Let (M,F) be aFolm,n-object

ARCHIVUM MATHEMATICUM (BRNO) Tomus 44 (2008), 115–118

CANONICAL 1-FORMS ON HIGHER ORDER ADAPTED FRAME BUNDLES

Jan Kurek and Włodzimierz M. Mikulski

Abstract. Let (M,F) be a foliatedm+n-dimensional manifold M with n-dimensional foliationF. LetV be a finite dimensional vector space overR.

We describe all canonical (Fol_m,n-invariant)V-valued 1-forms Θ :T P^r(M,F)

→V on ther-th order adapted frame bundleP^r(M,F) of (M,F).

All manifolds and maps are assumed to be of classC^∞.

A definition of foliations can be found in [2]. Let Folm,n be the category of foliated m+n-dimensional manifolds with n-dimensional foliations and their foliation respecting local diffeomorphisms. Let (M,F) be aFolm,n-object. Then we have an adaptedr-th order frame bundle

P^r(M,F) =

j₀^rϕ|ϕ: (R^m+n,F^m,n)→(M,F) is aFolm,n-map overM of (M,F) with the target projection, whereF^m,n=

{a}×Rⁿ

a∈R^m is the n-dimensional canonical foliation on R^m+n. We see thatP^r(M,F) is a principal bundle with the standard Lie groupG^r_m,n=P^r(R^m+n,F^m,n)₀ (with the multipli- cation given by the composition of jets) acting on the right onP^r(M,F) by the com- position of jets. Every Folm,n-mapψ: (M1,F1)→(M2,F2) induces a local fibred diffeomorphism (even a principal bundle local isomorphism)P^rψ:P^r(M1,F1)→ P^r(M2,F2) given byP^rψ(j₀^rϕ) =j₀^r(ψ◦ϕ).

Definition 1. LetV be a finite dimensional vector space overR. We recall that a Folm,n-canonicalV-valued 1-form Θ onP^ris a family ofFolm,n-invariantV-valued 1-forms Θ_(M,F):T P^r(M,F)→V onP^r(M,F) for anyFolm,n-object (M,F). The invariance means that theV-valued 1-forms Θ(M₁,F1)and Θ(M₂,F2)areP^rΦ-related (P^rΦ^∗Θ(M₂,F2)= Θ(M₁,F1)) for anyFolm,n-map Φ : (M1,F1)→(M2,F2).

It is rather-known the following Folm,n-canonical R^m+n-valued 1-form on P¹(M,F).

Example 1. For everyFolm,n-object (M,F) we define anR^m+n-valued 1-form θ_(M,F)onP¹(M,F) as follows. Consider the target projectionβ:P¹(M,F)→M

2000Mathematics Subject Classification:Primary: 58A20; Secondary: 58A32.

Key words and phrases:foliated manifold, infinitesimal automorphism, higher order adapted frame bundle, canonical 1-form.

Received July 24, 2007, revised October 2007. Editor I. Kolář.

116 J. KUREK AND W. M. MIKULSKI

given by β(j₀^rϕ) =ϕ(0), an element u=j₀¹ψ ∈P¹(M,F) and a tangent vector X =j₀¹c∈T_u P¹(M,F)

. We define the formθ=θ_(M,F)by

θ(X) =u⁻¹◦T β(X) =j₀¹(ψ⁻¹◦β◦c)∈T₀R^m+n =R^m+n.

Let us notice that if n = 0 then (M,F) = M and P¹(M,F) = P¹(M) and θ_(M,F)=θM is the well-known canonicalR^m-valued 1-form on the frame bundle P¹M.

To present a general example ofFolm,n-canonicalV-valued 1-forms onP^r we need the following lemma.

Lemma 1. Let(M,F)be aFolm,n-object. Then any vectorv∈TwP^r(M,F), w∈ P^r(M,F)

x,x∈M is of the formP^rX_w for some infinitesimal automorphism X ∈ X(M,F), where P^rX ∈ X P^r(M,F)

is the flow lifting ofX toP^r(M,F).

Moreoverj^r_xX is uniquely determined.

Remark 1. We inform that a vector fieldXonM is an infinitesimal automorphism of (M,F) iff the flow {ExptX} of X is formed by local Folm,n-maps (M,F)→ (M,F) or (equivalently) [X, Y] is tangent toF for anyY tangent toF. The space X(M,F) of all infinitesimal automorphisms of (M,F) is a Lie subalgebra inX(M).

Given an infinitesimal automorphism X ∈ X(M,F), the flow lifting P^rX is a vector field onP^r(M,F) such that if{Φ_t} is the flow ofX then{P^r(Φ_t)}is the flow ofP^rX. (Since ΦtareFolm,n-maps we can apply functorP^r.)

Proof of Lemma 1. We can of course assume that (M,F) = (R^m+n,F^m,n) and x= 0. SinceP^r(R^m+n,F^m,n) is in usual way a principal subbundle ofP^r(R^m+n), then by well-known manifold version of the lemma, we find X∈ X(R^m+n) such thatv=P^rXwand j₀^rX is determined uniquely. An infinitesimal automorphism Y ∈ X(R^m+n,F^m,n) gives P^rYw which is tangent to P^r(R^m+n,F^m,n). On the other hand the dimension ofP^r(R^m+n,F^m,n) and the dimension of the space of r-jets j₀^rY of Y ∈ X(R^m+n,F^m,n) are equal. Then the lemma follows from the dimension argument because flow operators are linear.

Example 2. Let λ: J₀^r−1 TInf−Aut(R^m+n,F^m,n)

→ V be an R-linear map, whereJ₀^r−1 TInf−Aut(R^m+n,F^m,n)

is the vector space of all (r−1)-jetsj₀^r−1X at 0 ∈ R^m+n of infinitesimal automorphisms X ∈ X(R^m+n,F^m,n). Given a Folm,n-object (M,F), we define aV-valued 1-form Θ^λ_(M,F):T P^r(M,F)→V on P^r(M,F) as follows. Letv ∈TwP^r(M,F), w=j^r₀ϕ∈(P^r(M,F))x, x∈M. By Lemma 1,v=P^rX_wfor some infinitesimal automorphismX∈ X(M,F), andj_x^rX is uniquely determined. Then it is determined the (r−1)-jetj₀^r−1 (ϕ⁻¹)_∗X

at 0 of the image (ϕ⁻¹)_∗X ofX byϕ⁻¹. We put

Θ^λ_(M,F)(v) :=λ j₀^r−1((ϕ⁻¹)∗X) .

Clearly, Θ^λ={Θ^λ_(M,F)}is a Folm,n-canonicalV-valued 1-form onP^r.

The main result of the present short note is the following classification theorem.

Theorem 1. AnyFolm,n-canonicalV-valued1-form onP^risΘ^λ for some unique R-linear mapλ:J₀^r−1 T_Inf−Aut(R^m+n,F^m,n)

→V.

CANONICAL 1-FORMS 117

In the proof of Theorem 1 we use the following fact.

Lemma 2. Let X, Y ∈ X(M,F) be infinitesimal automorphisms of(M,F) and x ∈ M be a point. Suppose that j_x^r−1X = j_x^r−1Y and Xx is not-tangent to F.

Then there exists a(locally defined) Folm,n-mapΦ : (M,F)→(M,F)such that j_x^r(Φ) =j_x^r(idM)andΦ∗X =Y near x.

Proof. A direct modification of the proof of Lemma 42.4 in [1].

Proof of Theorem 1. Let Θ be aFol_m,n-canonicalV-valued 1-form onP^r. We must defineλ:J₀^r−1 TInf−Aut(R^m+n,F^m,n)

→V by λ(ξ) := Θ_(Rm+n,F^m,n) P^rX˜_j^r

0(id_Rm+n)

for allξ∈J₀^r−1 TInf−Aut(R^m+n,F^m,n)

, where givenξin question, ˜X is a unique (a unique germ at 0 of) infinitesimal automorphism of (R^m+n,F^m,n) such that j₀^r−1X˜ = ξ and the coefficients of ˜X with respect to the basis of the canonical vector fields _∂x^∂_i ∈ X(R^m+n,F^m,n) (i= 1, . . . , m+n) are polynomials of degree

≤r−1.

We are going to show that Θ = Θ^λ. Because of theFol_m,n-invariance it remains to show that

(∗) Θ_(Rm+n,F^m,n)(v) = Θ^λ_(Rm+n,F^m,n)(v) for anyv∈T_j^r

0(id_Rm+n)P^r(R^m+n,F^m,n).

By the definition of λand Θ^λ we have (∗) for anyvof the formP^rX˜_j^r

0(id_Rm+n), where ˜X is an infinitesimal automorphism of (R^m+n,F^m,n) such that the coef- ficients of ˜X with respect to the basis of canonical vector fields on R^m+n are polynomials of degree≤r−1.

Now, let v be arbitrary in question. Then by Lemma 1, v is of the formv = P^rX_j^r

0(id_Rm+n) for some infinitesimal automorphismX of (R^m+n,F^m,n). Clearly (because of a density argument), we can additionally assume thatX0is not tangent to F^m,n. Let ˜X be an infinitesimal automorphism of (R^m+n,F^m,n) such that j₀^r−1X˜ =j^r−1₀ Xand the coefficients of ˜Xwith respect to the basis of constant vector fields onR^m+n are polynomials of degree≤r−1. Let ˜v=P^rX˜_j^r

0(id_Rm+n). Then (we have observed above) it holds Θ(R^m+n,F^m,n)(˜v) = Θ^λ_(Rm+n,F^m,n)(˜v). On the other hand by Lemma 2, there is aFolm,n-map Φ : (R^m+n,F^m,n)→(R^m+n,F^m,n) such thatj^r₀Φ =j₀^r(id_Rm+n) and Φ_∗X˜ =X near 0. Since j₀^rΦ = id, Φ preserves j₀^r(idR^m+n). Then since Φ_∗X˜ =X, Φ sends ˜vintov. Then because of the invariance of Θ and Θ^λwith respect to Φ, we obtain Θ(R^m+n,F^m,n)(v) = Θ(R^m+n,F^m,n)(˜v) = Θ^λ_(Rm+n,F^m+n)(˜v) = Θ^λ_(Rm+n,F^m,n)(v).

In the caser= 1, we haveJ₀⁰(TInf−Aut(R^m+n,F^m,n)) ˜=R^m+n. Then by Theo- rem 1, the vector space of Folm,n-canonical V-valued 1-forms on P¹ is (m+ n) dim(V)-dimensional. Then (because of a dimension argument) we have.

Corollary 1. AnyFolm,n-canonicalV-valued1-formΘ ={Θ_(M,F)} on P¹ is of the form

Θ_(M,F)=λ◦θ_(M,F):T P¹(M,F)→V

118 J. KUREK AND W. M. MIKULSKI

for some unique linear map λ: R^m+n→V, whereθ={θ_(M,F)} is the canonical R^m+n-valued 1-form on P¹ from Example 1.

Example 3. It is easy to see that

J₀^r−1 T_Inf−Aut(R^m+n,F^m,n)

˜

=R^m+n⊕ Lie(G^r−1_m,n).

Thus by Example 2 forλ= id_Rm+n⊕Lie(G^r−1_m,n)we have aFolm,n-canonicalR^m,n⊕ Lie(G^r−1_m,n)-valued 1-form

θ^r_(M,F):= Θ^{idRm+m⊕Lie(Gr−1m,n)}:T P^r(M,F)→R^m+n⊕ Lie(G^r−1_m,n)

on P^r. For r= 1, we haveθ¹ =θ as in Example 1. In particular, forn= 0 we obtain the well-known canonical R^m⊕ Lie(G^r−1_m )-valued 1-form

θ_M^r : P^rM →R^m⊕ Lie(G^r−1_m ) on ther-order frame bundleP^rM.

By similar arguments as for Corollary 1 we have.

Corollary 2. AnyFol_m,n-canonicalV-valued1-formΘ ={Θ_(M,F)} on P^r is of the form

Θ_(M,F)=λ◦θ_(M,F)^r :T P^r(M,F)→V

for some unique linear map λ: R^m+n ⊕ Lie(G^r−1_m,n) → V, where θ^r is as in Example 3.

In particular(forn= 0), any canonicalV-valued 1-formΘ ={ΘM} onP^rM is of the form

Θ_M =λ◦θ_M^r :T P^rM →V for some unique linear map λ:R^m⊕ Lie(G^r−1_m )→V.

Remark. Recently, we obtained (by a modification of the above paper) a similar result on gauge invariant vector valued 1-forms on higher order principal pro- longations of principal bundles. The paper will appear in Lobachevskii Math. J.

2008.

References

[1] Kolář, I., Michor, P. W., Slovák, J.,Natural Operations in Differential Geometry, Springer Verlag, 1993.

[2] Wolak, R. A.,Geometric structures on foliated manifolds, Publications del Departamento de Geometria y Topologia, Universidad de Santiago de Compostella76(1989).

Institute of Mathematics, Maria Curie-Sklodowska Univesity Pl. M. Curie-Sklodowskiej 1, Lublin, Poland

E-mail:[email protected]

Institute of Mathematics, Jagiellonian University ul. Reymonta 4, Kraków, Poland

E-mail:[email protected]