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Wiodzimierz M. Mikulski THERE EXISTS a BUNDLE FUNCTOR OF

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DEMONSTRATIO MATHEMATICA Vol. XXXVIII No 4 2005 Wiodzimierz M. Mikulski THERE EXISTS A BUNDLE FUNCTOR OF JET BUNDLE TYPE OF INFINITE ORDER IN THE FIRST FACTOR Abstract. It is known a complete description of all bundle functors on Mfm x Mf of finite order in the first factor. The most known example of such a bundle functor is the r-jet bundle functor JT on Mfm x Mf. In the present note an example of a bundle functor on Mfm x Mf of (essentially) infinite order in the first factor is constructed. The concept of bundle functor on the category Mf of manifolds and maps (or on the category Mfm of m-manifolds and local diffeomorphisms), the concept of natural transformations between bundle functors and the concept of order of bundle functor can be found in [1]. Let Mfm x Mf be the product of categories Mfm and Mf. The objects of Mfm x Mf are pairs (M,N), where M is an m-manifold and JV is a manifold. The morphisms (M,N) —• (M,N) are the pairs (<£>>/), where </?: M —> M is a local diffeomorphism between m-manifolds and f : N N is a map. Let TM be the category of fibered manifolds and their fibered maps. A bundle functor on MfmxMf is a covariant functor F : MfmxMf —• TM satisfying the following conditions (see [2]): (i) (Prolongation) For very m-manifold M and manifold N F(M,N) is a fibered manifold over M x N with projection p = PM,N = (a, ¡3) : F(M,N) —> M x N, and for every local diffeomorphism tp : M —M of two m-manifolds and every map / : N —> N of manifolds the induced mapping F(<p, f) : F(M, N) —> F(M, N) is a fibered manifold morphism over <pxf: MxN-^MxN. Key words and phrases: bundle functor, order of bundle functor, natural transforma- tion, jet. 1991 Mathematics Subject Classification: 58A05, 58A20. 962 W. M. Mikulski (ii) (Locality) If i = ¿1 x i2 : U = U\ x V\ —> M x N is an inclusion with open Ui and Vi, then F(U\,V{) = p~j^N{U) and F(ii,i2) is the inclusion of PU\N(U) into F(M,N). Prom now on let r be a natural number. A very important example of bundle functor on Mfm x Mf is the holonomic r-jets bundle functor Jr. More precisely, for every pair of a m- r manifold M and a manifold N we have the bundle J (M, N) = {jx(g) | g : M —> TV, x 6 M} of (holonomic) r-jets of maps M —• N. We have the source r projection a : J (M,N) —• M by a(jx(g)) = x and the target projection (3 : Jr(M,N) —>•_N by 0{jl{g)) = g(x)._Every pair of a local diffeomor- phism ip : M —> M and a map f : N —> N induces fibered map Jr(<p, f) : r r r 1 J (M,N) -» J (M,N) over y x / by J (<p, f)(fx(g)) = j^x){f o g o y^ ) . r Another example is the bundle functor J on Mfm x Mf of non- holonomic r-jets in the sense of Ehresman. To present a general example we recall that a group homomorphism r r 7n Tn H : G m —• N£(G) of the differential group G m = OTvJJ(R ,M )0 into the group N£(G) of natural equivalences of a bundle functor G on Mf is called N r smooth if for any manifold N the action H : G m x G(N) —> G(N) given N by H {£,V) = H(£)n(V) is smooth. EXAMPLE 1. ([2]) The associated, bundle. Let G be a bundle functor on Mf r and H : G m —» M£{G) be a group homomorphism that is smooth in the above sense. Then every G(N) is a G^-space and we put (G,H)(M,N) = Pr(M)[G(N),HN] , where Pr(M)[G(N), HN] is the associated bundle with Pr(M)=the r-th or- der frame bundle of an m-manifold M and fibre G(N). We define the source and the target projections a : (G, H)(M, N) —> M by a(<Jo(7), v>) = 7(0) r and p : (G,H)(M,N) ^ N by /?(< j 0(-y),v >) = y for all f0(7) € r P (M) and v € Gy(N), y € N. Every pair of a local diffeomorphism ip : M —> M and a map / : N —• N induce a fibered map (G, H)(tp, f) = (G,H)(M,N) - (G,H)(M,N) by r (G, #)(</>, /)(< j0 (7), t; >) =< X(<p o 7), G(/)(w) > . The correspondence (G, ii) : Mfm x Mf —> ^"jM is a bundle functor. Let F be a bundle functor on Mfmx Mf. We say that F is of order less or equal to r in the first factor if from jx((pi) — jxi^) it follows Fx(tp\, f) — Fx(<p2,f) for every local diffeomorphisms y>i,v?2 : M —»• M between m- manifolds, every map f : N —> N and every point x € M. The bundles Jr and Jr are of order r in the first factor. The associated bundle (G, H) is of order less or equal to r in the first factor. A bundle functor of jet bundle type 963 Let F be a bundle functor on Mfm x Mf of finite order r in the first fac- F F m tor. Functor F induces a bundle functor G _on Mf by G {N) = F0 (R , N) F F F and G (f) = F0(id&J) : G (N) G (N) for / : N N and a (smooth) F r F group homomorphism H : G m —»J\TS(G ) by F H UE{<P)Mv) = F(iPlidN)(v) for any manifold N any v G GF(N) and any jo(<p) G GJ^. We have the following classification theorem. THEOREM 1 ([2]). Every bundle functor F on Mfn x Mf of finite order r in the first factor is equivalent with the associated bundle (GF,HF). So, it is interesting the following question. Is there a bundle functor F on Mfm x Mf of (essentially) infinite order in the first factor ? The following example shows that the answer to the above question is positive. EXAMPLE 2. There exists a bundle functor on Mfm x Mf of (essentially) infinite order in the first factor. For any r = 1,2,3,... we define a vector bundle functor Gr : Mf VB by Gr(N) = T(N) x Vr and Gr(f) = T(f) x r r r m idvr : G (N) G (N) for any map / :N N, where T> = J£(M ,R) is the vector space of r-jets at 0 of maps Rm —> R. Let dr = dim(GZ{W)). We set G(N) = ®(ff«Gk(N)). FCE N For every manifold N, G(N) is a finite dimensional manifold because for k > dim(N) the bundle f\dk Gh(N) is zero-bundle. Hence the direct sum in the definition of G(N) is in reality a finite sum. For a mapping / : N —> N the induced mapping G(f) : G(N) —• G(N) is defined in the natural way from Adfc Gk(f). In this way we obtain vector bundle functor G on Mf of order 1. m m r The group G™ = invJg>{R ,R )0 of invertible oo-jets acts on V by the pull-back. Hence for r = 1,2,... we have a group homomorphism Hr : Gm N£vb(Gr) into the group of vector bundle natural equivalences of G by Hr(On : Gr(N) - Gr(N), £T(OAT(«,»/) = « 6 T(N), $ G rj G T>r, where the dot is the action. Hence for any £ G G^ and a manifold N we have a vector natural equivalence : G(N) G(N) which is dk k dk k dk k defined in natural way from /\ H {£)N : f\ G (N) /\ G (N). This defines a group homomorphism H : G^ —> J\f£vb(G). For any manifold N the corresponding action H of G^ on G(N) can be factorized by a smooth action of on G(N) for some finite In- Hence we can proceed similarly as in Example 1 for above G and H and obtain bundle functor (G, H) on Mfm x Mf. This bundle functor is of (essentially) 964 W. M. Mikulski infinite order in the first factor because the homomorphism H can not be l factorized by a group homomorphism G m —• M£(G) with finite I. References [1] I. Kolar et all, Natural Operations in Differential Geometry, Springer-Verlag, Berlin 1993. [2] I. Kolaf, W. M. Mikulski, On the fiber product preserving bundle functors, Diff. Geom. Appl. 11 (1999), 105-115. INSTITUTE OF MATHEMATICS JAGIELLONIAN UNIVERSITY ul. Reymonta 4 30-059 KRAK6W, POLAND e-mail: [email protected] Received June 24, 2004• .
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