October 11, 2004 14:39 Proceedings Trim Size: 9in x 6in orbits
REGULARITY OF PSEUDOGROUP ORBITS
PETER J. OLVER School of Mathematics University of Minnesota Minneapolis, MN 55455, USA E-mail: [email protected]
JUHA POHJANPELTO Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA E-mail: [email protected]
Let G be a Lie pseudogroup acting on a manifold M. In this paper we show that under a mild regularity condition the orbits of the induced action of G on the bundle J n(M, p) of nth order jets of p-dimensional submanifolds of M are immersed submanifolds of J n(M, p).
1. Introduction Lie pseudogroups, roughly speaking, are in nite dimensional counterparts of local Lie groups of transformations. The rst systematic study of pseu- dogroups was carried out at the end of the 19th century by Lie, whose great insight in the subject was to place the additional condition on the local transformations in a pseudogroup that they form the general solution of a system of partial di erential equations, the determining equations for the pseudogroup. Nowadays these Lie or continuous pseudogroups play an important role in various problems arising in geometry and mathemat- ical physics including symmetries of di erential equations, gauge theories, Hamiltonian mechanics, symplectic and Poisson geometry, conformal ge- ometry of surfaces, conformal eld theory and the theory of foliations. Since their introduction a considerable e ort has been spent on develop-
Work partially supported by grant DMS 01-03944 of the National Science Foundation.
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ing a rigorous foundation for the theory of Lie pseudogroups and the invari- ants of their action, and on their classi cation problem, see e.g. Refs. [5], [6], [7], [8], [9], [11] and the references therein. More recently, the au- thors of the paper at hand have employed a moving frames construction [3], [4] to establish a concrete theory for Lie pseudogroups amenable to practical computations. As applications, a direct method for uncovering the structure equations for Lie pseudogroups from the determining equa- tions for the in nitesimal generators of the pseudogroup action is obtained (see, in particular, the work [1] on the structure equations for the KdV– and KP–equations) and systematic methods for constructing complete sys- tems of di erential invariants and invariant forms for pseudogroup actions are developed. Moreover, the new methods immediately yield syzygies and recurrence relations amongst the various invariant quantities which are in- strumental in uncovering their structure, the knowledge of which is piv- otal e.g. in the implementation of Vessiot’s method of group splitting for obtaining explicit noninvariant solutions for systems of partial di erential equations. Let G be a Lie pseudogroup (a precise de nition will be given in Sec. 2) acting of a manifold M. The action of G on M naturally induces an action of G on the extended jet bundle J n(M, p) of nth order jets of submanifolds of M by the usual prolongation process. Our goal in this paper is to prove that under a mild regularity condition on the action of the pseudogroup G on M the orbits of G in J n(M, p) are immersed submanifolds for n su - ciently large. We were originally lead to the problem in connection of the research reported in Ref. [4] and the result is of importance in the the- oretical constructs therein. Interestingly, as we will see, the submanifold property of G orbits in J n(M, p) is closely related to local solvability of the determining equations for the in nitesimal generators of the pseudogroup action on M. The proof of our main result relies on classical work [12] on the structure of the orbits of a set of vector elds originally arising in the study of the accessibility question in the context of control theory. In Sec. 2 we cover some background material on Lie pseudogroups and discuss the regularity condition for pseudogroup actions needed in our main result. Sec. 3 is dedicated to the proof of the submanifold property of orbits of the action of G on the extended jet bundles J n(M, p). October 11, 2004 14:39 Proceedings Trim Size: 9in x 6in orbits
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2. Tameness of Lie Pseudogroups Let D = D(M) denote the pseudogroup of all local di eomorphisms of a (n) (n) manifold. We write jz ϕ for the nth order jet of ϕ ∈ D at z and : (n) (n) n D → M for the associated jet bundle, where (jz ϕ) = z stands for the (n) (n) (n) n source map. We furthermore write : D → M, (jz ϕ) = ϕ(z) = Z for the target map.
De nition 2.1. A subset G D is called a pseudogroup3 acting on M if
(1) the restriction id|O of the identity mapping to any open O M belongs to G; (2) if ϕ, ∈ G, then also the composition ϕ ∈ G where de ned; (3) if ϕ ∈ G, then also the inverse mapping ϕ 1 ∈ G. A pseudogroup G is called a Lie pseudogroup if, in addition, there exists N 1 so that the following conditions are satis ed for all n N: (4) G(n) D(n) is a smooth, embedded subbundle; n+1 (n+1) (n) (5) n : G → G is a bundle map; (n) (6) a local di eomorphism ϕ of M belongs to G if and only if z → jz ϕ is a local section of (n) : G(n) → M; (7) G(n) = prn N G(N) is obtained by prolongation.
We call the smallest N satisfying the above conditions the order of the pseudogroup, and unless otherwise speci ed, we will assume that n N in
what follows. Note that by (1) and (2), the restriction ϕ|O of a transfor- mation ϕ ∈ G to any open subset O of the domain of ϕ is again a member of the pseudogroup. Fix local coordinates (z1, . . . , zm) about some p ∈ M, and let (z, Z) = (z1, . . . , zm, Z1, . . . , Zm) denote the induced product coordinates about (p, p) ∈ D(0) = M M. Due to conditions (4) and (6) above, pseudogroup transformations are locally determined by a system (n) F (z, Z ) = 0, = 1, . . . , k, (1) of partial di erential equations, the determining equations for G. Here n N is xed and (z, Z(n)) stands collectively for the coordinates of D(n) induced by (z, Z). By De nition 2.1 the above equations are locally solv- (n) (n) able, that is, given a jet go = (zo, Zo ) satisfying (1), then there is a (n) (n) solution ϕ ∈ G of the equations so that jzo ϕ = go . Let X = X (M) denote the space of locally de ned vector elds on M. Thus the domain U(v) M of v ∈ X is an open subset of M. The vertical October 11, 2004 14:39 Proceedings Trim Size: 9in x 6in orbits
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lift V(n) of a vector eld v ∈ X to D(n) is the in nitesimal generator of the (n) (n) local one-parameter group t of transformations acting on D de ned (n) (n) (n) by t (jz ϕ) = jz ( t ϕ), where t stands for the ow map of v. Note that the domain of V(n) is (n) 1(U(v)). m a m a v v a V V a Pick = Pa=1 (z)∂z ∈ X and write = Pa=1 (Z)∂Z for the vertical counterpart of v. Then V(n) is simply given by the usual prolongation formula2,
m (n) (n) a (n) V z, Z D V z, Z ∂ a , ( ) = X X ( J )( ) ZJ (2) a=1 |J| n
where J = (j1, . . . , jp) is stands for a multi-index of integers, DJ =
j Dj1 Djp for the product of the total derivative operators Dj = ∂z + m a a Z ∂ a , and where the Z denote the components of the ber Pa=1 P|J| 0 Jj ZJ J (n) (n) coordinates on D induced by (z, Z). In particular, at the nth jet Iz of the identity mapping, Eq. (2) becomes
m (n) a V ∂ J v z ∂ a , (n) = X X z ( ) ZJ (3) Iz a=1 |J| n where we have again used the obvious multi-index notation. We denote the space of n jets of local di eomorphisms with source at a (n) (n) xed z ∈ M by D |z. It is easy to see that D |z is a regular submanifold (n) (n) (n) of D . Write Rh(n) for the right action of a jet h ∈ D on the source (n) (n) 1 (n) (n) (n) n ber D | (n)(h(n)) = ( (h )) by Rh(n) g = j (n)(h(n))(ϕ ), (n) n (n) n where h = j (n)(h(n)) , g = j (n)(h(n))ϕ. Then, by di erentiating the identity
(n) (n) (n) (n) Rh(n) t (g ) = t (Rh(n) g ),
(n) it is easy to see that V is Rh(n) -invariant, that is,
(n) (n) (n) (n) Rh(n) (V (g )) = V (Rh(n) g ), (4)
(n) (n) (n) (n) whenever (g ) = (h ). Note that the action of Rh(n) on V(n)(g(n)) is well de ned since V(n) is a vertical vector eld. Next let G be a Lie pseudogroup. A local vector eld v ∈ X on M is a G vector eld if its ow map t is a member of G for all xed t on some interval about 0. We denote the space of G vector elds by XG. (n) The in nitesimal determining equations L (z, jz v) = 0 for G vector elds v can be obtained by linearizing the determining equations (1) for G October 11, 2004 14:39 Proceedings Trim Size: 9in x 6in orbits
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(n) at Iz , that is, d L (z, j(n)v) = F (z, j(n) ) = 0 for all z ∈ U(v). (5) z dt z t |t=0 By (2), this is equivalent to the condition
(n) I(n) (V F )(z, z ) = 0 for all z ∈ U(v).
As a consequence of our de nition of a Lie pseudogroup, the in nitesimal determining equations completely characterize G vector elds.
Proposition 2.1. A local vector eld v ∈ X is a G vector eld if and only if
(n) L (z, jz v) = 0 for all z ∈ U(v) with some n N. (6)
Proof. We only need to show that a vector eld v satisfying (6) is a G vector eld. First note that equation (6) implies that V(n) is tangent to (n) (n) (n) G at Iz for all z ∈ U(v). Thus, by the right invariance (4), V is tangent to G(n) at any g(n) ∈ G(n) ∩ 1(U(v)), and, consequently,
(n) I(n) (n) t ( z ) ∈ G for all t. (7)
But (7) implies that
(n) (n) jz t ∈ G for all t and z ∈ U(v),
and consequently, t ∈ G for all t su ciently small and thus v is a G vector eld.
(n) (n) Recall that the lift t of the ow of v ∈ X to D is tangent to the (n) 10 bers D |z. We thus obtain canonical mappings g(n) from the space (n) (n) (n) of n-jets XZ , Z = (g ), of local vector elds into the tangent space (n) (n) (n) Tg(n) D |z of D |z at g by
(n) (n) (n) g(n) (jZ v) = V (g ). (8)
Proposition 2.2. The mappings
(n) (n) (n) (n) (n) (n) g(n) : XZ → Tg(n) D |z, where z = (g ), Z = (g ), are well de ned. October 11, 2004 14:39 Proceedings Trim Size: 9in x 6in orbits
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Proof. We only need to show that if v1, v2 are two local vector elds so (n) (n) that jZ v1 = jZ v2, then (n) (n) (n) (n) (n) (n) (n) V1 (g ) = V2 (g ) for all g with (g ) = Z. (9) (n) I(n) By (3), Eq. (9) holds for g = Z . But then, due to the invariance (4) (n) (n) of V1 , V2 under the right translations Rg(n) , Eq. (9) must hold for all g(n) with (n)(g(n)) = Z.
(n) (n) (n) Let G be a Lie pseudogroup and write G |z = G ∩ D |z. Note (n) (n) that G |z is a regular submanifold of G . In fact, the source mapping (n) (n) restricted to G |z is of maximal rank as is seen by observing that for (n) (n) ϕ ∈ G, the local section jz ϕ of G → M yields a local right inverse for (n). (n) (n) By the proof of Proposition 2.2, the mappings g(n) for g ∈ G (n) (n) (n) restrict to mappings from the n-jets of G vector elds XG,Z at Z = (g ) (n) (n) (n) into the tangent space Tg(n) G |z of the source ber G |z at g . De nition 2.2. A Lie pseudogroup G is called tame at order n provided that the mappings (n) (n) g(n) : XG,Z → Tg(n) G |z are isomorphisms for all g(n) ∈ G(n).
Remark 2.1. By Eqs. (2) and (3), the mappings g(n) are automatically monomorphisms. Thus, by the right invariance (4), G is a tame Lie pseu- (n) (n) dogroup provided that (n) maps X onto T (n) G |z for all z ∈ M. Iz G,z Iz Thus in particular, when G is tame, the dimension of the space of n jets of G vector elds at z ∈ M is constant in z. Proposition 2.3. A Lie pseudogroup G is tame at order n if and only if the nth order in nitesimal determining equations for G vector elds are locally solvable.
Proof. We can use (5) to identify the solution manifold of the nth order (n) in nitesimal determining equations with tangent vectors w ∈ T (n) G |z. Iz (n) By Remark 2.1, a Lie pseudogroup is tame if and only if any w ∈ T (n) G |z Iz can be represented by a solution of the equations, that is, provided that the in nitesimal determining equations are locally solvable.
Remark 2.2. While the local solvability of the determining equations for G is built into De nition 2.1 of a Lie pseudogroup, as far as we know, October 11, 2004 14:39 Proceedings Trim Size: 9in x 6in orbits
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the in nitesimal version of the equations does not necessarily possess this property. However, as of yet, we have been unable to construct an example of a Lie pseudogroup with in nitesimal determining equations that are not locally solvable.
3. Regularity of Orbits
De nition 3.1. Call g(n) ∈ G(n) reachable by G vector elds if there are v , . . . , v ∈ X with ow maps v1 , . . . , vs so that g(n) = j(n) ( v1 1 s G t t (n)(g(n)) t1 vs ts ) for some t1, . . . , ts.
(n) (n) (n) Write I = ∪z∈M Iz ∈ G for the image of the section generated by the identity mapping and denote the connected component of G(n) contain- (n) (n) ing I by Go . Theorem 3.1. Let G be a tame Lie pseudogroup at order n. Then any (n) (n) g ∈ Go is reachable by G vector elds.
(n) Proof. Denote by R|zo Go |zo the set of jets with source at zo that are reachable by G vector elds. Clearly R is non-empty. Our goal is to prove (n) that R|zo is both open and closed in Go |zo . (n) (n) (n) First, let go ∈ R|zo and write Zo = (go ). Choose a basis (n) w1, . . . , wr for T (n) G |z . By tameness, there are G vector elds vj ∈ XG go o
de ned in a neighborhood NZo of Zo in M so that (n) (n) wj = Vj (go ), j = 1, . . . , r. r i (n) Due to linearity, any w = a wi ∈ T (n) G |z is a linear combina- Pi=1 go o tion r i (n) (n) w = X a Vi (go ) i=1 (n) (n) of the vectors Vi (go ). Moreover, by shrinking NZo , if necessary, it is easy to verify that there is > 0 so that for any a = (a1, . . . , ar) with k a k < v r 1, the ow map t of v = a1v1 + + a vr is de ned on ( , ) NZo . By assumption, g(n) j(n) y1 ys o = zo ( t1 ts )
for some G vector elds y1, . . . , ys and for some t1, . . . , ts. It is clear that
there is a neighborhood Pzo of zo so that the composition v y1 ys z /2 t1 ts ( ) October 11, 2004 14:39 Proceedings Trim Size: 9in x 6in orbits
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r is de ned for all v = a1v1 + + a vr with k a k < 1 and z ∈ Pzo . Next consider the mapping
r (n) : {(a1, . . . , a )} → G |zo , r a , . . . , ar j(n) a1v1+ +a vr y1 ys , ( 1 ) = zo ( /2 t1 ts ) where k a k < 1. Then at a1 = = ar = 0, (∂ i ) = w . (10) a 2 i Hence the Jacobian of at a1 = = ar = 0 is non-degenerate, and so, (n) in particular, the image of contains an open neighborhood Q (n) of go go (n) (n) in G |z . Now by the de nition of , any g ∈ Q (n) is reachable by G o go (n) vector elds and consequently, R|zo Go |zo is open. (n) In order to show that R|zo is also closed in Go |zo , pick a sequence (n) (n) (n) {gi } R|zo of jets in R|zo converging to go ∈ Go |zo . By the rst part (n) (n) of the proof, there is a neighborhood Q (n) of go in Go |z so that any go o (n) (n) g ∈ Q (n) can be expressed as go
(n) v(n) (n) g = t (go ) (11) (n) for some G vector eld v. Choose i so large that g ∈ Q (n) . Then, by i go virtue of (11), we see that
(n) v(n) (n) go = t (gi ) (n) (n) for some G vector eld v. Consequently, go ∈ R|zo and R|zo Go |zo is closed. This completes the proof of the Theorem.
Next let V be a set of locally de ned vector elds on a manifold M. An orbit Op of V through p ∈ M is the set of points that can be reached from p by a composition of ow maps of vector elds in V,
q v1 vr p t , . . . , t R, v , . . . , v . Op = { = t1 tr ( ) | 1 r ∈ 1 r ∈ V }
We equip Op with the strongest topology that makes all the maps
t , . . . , t Rr v1 vr p ( 1 r) ∈ → t1 tr ( ) ∈ Op (12) continuous. One can easily verify that this topology is independent of the point p on the orbit. We call V everywhere de ned if every p ∈ M is contained in the domain of at least one v ∈ V. October 11, 2004 14:39 Proceedings Trim Size: 9in x 6in orbits
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Theorem 3.2. Let V be an everywhere de ned set of local, smooth vector elds on M and let Op be an orbit of V. Then Op admits a di erentiable structure compatible with the topology de ned by (12) in which it becomes an immersed submanifold of M.
Proof. A proof can be found in Ref. [12].
We write J n = J n(M, p) for the nth order extended jet bundle consisting of equivalence classes of p dimensional submanifolds of M under nth order contact and n : J n → M for the canonical projection. We denote the action of a local di eomorphism ϕ ∈ D on a jet z(n) ∈ J n by ϕ(n) z(n), where z = n(z(n)) is contained in the domain of ϕ. This action factors into an action of the di eomorphism jet bundle D(n) on J n in an obvious fashion. The in nitesimal generators of the action of D on J n are, by de nition, the prolongations2 to J n of local vector elds on M obtained in local coor- dinates by the usual prolongation formula. Similarly, for a Lie pseudogroup G the in nitesimal generators of the action of G on J n are, by de nition, prolongations of G vector elds to J n. We let g(n) X (J n) stand for the Lie algebra of in nitesimal generators of G. (n) (n) n We denote the g orbit of a point z ∈ J by Oz(n) and the orbit (n) n 0 of z under the action of G on J by Oz(n) . Then Oz(n) consists of the points
v1(n) vr (n) z(n), t1 tr
0 where each vi is a G vector eld. So obviously Oz(n) Oz(n) . Moreover, 0 (n) the orbit Oz(n) agrees with the orbit of z under the induced action of the pseudogroup jet bundle G(n), speci cally,
0 (n) (n) Oz(n) = G |z z . We call a Lie pseudogroup G connected if the subbundle G(n) D(n) is connected.
Theorem 3.3. Let G be a tame Lie pseudogroup at order n acting on M. Then the orbits of the action of G on J n are immersed submanifolds of J n.
Proof. First assume that G is connected. By virtue of tameness of G and the prolongation formula, the Lie algebra g(n) of in nitesimal generators is n (n) everywhere de ned on J . Thus by Theorem 3.2, the g orbits Oz(n) are October 11, 2004 14:39 Proceedings Trim Size: 9in x 6in orbits
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immersed submanifolds of J n. Hence, to conclude the proof of the Theorem 0 for connected pseudogroups, we only need to show that Oz(n) Oz(n) . For this, note that by Theorem 3.1, any jet g(n) ∈ G(n) can be expressed as
g(n) j(n) v1 vr , = z ( t1 tr ) (n) (n) where the vi are G vector elds and z = (g ). Consequently, g(n) z(n) v1 (n) vr (n) z(n), = t1 tr 0 which shows that Oz(n) Oz(n) . (n) Next assume that G is not connected and let G1 be a connected com- (n) (n) (n) ponent of G distinct from the connected component Go containing I . (n) (n) (n) (n) (n) Write G1 |z = G1 ∩ G |z and pick go ∈ G1 |z. Since G is tame, we can (n) (n) proceed as in the proof of Theorem 3.1 to show that any g ∈ G1 |z can be expressed as
g(n) v1 (n) vs (n) g(n) , = t1 tr ( o ) (n) n for some G vector elds v1, . . . , vr. This implies that the orbit of z ∈ J (n) (n) (n) (n) under G1 coincides with the orbit of go z under Go . This completes the proof of the Theorem.
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10. A. Rodrigues, The rst and second fundamental theorems of Lie for Lie pseudogroups, Amer. J. Math. 84, 265–282 (1962). 11. I. M. Singer, S. Sternberg, The in nite groups of Lie and Cartan. Part I (the transitive groups), J. Analyse Math. 15, 1–114 (1965). 12. H. J. Sussman, Orbits of families of vector elds and integrability of distri- butions, Trans. Amer. Math. Soc. 180, 171–188 (1973).