆ0, so that the sum on the right hand side vanishes, whence the result. Fix any such

`; according to Lemma 2.6 we may suppose that there exists q2nsuch

that [Xp;Xq]ˆg<sup>`</sup><sub>pq</sub>Y`, for otherwisea^`_pˆ0. Two cases arise: either q6ˆj orqˆj. In the first case, by (ii) in Corollary 2.5 necessarilyl_jˆl_q and hence l_jˆl_p, which gives X_q(l_q)ˆX_q(l_p). By (9) and (8), respectively, we obtain

2g^{`</sup><sub>qp</sub>a<sup>`}_pˆX

s2m

a^s_pqa^s_pˆg^{`</sup><sub>pq</sub>a<sup>`}_p;

the latter equality expressing [X_p;X_q]ˆg^{`</sup><sub>pq</sub>Y<sub>`</sub>. Therefore a^{`</sup><sub>p</sub>ˆ0 as de- sired. In the second case, [X<sub>p</sub>;X<sub>j</sub>]ˆg<sup>`}_pjY<sub>`</sub> and [X<sub>i</sub>;X<sub>j</sub>]ˆg<sup>k</sup><sub>ij</sub>Y<sub>k</sub>; by Lemma 2.8, we obtain that Xj(lj)ˆ2g<sup>`}_jpa^{`</sup><sub>p</sub> and Xj(lj)ˆ 2g<sup>k</sup><sub>ij</sub>a<sup>k</sup><sub>i</sub>, respectively, so 2g<sup>`}_jpa^`_pˆ 2g^k_ija^k_i. Arguing as above we finally get

g^{`</sup><sub>pj</sub>a<sup>`}_pˆX

`2m`6ˆk

a^{`</sup><sub>pj</sub>a<sup>`}_pˆ 2g^k_ija^k_i ˆ2g^{`</sup><sub>jp</sub>a<sup>`}_p:

Thereforea^`_pˆ0 as desired. p

LEMMA2.10. Let k; `2m with`6ˆk and let i;j;p2n with p6ˆj be such that[X_i;X_j]ˆg^k_ijY_kand[X_i;Xp]ˆg<sup>`</sup><sub>ip</sub>Y`, then a^s_iˆ0forany s2m.

PROOF. By Lemma2.7 and item (i) of Lemma2.8 0ˆX

s2m

a^k_sja^{`</sup><sub>s</sub>ˆg<sup>k</sup><sub>ij</sub>a<sup>`}_i‡a^k_pja^`_p 0ˆX

s2m

a^{`</sup><sub>sp</sub>a<sup>k</sup><sub>s</sub> ˆg<sup>`}_ipa^k_i ‡a^`_jpa^k_j:

We show now thata<sup>`</sup><sub>i</sub>ˆ0 whenever`6ˆk. This is obvious from the first equation ifa^k_pjˆ0. Suppose thena^k_pj6ˆ0. By item (ii) of Corollary 2.5, ne- cessarilyl_iˆl_p, which givesX_i(l_i)ˆX_i(l_p). By (9) and (8), respectively, we obtain

2g^{`</sup><sub>ip</sub>a<sup>`}_pˆg^{`</sup><sub>pi</sub>a<sup>`}_p:

Thereforea^{`</sup><sub>p</sub>ˆ0 and from the first equationa<sup>`}_iˆ0, as desired.

Next we show thata^k_i ˆ0. This is obvious from the second equation if a^{`</sup><sub>jp</sub>ˆ0. Suppose then a<sup>`}_jp6ˆ0. By item (ii) of Corollary 2.5, necessarily liˆlj, which givesXi(li)ˆXi(lj). By (9) and (8), respectively, we obtain

2g^k_ija^k_j ˆ g^k_ija^k_j:

Thereforea^k_j ˆ0 and from the second equationa^k_i ˆ0, as desired.

Finally, takes2mfor every s62 f`;kg. If there exists noq2nsuch that [X_i;X_q]ˆg^s_iqY_s, thena^s_iˆ0 by Lemma2.6. Conversely, if [X_i;X_q]ˆ

ˆg^s_iqY_sfor someq2nthen the previous argument withqin place ofpand

sin place of`givesa^s_i ˆ0. p

Observe that by Lemma 2.6, Lemma 2.7 and Lemma 2.10 the structure of themnmatrixaˆ(a^k_i) is subject to many constraints. Indeed, if we fix both the row indexkand the column indexiand we cannot find another column indexjfor which [X_i;X_j]ˆg^k_ijY_k, then Lemma2.6 says thata^k_i ˆ0.

Therefore for any fixed row k, the only column indicesi for whicha^k_i is possibly nonzero are those for which [X_i;X_j]ˆg^k_ijY_kfor somej. Take such a pair (i;j). By Lemma2.7 any othera^k_t vanishes. Hence for any fixed rowk there is at most a pair of column indicesiandjfor which the corresponding entries do not vanish, i.e., every row contains at most two nonzero entries.

But more is true: if for two different rows, say thek^thand the`^th, we find a coinciding column index, sayi, then by Lemma2.10 we have a^s_iˆ0 for everys. Therefore, every column contains at most one nonzero entry. It follows that, after reordering, we may assume that the matrixaˆ(a^k_i) ha s the following form

The stars refer to nonzero entries; evidently, each block of rows (those with two, those with one and those with no nonzero entries) could be empty. Next we investigate the functionsb^k_i.

LEMMA2.11. Let i;t2n and k2m. Then X_i(a^k_t)ˆ d_itb^k_t (10)

and if there is no j2n such that[X_i;X_j]ˆg^k_ijY_k, then X_i(a^k_i)ˆ b^k_i ˆ0.

PROOF. Sincenis step-two, using the definitions ofa^k_i,b^k_i, and (6) 0ˆ[Xi;Yk](ft)

ˆX_i(Y_k(f_t)) Y_k(X_i(f_t))

ˆX_i(a^k_t)‡d_itY_k(lt)

ˆXi(a^k_t)‡ditb^k_t;

thereby showing (10). It follows from Lemma2.6 that if there is noj2n such that [X_i;X_j]ˆg^k_ijY_k, then a^k_i ˆ0, in which case (10) implies

0ˆX_i(a^k_i)ˆ b^k_i. p

Observe that (10) describes X_i(a^k_j) for all possible triples (i;j;k)2 2nnm. Also, Lemma 2.6 says that if there is no j2n such that [X_i;X_j]ˆg^k_ijY_k, thena^k_i ˆ0; under these circumstances, because of (10), we havea fortiorithatb^k_i ˆ0.

LEMMA 2.12. If k2m and i;j2n are such that [Xi;Xj]ˆg^k_ijYk then b^k_i ˆb^k_j.

PROOF. Since [X_i;X_j]ˆg^k_ijY_k, (8) gives X_j(l_i)ˆX

`2m

a^{`</sup><sub>ij</sub>a<sup>`}_iˆg^k_ija^k_i: By the definitions ofb^k_i,b^k_j and by (9) and (10)

g^k_ijb^k_i ˆY_k(l_i)

ˆ[Xi;Xj](li)

ˆX_i(X_j(l_i))) X_j(X_i(l_i))

ˆX_i(g^k_ija^k_i) X_j(2g^k_ija^k_j)

ˆ g^k_ijb^k_i‡2g^k_ijb^k_j;

whereg^k_ij6ˆ0, hence the result. p

LEMMA2.13. Let i;j;t2n with i6ˆj and t62 fi;jg. Then X

`2m

a^{`</sup><sub>ij</sub>b<sup>`}_t ˆ0:

Thus, if k2m and i;j2n are such that[Xi;Xj]ˆg^k_ijYk, then b^k_t ˆ0for t62 fi;jg.

PROOF. ApplyXtto the left hand side of (8), use (10) and Lemma 2.7. p LEMMA2.14. Let k2m and i;j2n be such that[X_i;X_j]ˆg^k_ijY_k. If there exists p2n, p6ˆj, such thata^k_pi6ˆ0, then b^k_i ˆ0.

PROOF. By Lemma2.9 a^k_i ˆ0. Applying X_i and (10) to a^k_i ˆ0 we

obtain thatb^k_i ˆ0. p

LEMMA2.15. Let k; `2m with`6ˆk and i;j;p2n with p6ˆj be such that[X_i;X_j]ˆg^k_ijY_kand[X_i;Xp]ˆg<sup>`</sup><sub>ip</sub>Y`then

(i) b^s_i ˆ0forany s2m;

(ii) b^k_q ˆb^`_q ˆ0forany q2n.

PROOF. By Lemma2.10a^s_iˆ0. ApplyingXiand (10) toa^s_i ˆ0 we ob- tain thatb^s_i ˆ0 for anys2m. This proves (i). As for (ii), first of all we know from Lemma2.13 thatb^k_qˆ0 ifq62 fi;jgandb^`_qˆ0 ifq62 fi;pg. As for the remaining indices,by part (i) b^s_i ˆ0 for any s2m and by Lemma 2.12

b^k_j ˆb^k_i ˆ0 a ndb^{`</sup><sub>p</sub>ˆb<sup>`}_iˆ0. p

Lemma 2.11 tells us that the matrixbˆ(b^k_i) has the same general shape as the matrix aˆ(a^k_i). Furthermore, Lemma2.12 says that whenever a row contains two nonzero entries, they actually coincide. Finally, Lemma 2.15 implies that no row may contain a single nonzero entry. The general shape of thebmatrix is thus

LEMMA2.16. Forall t2n and all`;k2m c<sup>`k</sup>_t ˆ0. Forall i;j2n and all k2m

X_j(b^k_i)ˆ0:

(11)

PROOF. Assume firsti6ˆj. By Lemma2.11 we haveX_i(a^k_i)ˆ b^k_i and X_j(a^k_i)ˆ0. Hence,

X_j(b^k_i)ˆX_i(X_j(a^k_i)) X_j(X_i(a^k_i))ˆ[X_i;X_j](a^k_i)ˆX

`2m

a<sup>`</sup><sub>ij</sub>Y`(a^k_i)ˆX

`2m

a^{`</sup><sub>ij</sub>c<sup>`k}_i : In particular, if k2m and i;j2n are such that [Xi;Xj]ˆg^k_ijYk, then X_j(b^k_i)ˆg^k_ijc^kk_i and by Lemma 2.12 they are both equal toX_j(b^k_j). Similarly, X_i(b^k_i)ˆX_i(b^k_j)ˆ g^k_ijc^kk_j . Hence (11) will follow (for all possible indicesi;j andk) from the fact thatc<sup>`k</sup><sub>t</sub> ˆ0 for allt2nand all`;k2m.

Fix now k; `2m. By assumption (b) there exist i;j2n such that [Xi;Xj]ˆg<sup>k</sup><sub>ij</sub>Yk. We show next that for any `2m we have c^{`k</sup><sub>t</sub> ˆc<sup>k`}_t ˆ0 whenever t62 fi;jg. Indeed, by Lemma2.7 a^k_t ˆ0 for t62 fi;jg, so that using the definition we obtainc<sup>`k</sup><sub>t</sub> ˆY`(a^k_t)ˆ0 for any`2m. Further, by

definition and by (10)

g^k_ijc^{k`</sup><sub>t</sub> ˆg<sup>k</sup><sub>ij</sub>Yk(a<sup>`}_t)ˆXi(Xj(a^{`</sup><sub>t</sub>)) Xj(Xi(a<sup>`}_t))ˆ

0 t62 fi;jg X_j(b^`_i) tˆi

X_i(b^`_j) tˆj;

8>

><

>>

: whose first line proves our claim.

Hereafter we fix k2m and we assume that i;j2n are such that [X_i;X_j]ˆg^k_ijY_k. We show next thatc^kk_i ˆc^kk_j ˆ0. By definition ofb^k_i,c^kk_j , by (9) and Lemma 2.12

0ˆ[X_i;Y_k](l_i)

ˆX_i(b^k_i) Y_k(2g^k_ija^k_j)

ˆXi(b^k_j) 2Yk(g^k_ija^k_j)

ˆ g^k_ijc^kk_j 2g^k_ijc^kk_j ;

the latter equality following from Xi(b^k_j)ˆ g^k_ijc^kk_j , which was shown above. This proves thatc^kk_j ˆ0 and an analogous argument shows that c^kk_i ˆ0.

Finally we show that whenever`6ˆkwe havec<sup>`k</sup>_t ˆc^{k`</sup><sub>t</sub> ˆ0 fort2 fi; jg hence proving the lemma. To this end, we observe that by assumption (b) we may choose p;q2n such that [X<sub>p</sub>;X<sub>q</sub>]ˆg<sup>`}_pqY_` and two cases arise:

either fp;qg \ fi;jg ˆ ; or, by symmetry we may assume pˆi. In the first case, Lemma 2.13 impliesb^{`</sup><sub>i</sub>ˆb<sup>`}_j ˆ0 and hence computing as above further implies that

g^k_ijc^{k`</sup><sub>t</sub> ˆg<sup>k</sup><sub>ij</sub>Y<sub>k</sub>(a<sup>`}_t)ˆ

0 t62 fi;jg X_j(b^`_i) tˆi

Xi(b^`_j) tˆj 8>

><

>>

:

we get that the second and third lines vanish, whence c^{k`</sup><sub>t</sub> ˆ0 for all t2n. Similarly, by Lemma 2.13, b<sup>k</sup><sub>p</sub>ˆb<sup>k</sup><sub>q</sub>ˆ0 and hence computing as above that c<sup>`k}_t ˆ0 for all t2n. In the second case, that is pˆi, we compute that

g^k_ijc^{k`</sup><sub>t</sub> ˆg<sup>k</sup><sub>ij</sub>Y<sub>k</sub>(a<sup>`}_t)ˆX_i(X_j(a^{`</sup><sub>t</sub>)) X<sub>j</sub>(X<sub>i</sub>(a<sup>`}_t))ˆ

0 t62 fi;qg X_j(b^`_i) tˆi

0 tˆq

8>

<

>:

and by Lemma 2.15b^{`</sup><sub>i</sub>ˆ0, so thatc<sup>k`}_t ˆ0 for allt2n. Finally

g^{`</sup><sub>pq</sub>c<sup>`k}_t ˆg<sup>`</sup><sub>pq</sub>Y<sub>`</sub>(a^k_t)ˆ

0 t62 fi;jg X_q(b^k_i) tˆi

0 tˆj

8>

<

>:

and by Lemma 2.15b^k_i ˆ0, so thatc^`k_t ˆ0 for allt2n. p Intheend wehave that allY-derivatives and allX-derivatives ofa^k_tare zero for all choices of the indices, so the system ``closes up''. Indeed, all the deri- vativesofthe functionsg^k;ft;lt;a^k_t;b^k_t,for anyt2nandk2mareexpressedin terms of the functions themselves, as indicated by equations (5) through (11).

2.3.2 - The connection.

Recall that our strategy to prove rigidity is to introduce a connection on a vector bundle in such a way that finding a solution to the multicontact system is equivalent to finding a covariant constant section of the bundle. The bundle we are referring to is a trivial subbundle of the trivial r-plane bundle p:T!NoverN, whose total space isTˆR^r, withrˆm‡2n‡2mn. The idea is the following: once the multicontact structure has been selected, that is, the basesfX1;. . .;XngandfY1;. . .;Ymghave been fixed in such a way that assumptions (b) and (c) are satisfied, we choose only those directions along which the coordinates of a prolonged multicontact vector field are possibly non vanishing, as described by the lemmas of the previous sections.

The fiber coordinates will thus be indexed by arrays of the form g f a

l b 0

@

1 A

where, first of all,f;l2Rⁿ,g2R^m. As for the vectorsaandb, they will be labeledaˆ(a^k_i) a ndbˆ(b<sup>`</sup><sub>j</sub>) respectively, with indices (i;k) and (j; `) run- ning only in those index sets that correspond to non vanishing coordinates of multicontact vector fields. As already illustrated, the results of the previous section show that for eachk2mthere exists asetAk(consisting at most of two indices) such that a^k_pˆ0 ifp62A_k, whenever a^k_p is the prolonged co- ordinate of a multicontact vector field. Notice that ifi;j2A_kit may well be thata^k_i 6ˆa^k_j. We thus introduce the index set

Aˆ f(i;k)2nm:i2A_kg

in order to parametrize thea-vector, that is we takeaˆ(a^k_i) with (i;k)2A.

Similarly, for each`2mthere exists asetB~`(consisting at most of two indices) such thatb<sup>`</sup><sub>q</sub> ˆ0 ifq62B~`, wheneverb<sup>`</sup><sub>q</sub>is the prolonged coordinate of amulticontact vector field. However, ifi6ˆjare inB~`, thenb^{`</sup><sub>i</sub>ˆb<sup>`}_j. We thus define anew index setB_`consisting at most of a single index in order to achieve a parametrization of the b-vector, by choosing one of i or j.

Finally, we introduce the index set

Bˆ f(j; `)2nm:j2B`g

and considerbˆ(b<sup>`</sup><sub>j</sub>) with (j; `)2B. Hencea2R^c(A)andb2R^c(B), where c(A) a ndc(B) are the cardinalities ofAandB, respectively. The bundle we shall be concerned with is thereforep:E!N where the total space is EˆR^s, withsˆm‡2n‡c(A)‡c(B).

The sections S2G(E) of E are mappings S:N!E such that pSˆid_Nthat we arrange in arrays as follows

x7!S(x)ˆ f(x) a(x) g(x) l(x) b(x) 0

@

1 A2E (12)

wheregis acolumn vector inR^m,f,lare s row vectors inRⁿanda2R^c(A) and b2R^c(B) are vectors whose entries are arranged in (mostly empty) nmmatrices. For practical purposes, it is convenient to think ofEa s a subbundle of T and hence to regard a and b as nm matrices whose entries are zero in the complement of the index setsAandB. Omitting the dependence onx, one may thus think

and remember that mosta's andb's are zero forS2G(E).

An affine connection on N is an R-bilinear map r:X(N)G(E)!

!G(E), writtenr(X;S)ˆ rXS, with the following properties:

(i) r_WXSˆWr_XSfor allW2 F(N) a llX2X(N) and allS2G(E);

(ii) rX(WS)ˆX(W)S‡WrXSfor allW2 F(N), allX2X(N) and all S2G(E).

We will define aconnection onEby specifying its valuesrXiSfor all i2nandrYkSfor allk2mand then using (i). We shall do so by using the right hand sides of the various (affine) equations that have been obtained in the previous section. In order to write explicitly the definition, we para- metrizeSas in (12). Instead of writingr_XiSandr_YkSas large arrays, we introduce the dual bases (g^s),f_t,l_t, (a^s_t)and (b^s_t). Here is the definition of the various components ofrYkS:

(g^s)(rYkS)ˆYk(g^s)‡dks(li‡lj) f_t(r_YkS)ˆ Y_k(f_t) a^k_t if (t;k)2A

Y_k(f_t) if (t;k)62A

l_t(rYkS)ˆ Y_k(l_t) b^k_t if (t;k)2B Y_k(l_t) if (t;k)62B

(a^s_t)(rYkS)ˆYk(a^s_t) (b^s_t)(r_YkS)ˆY_k(b^s_t):

(13)

For the sake of precision we remark that in the first line one picks any pair (i;j) with the property that [Xi;Xj]ˆg^k_ijYk (with g^k_ij6ˆ0), certainly existing by our hypotheses, and the sum l_i‡l_j is independent of such choice; further, the last two lines are defined for (t;s) in A and B, respectively. Similarly, the definition of the various components of r_XiS is:

(g^s)(rXiS)ˆX_i(g^s) X

t2n

a^s_tift

f_t(rXiS)ˆXi(ft)‡ditlt

l_t(rXiS)ˆ

Xi(lt) X

(t;`)2A

a^{`</sup><sub>ti</sub>a<sup>`}_t if t6ˆi

X_i(l_i) 2g^k_ija^k_j if tˆiand (j;k)2A Xi(li) if tˆiand (j;k)62A 8>

><

>>

:

(a^s_t)(rXiS)ˆXi(a^s_t)‡ditb^s_t (b^s_t)(r_XiS)ˆX_i(b^s_t):

(14)

Here again the last two lines are defined for (t;s) inAandB, respectively; as for the value ofl_t(rXiS), the indexjappearing in the right hand side is an index such that [X_i;X_j]ˆg^k_ijY_k(withg^k_ij6ˆ0), certainly existing by our hy- potheses.

PROPOSITION2.17. Formulae(13) and(14)define an affine connec- tion onE.

PROOF. LetXbe any vector field in the basis and letSbe asection. We temporarily change notation and write Sas a vector with componentsS_i withiranging in the index setIand observe that formulae (13) and (14) are all of the form

rX(S)

… †_iˆX(S_i)‡X

j2I

C_jS_k(j);

where j7!k(j) is some mapping ofI andC_j is apossibly zero constant.

Property (i) is automatic from our definition. As for (ii), for any smoothc we have

rX(cS)

… †_iˆX(cSi)‡X

j2I

CjcSk(j)

ˆX(c)Si‡cX(Si)‡X

j2I

CjcSk(j)

ˆX(c)S_i‡c…r_X(S)†_i;

as desired. p

2.3.3 - End of proof.

Summarizing, the vector field (3) is a multicontact vector field if and only if the functionsff_t;l_t;g^s;a^s_t;b^s_t :t2n;s2mgsatisfy the equations (5) through (11) and this in turn happens if and only if the sectionSdefined by (12) is a covariant constant section of the bundlep:E!Nwith respect to the connectionr, that is, if it satisfiesrXSˆ0 for anyX2X(N). Recall that the bundle we shall be concerned with is thereforep:E!N where the total space isEˆR^s, withsˆm‡2n‡c(A)‡c(B). Thus, there are at mostslinearly independent such sections, and this proves that the space of multicontact vector fields is at mosts-dimensional.

There are noa priori conditions on thef;l2Rⁿ,g2R^m. Therefore, they may contribute up to 2n‡m dimensions. Moreover, we consider aˆ(a^k_i) with (i;k)2A and bˆ(b<sup>`</sup><sub>j</sub>) with (j; `)2B. Hencea2R^c(A) and b2R^c(B), wherec(A) a ndc(B) are the cardinalities ofAandB, respectively.

We achieve the total number ofs.

In the next section we improve our result by proving that the connec- tionris actually flat.