SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS

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1 SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS Ivan Kolář Abstract. Let F be a fiber product preserving bundle functor on the category FM m of the proper base order r. We deduce that the r-th principal gauge prolongation W r P of a principal bundle P is a reduction of the principal bundle W F P constructed in [1]. In certain special cases, we solve the problem up to what extent F is determined by the underlying group homomorphism. According to [7], every fiber product preserving bundle functor F on the category FM m of fibered manifolds with m-dimensional bases and fibered morphisms with local diffeomorphisms as base maps can be characterized by a triple (A, H, t), where A is a Weil algebra, H : G r m Aut A is a group homomorphism of the r-th jet group in dimension m into the group of all algebra automorphisms of A and t: D r m A is an equivariant algebra homomorphism, D r m = J r 0(R m, R). In [1], the principal bundle W F P was defined for every principal bundle P and it was deduced that the F -prolongation of an associated bundle E = P [S] is an associated bundle F E = W F P [T A S]. In Definition 1 of the present paper, we introduce the concept of the proper base order of F. Lemma 1 is a general algebraic result concerning equivariant algebra homomorphisms D r m A. Using Lemma 1, we deduce, in Proposition 1, that if r is the proper base order of F, then the canonical map J r Y F Y is injective for every fibered manifold Y. This implies easily that the r-th principal gauge prolongation W r P of P, [2], [6], is a reduction of W F P. Hence F E can be interpreted as a fiber bundle associated to W r P as well. The second part of the present paper is devoted to the following problem. Given H : G r m Aut A, what are all possible t: D r m A such that (A, H, t) is a fiber product preserving bundle functor of FM m? In Section 4 we deduce that for holonomic and semiholonomic r-jets, 2000 Mathematics Subject Classification : 58A20, 58A32. Key words and phrases : principal gauge-like prolongation, Weil algebra, jet bundle, semiholonomic r-jet category. The author was supported by the Ministry of Education of the Czech Republic under the project MSM and by GACR under the grant 201/09/

2 2 IVAN KOLÁŘ r 2, the only two possibilities are the r-th horizontal and the r-th vertical prolongation of a fibered manifold. In the case of nonholonomic r-jets, there are more possibilities. In Section 5 we describe completely the case r = 2. All manifolds and maps are assumed to be infinitely differentiable. Unless otherwise specified, we use the terminology and notations from the book [6]. 1. Decomposition lemma. By [4], the group Aut D r m of all algebra automorphisms of D r m coincides with G r m. The corresponding map G r m D r m D r m is the jet composition (g, X) X g, g G r m, X D r m. Consider an arbitrary Weil algebra A and a group homomorphism H : G r m Aut A. An algebra homomorphism µ: D r m A is called H-equivariant, if µ(x g) = H(g) ( µ(x) ), X D r m, g G r m. In particular, the jet projection πs r : G r m G s m, s r, is a group homomorphism and the jet projection βs r : D r m D s m is a πs-equivariant r algebra homomorphism. Lemma 1. For every H-equivariant algebra homomorphism µ: D r m A, there exists s r and an injective H s -equivariant algebra homomorphism i: D s m A such that µ = i β r s, where H s : G s m Aut A is the group homomorphism determined by H = H s π r s. Proof. Write I = Ker µ and L r m,m = J0(R r m, R m ) 0. Then G r m L r m,m is open and dense and the jet composition D r m L r m,m D r m is a polynomial map. We know that X I and g G r m implies X g I. By continuity, this holds for every g L r m,m. Now we can apply our result from [3]. Write Ĩ for the inverse image of I in the algebra E(m) of germs of smooth functions on R m at 0. Then Ĩ has the substitution property: if ξ Ĩ and γ is a germ of an origin preserving smooth map R m R m, then ξ γ Ĩ. By Lemma 2 from [3], Ĩ = ms+1 (m), where m(m) is the maximal ideal of E(m). Hence I = m s+1 (m) / m r+1 (m) =: Is r. Write Ks r = Ker πs. r Then we have i: D s m A, i(x + Is r ) = µ(x) and H s (gks r ) ( i(x + Is r ) ) = i(x g + Is r ). 2. Fiber product preserving bundle functors. We reformulate the description of a fiber product preserving bundle (in short: f.p.p.b.) functor F on FM m, [7], by using an additional concept of the proper base order of F. By Section 13 of [7], F has finite order. The construction of product bundles and product morphisms defines an injection

3 SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS3 i: Mf m Mf FM m, where Mf denotes the category of all manifolds and Mf m means the category of m-dimensional manifolds and local diffeomorphisms. In general, a bundle functor Φ on Mf m Mf is said to be of order r in the first factor, if for every g, g : M M and f : N N, j r xg = j r xg implies Φ x (g, f) = Φ x (g, f): Φ x (M, N) Φ g(x) (M, N ), x M, [7]. Consider the case F := F i. Definition 1. The minimal order of F in the first factor is called the proper base order of F. We recall that the product preserving bundle functors on Mf are in bijection with the functors T A determined by a Weil algebra A and the natural transformations T A 1 T A 2 are in bijection with the algebra homomorphisms µ: A 1 A 2, [4], [6]. We write µ N : T A 1 N T A 2 N for the value of µ on N. According to [7], F induces a functor F 0 on Mf by F 0 N = F 0 (R m, N) and F 0 f : F 0 N 1 F 0 N 2 is the restriction and corestriction of F (id R m, f), f : N 1 N 2. Since F 0 preserves products, it is a Weil functor T A. Consider further an origin preserving diffeomorphism ϕ of R m. If r is the proper base order of F, then the restriction and corestriction of F (ϕ, id N ) over 0 R m depends on g = j0ϕ r G r m only. The rule g F 0 (ϕ, id N ) =: H(g) N : T A N T A N defines a group homomorphism H : G r m Aut A. So we have an action H N of G r m on T A N and each T A f : T A N 1 T A N 2 is an H-equivariant map. Then F (M, N) coincides with the associated bundle P r M[T A N, H N ] and F (g, f) is the corresponding morphism P r g[t A f] of associated bundles, see e.g. [6]. Conversely, A and H : G r m Aut A define a bundle functor (A, H) on Mf m Mf by (1) (A, H)(M, N) = P r M[T A N, H N ], (A, H)(g, f) = P r g[t A f]. Clearly, r is the minimal order of (A, H) in the first factor, iff H cannot be factorized through a group homomorphism G r 1 m Aut A. Further, every section σ of a fibered manifold p: Y M can be interpreted as a base preserving morphism σ : M pt Y, where pt denotes a singleton. Then F σ is identified with a section of F Y M. Lemma 2. If r is the proper base order of F, then (F σ)(x) depends on j r xσ only, x M. Proof. By locality, we may assume Y = M N and σ is a constant section x (x, c), c N. Then (1) implies our claim. Thus we obtain a natural transformation t Y : J r Y F Y, t Y (j r xσ) = (F σ)(x).

4 4 IVAN KOLÁŘ A simple analysis shows that t is equivalent to an equivariant algebra homomorphism t: D r m A, [7]. In the product case Y = M N, we have t N : T r mn T A N and t M N : P r M[T r mn] P r M[T A N], t M N ( {u, X} ) = {u, tn (X)}, u P r M, X T r mn. Conversely, every such triple (A, H, t) defines a f.p.p.b. functor F on FM m by F (M N M) = (A, H)(M, N) = P r M[T A N] and F Y F (M Y M) = P r M[T A Y ] is the subset characterized by (2) t N (u) = T A p(x), {u, X} P r M[T A Y ], where we use P r M TmM. r For an FM m -morphism f : Y 1 Y 2 with base map f, F f : F Y 1 F Y 2 is the restriction and corestriction of P r f[t A f]. Clearly, r is the proper base order of F, iff t: D r m A cannot be factorized through an algebra homomorphism D r 1 m A. Proposition 1. If r is the proper base order of F, then t Y : J r Y F Y are injective maps. Proof. This follows directly from Lemma 1. Since r is the proper base order, we have r = s. The simpliest examples of f.p.p.b. functors are the vertical Weil bundle V A Y = T A (Y x ), x M the r-th jet prolongation J r Y =: Jh r Y, that we call the r-th horizontal prolongation in this general context, the r-th vertical jet prolongation Jv r Y = Jx(M, r Y x ) x M and their iterations. In general, if F 1 and F 2 are two f.p.p.b. functors on FM m, then the functor F 3 Y = F 1 Y M F 2 Y, F 3 f = F 1 f f F 2 f preserves fiber products as well. If F i = (A i, H i, t i ), i = 1, 2, 3, then A 3 is the sum A 1 A 2, i.e. the subset of A 1 A 2 formed by all pairs with equal real parts, [4]. Further we have Aut A 1 Aut A 2 Aut (A 1 A 2 ), r 3 = max(r 1, r 2 ), H 3 (g) = ( H 1 (π r 3 r 1 g), H 2 (π r 3 r 2 g) ), g G r 3 ( m and t 3 (X) = t1 (β r 3 r 1 X), t 2 (β r 3 r 2 X) ), X D r 3 m. We write F 3 = F 1 F 2 and we say F 3 is the fiber product of F 1 and F 2. The natural transformations (A 1, H 1, t 1 ) (A 2, H 2, t 2 ) are in bijection with equivariant algebra homomorphisms µ: A 1 A 2 satisfying t 2 = µ t 1, [7]. We write µ Y : F 1 Y F 2 Y and we have µ Y ( {u, X} ) = { u, µy (X) }, u P r M, X T A 1 Y,

5 SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS5 where µ Y : T A 1 Y T A 2 Y is the manifold version of µ. In the product case Y = M N, we have µ M N ( {u, X} ) = { u, µn (X) }, u P r M, X T A 1 N. 3. F -prolongation of associated bundles. The r-th principal gauge prolongation of a principal bundle P (M, G) is the bundle W r P of r-jets of local principal bundle isomorphisms j r (0,e)ψ, ψ : R m G P, 0 R m, e G. This is a principal bundle over M with structure group W r mg = W r 0 (R m G), [6]. We have W r P = P r M M J r P and W r mg = G r m T r mg with the group composition (g 1, X 1 ) (g 2, X 2 ) = ( g 1 g 2, T r mγ(x 1 g 2, X 2 ) ), where γ : G G G is the group composition of G. For an arbitrary fibered manifold p: Y M, one defines W r Y = P r M M J r Y. In [1], the authors introduced formally W F Y = P r M M F Y. For every natural transformation µ Y : F 1 Y F 2 Y, they defined W µ Y : W F 1 Y W F 2 Y, {u, X} { u, µ Y (X) }, u P r M, X F 1 Y. In particular, W t Y : W r Y W F Y. Hence Proposition 1 implies Corollary. If r is the proper base order of F, then W t Y : W r Y W F Y are injective maps. For a principal bundle P (M, G), W F P is a principal bundle over M, the structure group of which is W A H G = Gr m H T A G with the group composition (g 1, X 1 ) (g 2, X 2 ) = ( g 1 g 2, T A γ ( H G (g 1 2 )(X 1 ), X 2 )), [1]. The map id G r m t G : W r mg = G r m T r mg G r m T A G = W A H G is a group homomorphism. Then Proposition 1 implies easily Proposition 2. If r is the proper base order of F, then W t P : W r P W F P is a reduction to the subgroup W r mg W A H G. According to [1], a left action l : G S S induces a left action W A H l : W A H G T A S T A S, W A H l ( (g, X), Z ) = H(g) S ( T A l(x, Z) ), g G r m, X T A G, Z T A S. Consider the associated bundle E = P [S, l]. Its frame map (u, {u, y}) y, u P, y S can be interpreted as a base preserving morphism ϕ: P M E M S.

6 6 IVAN KOLÁŘ Applying F, we obtain F ϕ: F P M F E P r M[T A S]. Then F E is an associated bundle W F P [T A S, WH A l], the frame map of which ϕ F : W F P M F E T A S is of the form ϕ F ( (u, X), Z ) = ũ 1 ( F ϕ(x, Z) ), u P r M, X F P, Z F E, where ũ is the frame map of P r M[T A S] corresponding to u P r M, [1]. Adding the group homomorphism id G r m t G : WmG r WH A G, we obtain a left action WH r l : W mg r T A S T A S, W r Hl ( (g, X), Z ) = H S (g)t A l ( t G (X), Z ). Then Proposition 2 implies directly Proposition 3. Let E = P [S, l] be an associated bundle and r be the proper base order of F. Using WP t : W r P W F P, we can interpret F E as an associated bundle W r P [T A S, WH r l] with the frame map ϕ r F : W r P M F E T A S, ϕ r F (( u, t P (X) ), Z ) = ũ 1( F ϕ ( t P (X), Z )). One verifies easily that the frame map j (0,e) r ψ : F xe T A S can be constructed in the following geometric way. We have F (R m S) = P r R m [T A S] = R m T A S, where we use the identification P r R m = R m G r m defined by the translations on R m. The local principal bundle isomorphism ψ : R m G P induces a local isomorphism of associated bundles ψ S : R m S E. Applying F, we obtain F ψ S : R m T A S F E. This map is restricted and corestricted into a diffeomorphism {x} T A S F x E, that is the inverse map to j (0,e) r ψ. 4. Algebra homomorphisms compatible with H. The bundles of nonholonomic r-jets J r (M, N) J r (M, N) represent an important generalization of the classical bundles of holonomic r-jets, [4]. C. Ehresmann defined the composition of nonholonomic r-jets Z X J r x(m, Q) z for every X J r x(m, N) y and Z J r y(n, Q) z, that coincides with the classical jet composition in the holonomic case, see e.g. [4]. In [5], we introduced the general concept of nonholonomic r-jet category C as a subcategory J r (M, N) C(M, N) J r (M, N). For every m, we define a Lie group G C m = inv C 0 (R m, R m ) 0, [5], and a Weil algebra D C m = C 0 (R m, R) = R N C m, N C m = C 0 (R m, R) 0. We

7 SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS7 have the canonical injections i C m : G r m G C m and ι C m : D r m D C m in both group and algebra cases. The jet composition determines an injection G C m Aut D C m. Definition 2. For a fibered manifold p: Y M, we define its horizontal C-prolongation C h Y = { X C(M, Y ), (j r βxp) X = j r αx id M } and its vertical C-prolongation C v Y = C x (M, Y x ) C(M, Y ). x M Both C h and C v are f.p.p.b. functors on FM m with A = D C m. In both cases, the underlying group homomorphism is i C m : G r m G C m. In the first case, the algebra homomorphism is ι C m : D r m D C m, while in the second one it is the zero homomorphism D r m D C m, (x, n) (x, 0), x R, n N r m. This leads us to the following GENERAL PROBLEM. Let H : G r m Aut A be a group homomorphism. What are all possible algebra homomorphisms t: D r m A such that (A, H, t) is a f.p.p.b. functor on FM m? IN OTHER WORDS: up to what extent a f.p.p.b. functor F on FM m is determined by its restriction F = F i to Mf m Mf? First of all, we discuss the case A = D r m and H = id G r m =: i r m. Lemma 3. For r 2, the only two i r m-equivariant algebra homomorphisms D r m D r m are id D r m and the zero homomorphism. For r = 1, all possibilities are (x, n) (x, kn), k R. Proof. Consider first the case r = 2. Write x i, x ij for the canonical coordinates on N 2 m. By standard representation results, [6], a G 1 m- equivariant linear map f : N 2 m N 2 m is of the form x i = k 1 x i, x ij = k 2 x ij. The equivariancy of f with respect to (δ i j, a i jl ) G2 m means k 2 (x ij + a l ijx l ) = k 2 x ij + a l ijk 1 x l. This implies k 1 = k 2 =: k. The homomorphism condition yields (kx i )(kx j ) = k(x i x j ). Hence k 2 = k, i.e. k = 0, 1. Further, for r = 1 the homomorphism condition is automatically satisfied, because of nn = 0 for all n, n N 1 m. This yields our second claim. For r > 2, a standard recurrence procedure leads to the first claim.

8 8 IVAN KOLÁŘ In Section 2 we introduced the functors Jh r and J v r. If r = 1, we further take into account that Jx(M, 1 N) y = T y N Tx M is a vector space. Hence we can define J 1,k Y = { } X J 1 (M, Y ), T p(x) = k id TαX M for every k R. Clearly, J 1,0 Y = J 1 v Y and J 1,1 Y = J 1 h Y. Then Lemma 3 yields directly Proposition 4. For r 2, the only two f.p.p.b. functors with the underlying group homomorphism i r m are Jh r and J v r. If r = 1, all of them form a one-parameter family J 1,k, k R. An important subcategory of J r is the category J r of semiholonomic r-jets, [4]. In general, a nonholonomic r-jet category C will the called semiholonomic, if C(M, N) J r (M, N) for every M and N. If C is semiholonomic, then we find easily from the proof of Lemma 3 that the values of every i C m-equivariant algebra homomorphism D r m D C m lie in D r m D C m. Then Lemma 3 implies that for r 2 the only two possibilities are i C m and the zero homomorphism. Thus we have proved Proposition 5. If C is a semiholonomic r-jet category, r 2, then the only two f.p.p.b. functors on FM m with underlying group homomorphism i C m : G r m G C m Aut D C m are C h and C v. We remark that the construction of fiber products of f.p.p.b. functors clarifies that we can have more possibilities in some further cases. For example, consider the algebra D r m D r m and the injection δ r m : G r m Aut (D r m D r m) from Section 2. Then J r h J r h, J r h J r v and J r v J r v are f.p.p.b. functors on FM m with the underlying group homomorphism δ r m. 5. The nonholonomic case. To outline what can happen in the case of the category C = J r of all nonholonomic r-jets, we discuss the order r = 2 in detail. We write G 2 m = inv J 2 0 (R m, R m ) 0, D 2 m = J 2 0 (R m, R) and ĩ 2 m : G 2 m G 2 m, ι 2 m : D 2 m D 2 m for the canonical injections. We have N 2 m = (x i, x ij ), Ñ 2 m = (y i, y 0i, y ij ). Lemma 4. All ĩ 2 m-equivariant algebra homomorphisms D 2 m D 2 m form two one-parameter families I and II of the common form (3) y i = kx i, y 0i = lx i, y ij = klx ij with I : k R, l = 1 and II : k = 0, l R. Proof. By the classical representation theory, a G 1 m-equivariant linear map N 2 m Ñ 2 m is of the form y i = kx i, y 0i = lx i, y ij = hx ij.

9 SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS9 The equivariancy with respect to (δ i j, a i jl ) G2 m implies h = k as in the proof of Lemma 3. The homomorphism condition yields k = kl. We geometrize this result by using the functor J 1,k. We recall J 1,0 = J 1 v and J 1,1 = J 1 h. Proposition 6. All f.p.p.b. functors on FM m with the underlying group homomorphism ĩ 2 m : G 2 m G 2 m Aut D 2 m form two one-parameter families I J 1,k Jh 1 and II J v 1 J 1,l. Proof. We apply the general description of the iteration of two f.p.p.b. functors on FM m from [1]. If F = (A, H, t) and E = (B, K, u), K : G s m Aut B, u: D s m B are two such functors, then the Weil algebra of F E is A B and the algebra homomorphism v : D r+s m Aut (A B) is of the form v = t B Tmu r ι r,s m, where t B : TmB r T A B = A B is the value of t on B, Tmu: r TmD r s m = D r m D s m TmB r and ι r,s m : D r+s m D r m D s m is the canonical injection (we do not need the explicit formula for the group homomorphism of F E). In particular, this implies D 2 m = D 1 m D 1 m. Write x, x i for the canonical coordinates on D 1 m. In the case I, u or t is of the form x = x, y i = kx i or x i = x i respectively. Then the coordinate expression of Tmu 1 is y 0i = x i, y ij = kx ij. This corresponds to (3) with l = 1. In the case II, u or t is of the form x = x, x i = 0 or y 0i = lx i, respectively. Then the coordinate expression of Tmu 1 is y i = 0, y ij = 0. This corresponds to (3) with k = 0. Clearly, for k = l = 1 or k = l = 0 we obtain J 2 h or J 2 v, respectively. References [1] M. Doupovec, I. Kolář, Iteration of fiber product preserving bundle functors, Monatsh. Math. 134 (2001), [2] L. Fatibene, M. Francaviglia, Natural and Gauge Natural Formalism for Classical Fields Theories, Kluwer, [3] I. Kolář, An abstract characterization of jet spaces, Cahiers Topol. Géom. Diff. Catégoriques 34 (1993), [4] I. Kolář, Weil bundles as generalized jet spaces, in: Handbook of Global Analysis, Elsevier, Amsterdam (2008), [5] I. Kolář, On special types of nonholonomic contact elements, to appear. [6] I. Kolář, P. W. Michor, J. Slovák, Natural Operations in Differential Geometry, Springer Verlag, [7] I. Kolář, W. M. Mikulski, On the fiber product preserving bundle functors, Differential Geom. Appl. 11 (1999),

10 10 IVAN KOLÁŘ Institute of Mathematics and Statistics Masaryk University Kotlářská 2, CZ Brno Czech Republic

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