summary:Let $(M,\mathcal{F})$ be a foliated $m+n$-dimensional manifold $M$ with $n$-dimensional foliation $\mathcal{F}$. Let $V$ be a finite dimensional vector space over $\mathbf{R}$. We describe all canonical (${\mathcal{F}}\mbox {\it ol}_{m,n}$-invariant) $V$-valued $1$-forms $\Theta \colon TP^r(M,{\mathcal{F}}) \rightarrow V$ on the $r$-th order adapted frame bundle $P^r(M,\mathcal{F})$ of $(M,\mathcal{F})$

Kurek, Jan

Mikulski, Włodzimierz M.

English

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumJan Kurek; Włodzimierz M. MikulskiCanonical 1-forms on higher order adapted frame bundlesArchivum Mathematicum, Vol. 44 (2008), No. 2, 115--118Persistent URL: http://dml.cz/dmlcz/116928Terms of use:© Masaryk University, 2008Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVUM MATHEMATICUM (BRNO)Tomus 44 (2008), 115–118CANONICAL 1-FORMS ON HIGHER ORDER ADAPTEDFRAME BUNDLESJan Kurek and Włodzimierz M. MikulskiAbstract. Let (M,F) be a foliated m + n-dimensional manifold M withn-dimensional foliation F . Let V be a finite dimensional vector space over R.We describe all canonical (Folm,n-invariant) V -valued 1-forms Θ: TP r(M,F)→ V on the r-th order adapted frame bundle P r(M,F) of (M,F).All manifolds and maps are assumed to be of class C∞.A definition of foliations can be found in [2]. Let Folm,n be the category offoliated m + n-dimensional manifolds with n-dimensional foliations and theirfoliation respecting local diffeomorphisms. Let (M,F) be a Folm,n-object. Thenwe have an adapted r-th order frame bundleP r(M,F) ={jr0ϕ | ϕ : (Rm+n,Fm,n)→ (M,F) is a Folm,n-map}over M of (M,F) with the target projection, where Fm,n ={{a}×Rn}a∈Rm is then-dimensional canonical foliation on Rm+n. We see that P r(M,F) is a principalbundle with the standard Lie group Grm,n = P r(Rm+n,Fm,n)0 (with the multipli-cation given by the composition of jets) acting on the right on P r(M,F) by the com-position of jets. Every Folm,n-map ψ : (M1,F1)→ (M2,F2) induces a local fibreddiffeomorphism (even a principal bundle local isomorphism) P rψ : P r(M1,F1)→P r(M2,F2) given by P rψ(jr0ϕ) = jr0(ψ ◦ ϕ).Definition 1. Let V be a finite dimensional vector space over R. We recall that aFolm,n-canonical V -valued 1-form Θ on P r is a family of Folm,n-invariant V -valued1-forms Θ(M,F) : TP r(M,F)→ V on P r(M,F) for any Folm,n-object (M,F). Theinvariance means that the V -valued 1-forms Θ(M1,F1) and Θ(M2,F2) are P rΦ-related(P rΦ∗Θ(M2,F2) = Θ(M1,F1)) for any Folm,n-map Φ: (M1,F1)→ (M2,F2).It is rather-known the following Folm,n-canonical Rm+n-valued 1-form onP 1(M,F).Example 1. For every Folm,n-object (M,F) we define an Rm+n-valued 1-formθ(M,F) on P 1(M,F) as follows. Consider the target projection β : P 1(M,F)→M2000 Mathematics Subject Classification: Primary: 58A20; Secondary: 58A32.Key words and phrases: foliated manifold, infinitesimal automorphism, higher order adaptedframe bundle, canonical 1-form.Received July 24, 2007, revised October 2007. Editor I. Kolář.116 J. KUREK AND W. M. MIKULSKIgiven by β(jr0ϕ) = ϕ(0), an element u = j10ψ ∈ P 1(M,F) and a tangent vectorX = j10c ∈ Tu(P 1(M,F)). We define the form θ = θ(M,F) byθ(X) = u−1 ◦ Tβ(X) = j10(ψ−1 ◦ β ◦ c) ∈ T0Rm+n = Rm+n .Let us notice that if n = 0 then (M,F) = M and P 1(M,F) = P 1(M) andθ(M,F) = θM is the well-known canonical Rm-valued 1-form on the frame bundleP 1M .To present a general example of Folm,n-canonical V -valued 1-forms on P r weneed the following lemma.Lemma 1. Let (M,F) be a Folm,n-object. Then any vector v ∈ TwP r(M,F), w ∈(P r(M,F))x, x ∈M is of the form PrXw for some infinitesimal automorphismX ∈ X (M,F), where PrX ∈ X(P r(M,F))is the flow lifting of X to P r(M,F).Moreover jrxX is uniquely determined.Remark 1. We inform that a vector field X on M is an infinitesimal automorphismof (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 to F for any Y tangent to F . The spaceX (M,F) of all infinitesimal automorphisms of (M,F) is a Lie subalgebra in X (M).Given an infinitesimal automorphism X ∈ X (M,F), the flow lifting PrX is avector field on P r(M,F) such that if {Φt} is the flow of X then {P r(Φt)} is theflow of PrX. (Since Φt are Folm,n-maps we can apply functor P r.)Proof of Lemma 1. We can of course assume that (M,F) = (Rm+n,Fm,n) andx = 0. Since P r(Rm+n,Fm,n) is in usual way a principal subbundle of P r(Rm+n),then by well-known manifold version of the lemma, we find X ∈ X (Rm+n) suchthat v = PrXw and jr0X is determined uniquely. An infinitesimal automorphismY ∈ X (Rm+n,Fm,n) gives PrYw which is tangent to P r(Rm+n,Fm,n). On theother hand the dimension of P r(Rm+n,Fm,n) and the dimension of the space ofr-jets jr0Y of Y ∈ X (Rm+n,Fm,n) are equal. Then the lemma follows from thedimension argument because flow operators are linear. Example 2. Let λ : Jr−10(TInf−Aut(Rm+n,Fm,n))→ V be an R-linear map,where Jr−10(TInf−Aut(Rm+n,Fm,n))is the vector space of all (r − 1)-jets jr−10 Xat 0 ∈ Rm+n of infinitesimal automorphisms X ∈ X (Rm+n,Fm,n). Given aFolm,n-object (M,F), we define a V -valued 1-form Θλ(M,F) : TP r(M,F)→ V onP r(M,F) as follows. Let v ∈ TwP r(M,F), w = jr0ϕ ∈ (P r(M,F))x, x ∈ M . ByLemma 1, v = PrXw for some infinitesimal automorphism X ∈ X (M,F), and jrxXis uniquely determined. Then it is determined the (r − 1)-jet jr−10((ϕ−1)∗X)at 0of the image (ϕ−1)∗X of X by ϕ−1. We putΘλ(M,F)(v) := λ(jr−10 ((ϕ−1)∗X)).Clearly, Θλ = {Θλ(M,F)} is a Folm,n-canonical V -valued 1-form on P r.The main result of the present short note is the following classification theorem.Theorem 1. Any Folm,n-canonical V -valued 1-form on P r is Θλ for some uniqueR-linear map λ : Jr−10(TInf−Aut(Rm+n,Fm,n))→ V .CANONICAL 1-FORMS 117In the proof of Theorem 1 we use the following fact.Lemma 2. Let X,Y ∈ X (M,F) be infinitesimal automorphisms of (M,F) andx ∈ M be a point. Suppose that jr−1x X = jr−1x Y and Xx is not-tangent to F .Then there exists a (locally defined) Folm,n-map Φ: (M,F) → (M,F) such thatjrx(Φ) = jrx(idM ) and Φ∗X = Y near x.Proof. A direct modification of the proof of Lemma 42.4 in [1]. Proof of Theorem 1. Let Θ be a Folm,n-canonical V -valued 1-form on P r. Wemust define λ : Jr−10(TInf−Aut(Rm+n,Fm,n))→ V byλ(ξ) := Θ(Rm+n,Fm,n)(PrX̃jr0 (idRm+n ))for all ξ ∈ Jr−10(TInf−Aut(Rm+n,Fm,n)), where given ξ in question, X̃ is a unique(a unique germ at 0 of) infinitesimal automorphism of (Rm+n,Fm,n) such thatjr−10 X̃ = ξ and the coefficients of X̃ with respect to the basis of the canonicalvector fields ∂∂xi ∈ X (Rm+n,Fm,n) (i = 1, . . . ,m+ n) are polynomials of degree≤ r − 1.We are going to show that Θ = Θλ. Because of the Folm,n-invariance it remainsto show that(∗) Θ(Rm+n,Fm,n)(v) = Θλ(Rm+n,Fm,n)(v)for any v ∈ Tjr0 (idRm+n )Pr(Rm+n,Fm,n).By the definition of λ and Θλ we have (∗) for any v of the form PrX̃jr0 (idRm+n ),where X̃ is an infinitesimal automorphism of (Rm+n,Fm,n) such that the coef-ficients of X̃ with respect to the basis of canonical vector fields on Rm+n arepolynomials of degree ≤ r − 1.Now, let v be arbitrary in question. Then by Lemma 1, v is of the form v =PrXjr0 (idRm+n ) for some infinitesimal automorphism X of (Rm+n,Fm,n). Clearly(because of a density argument), we can additionally assume that X0 is not tangentto Fm,n. Let X̃ be an infinitesimal automorphism of (Rm+n,Fm,n) such thatjr−10 X̃ = jr−10 X and the coefficients of X̃ with respect to the basis of constant vectorfields on Rm+n are polynomials of degree ≤ r − 1. Let ṽ = PrX̃jr0 (idRm+n ). Then(we have observed above) it holds Θ(Rm+n,Fm,n)(ṽ) = Θλ(Rm+n,Fm,n)(ṽ). On theother hand by Lemma 2, there is a Folm,n-map Φ: (Rm+n,Fm,n)→ (Rm+n,Fm,n)such that jr0Φ = jr0(idRm+n) and Φ∗X̃ = X near 0. Since jr0Φ = id, Φ preservesjr0(idRm+n). Then since Φ∗X̃ = X, Φ sends ṽ into v. Then because of the invarianceof Θ and Θλ with respect to Φ, we obtain Θ(Rm+n,Fm,n)(v) = Θ(Rm+n,Fm,n)(ṽ) =Θλ(Rm+n,Fm+n)(ṽ) = Θλ(Rm+n,Fm,n)(v). In the case r = 1, we have J00 (TInf−Aut(Rm+n,Fm,n))=̃Rm+n. Then by Theo-rem 1, the vector space of Folm,n-canonical V -valued 1-forms on P 1 is (m +n) dim(V )-dimensional. Then (because of a dimension argument) we have.Corollary 1. Any Folm,n-canonical V -valued 1-form Θ = {Θ(M,F)} on P 1 is ofthe formΘ(M,F) = λ ◦ θ(M,F) : TP 1(M,F)→ V118 J. KUREK AND W. M. MIKULSKIfor some unique linear map λ : Rm+n → V , where θ = {θ(M,F)} is the canonicalRm+n-valued 1-form on P 1 from Example 1.Example 3. It is easy to see thatJr−10(TInf−Aut(Rm+n,Fm,n))=̃Rm+n ⊕ Lie (Gr−1m,n) .Thus by Example 2 for λ = idRm+n⊕Lie (Gr−1m,n) we have a Folm,n-canonical Rm,n ⊕Lie (Gr−1m,n)-valued 1-formθr(M,F) := ΘidRm+m⊕Lie (Gr−1m,n) : TP r(M,F)→ Rm+n ⊕ Lie (Gr−1m,n)on P r. For r = 1, we have θ1 = θ as in Example 1. In particular, for n = 0 weobtain the well-known canonical Rm ⊕ Lie (Gr−1m )-valued 1-formθrM : P rM → Rm ⊕ Lie (Gr−1m )on the r-order frame bundle P rM .By similar arguments as for Corollary 1 we have.Corollary 2. Any Folm,n-canonical V -valued 1-form Θ = {Θ(M,F)} on P r is ofthe formΘ(M,F) = λ ◦ θr(M,F) : TP r(M,F)→ Vfor some unique linear map λ : Rm+n ⊕ Lie (Gr−1m,n) → V , where θr is as inExample 3.In particular (for n = 0), any canonical V -valued 1-form Θ = {ΘM} on P rMis of the formΘM = λ ◦ θrM : TP rM → Vfor some unique linear map λ : Rm ⊕ Lie (Gr−1m )→ V .Remark. Recently, we obtained (by a modification of the above paper) a similarresult 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, SpringerVerlag, 1993.[2] Wolak, R. A., Geometric structures on foliated manifolds, Publications del Departamento deGeometria y Topologia, Universidad de Santiago de Compostella 76 (1989).Institute of Mathematics, Maria Curie-Sklodowska UnivesityPl. M. Curie-Sklodowskiej 1, Lublin, PolandE-mail: kurek@hektor.umcs.lublin.plInstitute of Mathematics, Jagiellonian Universityul. Reymonta 4, Kraków, PolandE-mail: mikulski@im.uj.edu.pl

Canonical 1-forms on higher order adapted frame bundles

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Canonical 1-forms on higher order adapted frame bundles

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Institute of Mathematics AS CR, v. v. i.