D-modules, contact valued calculus and Poincare-Cartan form

Czechoslovak Mathematical Journal, 49 (124) (1909), Praha

^-MODULES, CONTACT VALUED CALCULUS AND POINCARE-CARTAN FORM RICARDO J. ALONSO BLANCO, Salamanca (Received October 30, 1996)

1. INTRODUCTION AND NOTATION The geometric formulation of the calculus of variations has been carried out in the last decades in different fashion (see, for instance, [1], [2], [3], [4], [6], [7], [9 [12], [13], [16]). These methods, used in the process of generalizing from mechanics to the higher order case and several variables, may seem somewhat artificial. For instance, in the construction of a general Poincare-Cartan form, one utilizes a method involving determination of coefficients under certain prescribed conditions, and other adhoc techniques. It is not always clear what the real significance is of the role played by new objects (e.g. a linear connection on the base manifold) that are extrinsic t the problem. We offer a new approach that unifies the classical methods and their generalizations to field theories of higher order and the study of the variational bicomplex in a natural and rather simple way. We make use of two basic tools; first, a decomposition of operators that is trivial in the first order case. This is where the above mentioned connection comes in. Second, we introduce a formal covariant derivation law which is the geometric analogue of the derivative of variations with respect to time in mechanics. For a differentiable manifold M, we make the key remark that the algebra nn of differential forms of maximal degree is a right module over the ring @ of differential operators. With each linear connection on M we associate a morphism whose effect is to decompose the higher order tangent fields &k, k > I (^k C @) as a composition of ordinary tangent fields with differential operators of order k — 1. With this we achieve a factorization, via the exterior differential, of the action of &k on the *&module Hn. Mutatis mutandis, it is possible to generalize the method to the valued 585

case Qn®^/, when sd is a left ^-module through the action of the covariant derivative associated with a derivation law. Let ^ be the contact module on the jet spaces of a fibred manifold with base manifold M. By making use of the formal or total derivative, we define a covariant derivation law on M with values in M is a projection between differentiable manifolds, we will write TV(N/M) for the bundle of vertical tangent vectors of TT; if F —» M and G —> M are vector bundles, we will write FN for the 586

pullback TT*F and F <S>M G for the tensor product ; when there is no risk of confusion we will understand that the tensor product is taken before a suitable pullback. Calligraphic letters will be reserved for the modules of sections of the corresponding vector bundles; for instance, &M will mean the ^°°(M)-module of sections of TM, i.e. the module of tangent vector fields. As an exception np(M) will mean the •^(M^-module of sections of A^Af.

2. GRADUATION OF DIFFERENTIAL OPERATORS Let M be a differentiate manifold, dimM = n and mx the maximal ideal of functions vanishing at a point a; of M. A differential operator of order k on M is, by definition, an D?-linear morphism P: ^°°(M) that sends m£+1 to mx for each re e M. We will denote the set of differential operators of order k on M by ®k. Let p: M x IR —> M be the projection onto the first factor, Jkp the fiber bundle of A-jets of sections of p (i.e. the functions on M) and ^kp the module of the corresponding sections. The map jk: t^°°(M) —» ^kP, associating with every function its k-ih Taylor expansion, represents @k in the following sense: for every P € @>k there is a unique morphism P: ^kp —> ^°°(M) such that P = P o jk In other words, @k is the dual ^"(MJ-module of a?kp. Therefore @k is locally free, with rank (n*k), and, if (xi,... ,xn) is a local chart on M, @k is generated by da = (gf^)"1 o • • • o (gf^) Qn as a = (ai,... ,a n ) runs through the multi-indexes |a| ^ k (d° means the constant function 1). If k < r, then there is a natural immersion @k C ®T (observe that 0° = ^f°°(M)). Besides the structure mentioned, every 3>k has a second canonical ^^(MJ-module structure. If g 6 ^>00(M) and P € Sk, we define the operator g * P € @k according to the following rule: (g * P)f = P(g/)for every / e 00(M). This new structure for 3>k will be denoted by "W*. The ^°° (M)-module Hfi is also locally free with the same local generators. The subbundle T*>kM of <j?kp comprised of those fc-jets whose projections over M x IR are zero is, by definition, the fe-th cotangent fibre bundle of M. The fibre of each a; e M is T*'kM = m!C/m£+1 C ^°°(M)lmkx+l = Jkp. The module of sections of T*'fc will be denoted by ^*'fc. The dual of T*'k is called the fc-th tangent fiber bundle of M. We will denote it by TkM. Equivalently, TkM is the incident of R in Jkp. The module of sections of TkM will be denoted by £TkM. The elements of &kM can be characterized by Leibniz's rule of derivations of products. On the other hand &kM is, at the same 587

time, a quotient and a submodule of @k; in fact we have the decomposition

where we put P = P(l) + (P - P(l)) for each P 6 3>k. For the sake of simplicity, we will write &k instead of &kM and & instead of &1M. For k ^ 0, the quotient of the k/^k~l has the fibre (m^/m*"1"1)* c± Sk(mx/ml)* at each point x 6 M, where ( )* means taking the dual and 5fc means the fc-th tensor symmetric product. From the above, making use of the isomorphism (tn x /m^)* ~ Tx, we infer the exact sequence The image under the projection @k —>• Sh & of an element P € 3>k is known as the symbol of P. Remark 2.1. The same exact sequence holds for the tangent bundles (taking into account the inclusions 3?k C @h):

We can consider C/°°(M) and Sk& as the homogeneous minimal and maximal degree components of ^ fc , respectively. We will show how to define the remaining components for a specific linear connection. Lemma 2.2.

A linear connection V on M gives the module @k a graduation k

via an isomorphism © Sk& ~ 2>k (where S°& = ^°°(M)). r=0

Proof.

It suffices to find sections sr for the exact sequences

For r = 1, the sequence is

and we take the section si: ^ —> @l as the natural inclusion. For r > 1 we define the section sr: Sr' & —>• 3>r by the rule

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where £>i • • • Dr is the symmetric product of r tangent vector fields Di and D? is the covariant derivative that V induces over Sr~1?7. It is easy to check that each sr is well defined, ^""'(MJ-linear and it is also a section of $r —>• Sr&. See [5] for another construction of this graduation of 3>k. We obtain the expression of this in coordinates by letting (zi,..., xn) be a local chart for M, dt = gfr, d*a = d"1 • • -9° 6 Sr&, da = df1 o • • • o d£ e 0r with a = (ai,..., a n ), |a| = r and V the local linear connection denned by (xi,..., xn)Then sr(dta) = da.

3. DECOMPOSITION MORPHISM OF HIGHER ORDER TANGENT BUNDLES If P is a differential operator of order k - I and D is a tangent vector field, the composition P o D belongs to &k. This type of compositions defines a Sf°°(M)module morphism that we will explain in what follows. Define a map

where G(P ® D) = P o D and extending by linearity. The map G is well defined by the choice of the structure for f^*"1. However, G is not ^>00(M)-linear. Now let jYk be the ^°°(M)-module obtained from Sik~lM O & by defining the product by functions as follows: for each / 6 EndR jtf defining an R-linear action of ^ on f^ such that: 1) [X(D), a(f)] = a(Df) (resp. -a(Df)), 2) (x(Di),x(D2)} = x((Di,D2}) (resp. -x([I>2,Z?i]);, 3) «(/) o X(D) = x(fD) (resp. X(D) ° a(f) = x(fD)) for every £>i, D2, D3 6 ff and every f 6 ^°°(M). Then there exists a unique structure of the left (resp. right) ^-module enlarging the actions a and xCorollary 4,3. The exterior algebra $ln(M) comprised of the forms of maximal degree possesses a unique structure of the right ^-module that extends its structure, a, of the tf00 (M)-module, and also extends the action of ^ given by LJ • D = —Lou for any a; € fln(M),D 6 & (Lo being the Lie derivative). Corollary 4.4. Let £^\, stfi be two left ^-modules and & a right ^-module. Then: 1) The action of t? on the ft00(M)-module $4\ ®M ^i given by D(a\ 02) = Dai ® ^2 + «i ® Da,2 for any D € &, a\ € J&i, 0,2 6 J&2, extends to a unique structure of the left ^-module on stfi M -#22) The action of & on the ^°° (M)-module SB ®M £& given by (b ® o) • D = b • D a — b®Da for any D £ & ,b £ &, a € ^ 2 , extends to a unique structure of the right ^-module on & ®M ^2-

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5. FACTORIZATION OF THE ACTION OF HIGHER ORDER TANGENT FIELDS ON fin Let D be a (tangent) vector field on M and u> a form of maximal degree. Then for the action denned in 4.3 we have w • D = —LouJ = d(-irju) (irju being the inner contraction of u> with D), in other words, we can build an explicit integral for the exact form u> • D. We want now to extend this result for the action of higher order vector fields P e &k> k > 1. The key to this resides in the decomposition morphism (Theorem 3.2). Theorem 5.1. Given a linear connection V on M, for each P € ^k, there is an K-h'near map $p: ftn(M) —» H n ~ 1 (M) that makes the next diagram commutative:

where -P is the action of P and d is the exterior differential. In other words, for any u> £ n n (M), $pw is an integral ofui • P. P r o o f . If Hv(P) = Y,Q®D £ ^k then we can define $ pw = - £ J D ( W - < ? ) which gives

where we have made use of the fact that LD = dio on fP(M). Remark 5.2. When P is of order 1 or 2, the decomposition morphism is independent of the linear connection, as a result of which $p is also independent. On the other hand, provided that dim M = 1, V is another connection and $p is the associated operator, then $p can only differ from $p in a constant term. Both $P and $p depend linearly on P, thus we deduce that this constant term is zero. Therefore, if dim M = 1 for any order fc, $p is also independent of the connection. Remark 5.3. The result obtained in the theorem is automatically generalizable to the following ('valued') case: let st be a (M)-module with a covariant derivation law such that the Lie covariant derivative converts £/ into a left ^-module (Lemma 4.2). The • fin(M) be the morphism associated with the action of &k C @ on fin(M). Given a fixed linear connection there exists a map $ that makes the next diagram commutative:

P r o o f . Define $(P®w) = $pw with P e &k, u 6 fin(M) and $P the operator denned in 5.1. Extending $ by linearity the proof is complete. Remark 5.5. In the same way as 5.1, Corollary 5.4 is valid in the 'valued' case.

6. JET SPACES AND VERTICAL LIFT Let TT: E —> M be a fibred manifold and Jk = Jk(E/M) the space of fc-jets of sections of TT. It is known that there is an isomorphism

i.e., if p1 is a 1-jet of J1 that projects over p 6 E and x G M, with each pair (w x , Dp), comprised by ujx £ T*(M) and Dp 6 T£(E/M), we can associate canonically a tangent vector in T£(Jl/E). It will be shown in this section how to generalize this operation in higher orders. The result obtained is equivalent to [7] ((14) of §1). Nevertheless, our calculations make use of an alternative construction of the jet spaces [14], following Weil's suggestions (see [20] and [8]). In agreement with [14], a fc-jet pk € Jk over a point x 6 M is a morphism of algebras tf°°(E) —> fc+1 shall be called the vertical lift. To express this in coordinates, let (xi,...,xn) be a local chart on M and let (j/ii • • • 12/m) be local coordinates on the fibres of TT : E —> M. These charts produce the usual fibred coordinates (EJ, j/j a ), i = 1,... ,n, j = 1,... m, |a| < k on Jk. Then

where 9" = (jf^-i o ... o (^-)«-, a = (a,,...,an). 7. THE FORMAL DERIVATIVE

In this section the concept of 'formal' (also known as 'total') derivative will be defined (refer to [10] or [15]). By means of the immersion J fc+1 c J1 J fe , we associate with each pk+l e Jk+l an element of J1 Jk that will be denoted by pk>1. Following [14], pk'1 is a morphism tf°°(Jk) —>• «700(M)/m^. Thus for each function / e ^°°(Jfe) we have pM/ € if 00 (M)/m2. 594

Remark 7.1. With the usual construction of jet spaces, pk+l is the (k + l)-jet of some local section s in x of TT: E —»• M, i.e. pk+l = jk+1s. Let s = j fe s be the local fc-jet prolongation of s. Then pk>l = j'^s and pk'lf is the class of the function / o s modulo m£. Definition 7.2. Each tangent vector field D 6 ^ produces a derivative

denned by the rule (Df)pk+l = Dx(pk>1 /) 6 ^°°(M)/mx ~ R. We will call D the holonomic lift of D to the jets. Similarly, we can consider D as a section of (TJk)jb+i or as a map Jk+1 —t TJk over Jk. Later on we will use the notation D = bk+i(D). Remark 7.3. The holonomic lifts to different orders are compatible with each other. This fact allows us to use the notation D without reference to k. Let ( x i , y j a ) be fibred coordinates as above. Then ^ = ^- + 53 Vja+eiQ^— • |a|pfc+i isinT p fcJ f e ). Definition 7.5. We will denote the formal Lie derivative with respect to the vector field D on M as the derivative

defined by Cartan's formula Lg = igd + dig . The formal Lie derivative verifies the usual properties of the ordinary Lie derivative (see [10], [15]).

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8. DIFFERENTIAL CALCULUS WITH VALUES IN THE CONTACT MODULE In this section we will recall the definition of the contact module and then see how the restriction of the formal Lie derivative (Definition 7.5) produces a covariant derivation law. The holonomic lift, &&: T(M)jk —> T(Jk~l)jk, is a canonical splitting of the exact sequence

Definition 8.1. The retract 6k associated with && is known as the structure form of Jk. Through pull-back to Jk, Ok defines a section of the bundle T* Jk ®jt.-. T v (J k ~ l /M). With the above notation, in a fibred local chart ( x i , y j a ) we have

As is known, a section s of the projection Jk —>• M is holonomic (i.e., is the fc-jet prolongation of some section of ?r: E —> M) if and only if 6k vanishes on the image of s. Definition 8.2. We will call the (^'00(Jk)-module comprised by those 1-forms of Jk that become zero over the holonomic sections as the contact module of J*, denoted by %. The module % is generated by the components of Ok (the above I0ja-s'). The fibre bundle associated with ^ will be denoted by Ck. It is useful to observe that the dual map of 6k produces an immersion

which identifies the dual of Tv(Jk 1/M)Jk with Ck. In particular, the holonomic lifts of tangent vectors on M are incident with Ck • Denote the natural projections Jk —> Jr, 0 ^ r < k, by irk. Each ^ with r < k induces a submodule of % via the pull back by ifk. Definition 8.3. We will call the injective limit of the system (^,71-*) the contact module *&:

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The restriction of the formal Lie derivative in Definition 7.5 to ^fc C ft1(Jk) takes values in ^t+i. Indeed, if s is a local section of •n: E —> M, jfc+1s is the (k + l)-prolongation and a G %, then because i^a = 0 we have (j fc+1 s)*Lgcr = (jk+1s)*igda = iDd(jks)*a = iDdQ = 0, hence Lga € tfk+iIn a local chart (xi,yja) as above, we have L~g~9ja = Oja+a • Qa-.i

Because of the compatibility of the holonomic lifts with the projections TT£ it is possible to define, for any tangent vector field D on M, the formal Lie derivative

Recall that ^ is, in particular, a ^°° (M)-module. Proposition 8.4. The assignment D —> Lg produces a covariant derivation law on the • EndR ^ by X(D) = LD for any D 6 &. We must check properties 1), 2) and 3) of Proposition 4.2 as follows: For D,Di,Da 6 &, f € V°°(M) and yeV, 1) [x(D),a(f)]a = LD(fa)- fLDa = Df • a +JLDa - fLDa = a(Df)a. 2) [x(Di),x(D2)}cr = LDlLD^a - LD2LDlff = L[DltDz]a = x([Di,D2])a. 3) &(f) ° x(D)a = fLov = fioda = ifDda = L5t>a = x ( f D ) a . Corollary 8.7. fin(M) ®M *& has a right ^-module structure (n = dim M). P r o o f . According to Corollaries 4.3 and 4.4, the former structure is obtained by extending the following action of &\ for any w € f l n ( M ) , a € *& and D € ST we put (u<S>a)-D = ui-D®cr — u<S> D • a = -Lpt^ ® a - u> ® LDCT = — LD(W ® i = z'a;<j. Finally, we make use of the combinatorial identity

to obtain

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The above expression coincides with that obtained in [12] and [16]. Theorem 9.8. dX = H - dH®. P r o o f . This is a direct consequence of Theorem 9.5 and the relations (rjod]Q1 = (dHorj)el=dHQi, da\ = 0. Remark 9.9. The property enunciated in the last theorem or, equivalently, the correspondent one in Theorem 9.5 is known as 'decomposition formula' in [7]. This formula makes it possible to find the equations for the critical sections in a variational problem defined for a given Lagrangian density (see [7]). As demonstrated in [6], any other form verifying 'decomposition formula' as 61 is of the form &i + G, where G <E (^j^-' ®M n n ~ 2 (M). Remark 9.10. There is no total agreement on this topic. One can consult [4] and [13] for a discussion concerning the construction of the Poincare"-Cartan form. See [1], [2], [3], [9], [11], [12] and [16] for other references of the geometric formula the calculus of variations (naturally, the above list is not exhaustive). 10. APPLICATION TO THE EULER-LAGRANGE RESOLUTION The ^-valued differential calculus of §8 allows us to define intrinsically the elements of the Euler-Lagrange resolution. We will follow Tulczyjew's paper [19] (it can also be consulted in [17]). We fix the projection TT: E —> M and work on the infinity jet space J°° = proj lim Jk. Let us consider the tensor products $r's = Ar1 is known as the Helmholtz-Sonin operator (inverse problem of the calculus of variations). Acknowledgements. The author wishes to thank Prof. Munoz Diaz and Prof. Rodriguez Lombardero for stimulating discussions and continuous encouragement during all stages of this work. 605

References [1] P.L. Garcia: The Poincare-Cartan invariant in the calculus of variations. Symp. Math. XIV (1974), 219-246. [2] P.L. Garcia and J. Munoz: On the geometrical structure of higher order variational calculus. Proceedings of the IUTAM-ISIMM. Symposium on Modern Developments in Analytical Mechanics (Bologna). Tecnoprint, 1983, pp. 127-147. [3] H. Goldschmidt and S. Sternberg: The Hamilton-Cartan formalism in the calculus of variations. Ann. Inst. Fourier (Grenoble) 23 (1973), 203-267. [4] M. Gotay: An exterior differential system approach to the Cartan form. Geometric Symplectique et Physique Mathematique (Boston). P. Donato et al., Boston, 1991, pp. 160-188. [5] H. Hess: Symplectic connections in geometric quantization and factor orderings. Ph.D. thesis, Berlin, 1981. [6] M. Horde and I. Kolar: On the higher order Poincare-Cartan forms. Czechoslovak Math. J. 33 (1983), 467-475. [7] /. Koldr: A geometric version of the higher order Hamilton formalism in fibered manifolds. J. Geom. Phys. 1 (1984), 127-137. [8] /. Koldr, P. Michor and J. Slovak: Natural Operations in Differential Geometry. Springer-Verlag, Berlin, 1993. [9] D. Krupka: A geometric theory of ordinary first order variational problems in fibered manifolds I. Critical sections. J. Math. Anal. Appl. 49 (1975), 180-206. [10] A. Kumpera: Invariants differentiels d'un pseudogroupe de Lie, I. J. Differential Geom. 10 (1975), 289-345. [11] B. Kupershmidt: Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalism. Lect. Notes in Math. 775. 1980, pp. 162-217. [12] J. Munoz: Poincare-Cartan forms in higher order variational calculus on fibred manifolds. Rev. Mat. Iberoamericana 1 (1985), no. 4, 85-126. [13] P.J. Olver: Equivalence and the Cartan form. Acta Appl. Math. 31 (1993), 99-136. [14] J. Rodriguez: Sobre los espacios de jets y los fundamentos de la teoria de los sistemas de ecuaciones en derivadas parciales. Ph.D. thesis, Salamanca, 1990. [15] C. Ruiz: Prolongament formel des systemes differentiels exterieurs d'ordre superieur. C. R. Acad. Sci. Paris Ser. I Math. 285 (1977), 1077-1080. [16] D. Saunders: An alternative approach to the Cartan form in Lagrangian field theories. J. Phys. A 20 (1987), 339-349. [17] D. Saunders: The Geometry of Jet Bundles. Lecture Notes Series, vol. 142. London Mathematical Society, Cambridge University Press, New York, 1989. [18] J.P. Schneiders: An introduction to the D-Modules. Bull. Soc. Roy. Sci. Liege 63 (1994), 223-295. [19] W.M. Tulczyjew: The Euler-Lagrange resolution. Lect. Notes in Math. 836. 1980, pp. 22-48. [20] A. Weil: Theorie des points proches sur les varietes differentiables. Colloque de Geometric Differentielle, C.N.R.S. (1953), 111-117. Author's address: Departamento de Matematica Pura y Aplicada, Universidad de Salamanca, Plaza de la Merced, 1-4. E-37008 Salamanca, Spain, e-mail: ricardoQgugu.usal.es.

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