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 rth 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 mdimensional 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 rth 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 rth 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 rjets, 2000 Mathematics Subject Classification : 58A20, 58A32. Key words and phrases : principal gaugelike prolongation, Weil algebra, jet bundle, semiholonomic rjet 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 rth horizontal and the rth vertical prolongation of a fibered manifold. In the case of nonholonomic rjets, 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 Hequivariant, 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 πsequivariant r algebra homomorphism. Lemma 1. For every Hequivariant 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 mdimensional 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 Hequivariant 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 rth jet prolongation J r Y =: Jh r Y, that we call the rth horizontal prolongation in this general context, the rth 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 rth principal gauge prolongation of a principal bundle P (M, G) is the bundle W r P of rjets 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 rjets J r (M, N) J r (M, N) represent an important generalization of the classical bundles of holonomic rjets, [4]. C. Ehresmann defined the composition of nonholonomic rjets 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 rjet 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 Cprolongation C h Y = { X C(M, Y ), (j r βxp) X = j r αx id M } and its vertical Cprolongation 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 mequivariant 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 oneparameter family J 1,k, k R. An important subcategory of J r is the category J r of semiholonomic rjets, [4]. In general, a nonholonomic rjet 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 mequivariant 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 rjet 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 rjets, 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 mequivariant algebra homomorphisms D 2 m D 2 m form two oneparameter 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 mequivariant 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 oneparameter 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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