CLASSIFICATION OF PRINCIPAL CONNECTIONS NATURALLY INDUCED ON W 2 P E. Jan Vondra

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1 ARCHIVUM MATHEMATICUM BRNO Tomus , CLASSIFICATION OF PRINCIPAL CONNECTIONS NATURALLY INDUCED ON W 2 P E Jan Vondra Abstract. We consider a vector bundle E M and the principal bundle P E of frames of E. Let K be a principal connection on P E and let Λ be a linear connection on M. We classify all principal connections on W 2 P E = P 2 M M J 2 P E naturally given by K and Λ. Introduction Principal connections on principal prolongation W r P of a principal bundle P naturally given by a principal connection Γ on P and a linear connection Λ on M considered as a principal connection on the principal frame bundle P M play an important role in gauge invariant field theory, [4]. This problem was studied by many authors, see for instance [, 2, 8, 0]. Complete classification is known for the first prolongation W P but for general r the problem is still open. In this paper we give the full classification of principal connections for the linear gauge group GLn playing the role the structure group and for the order r = 2. We use the terminology and methods of the theory of natural and gauge-natural bundles and natural operators, see [3, 9,, 2, 3]. We denote by PB m G the category of principal G-bundles with m-dimensional bases and principal bundle morphisms over diffeomorphisms of bases and by FM m the category of fibered manifolds with m-dimensional bases and fibered morphisms over diffeomorphisms of bases. Then a gauge-natural bundle functor is a functor from the category PB m G to the category FM m transforming the diagram for a morphism in PB m G to a commutative diagram in FM m. Moreover, such functor has to satisfy some locality condition. All manifolds and maps are assumed to be smooth.. Principal bundles and principal connections.. Principal bundle and its principal prolongation. We consider a principal bundle P = P, M, π; G with a structure group G. We denote by x, z a fibered coordinates on P, =,..., dim M, a =,..., dim G Mathematics Subject Classification: primary 53C05; secondary 53C0, 58A20, 58A32. Key words and phrases: natural bundle, gauge-natural bundle, natural operator, principal bundle, principal connection. This research has been supported by the Grant agency of the Czech Republic under the project 20/05/H005.

2 536 J. VONDRA By adp we denote the vector bundle associated to P with respect to the adjoint action of G on its Lie algebra g. We denote by B a a base of g and we denote by x, u a the induced fiber linear local coordinates on adp. Let us denote by c a bd the related structure constants, i.e. [B b, B d ] = c a bd B a. For the right action r g : P P given by an element g G we consider the tangent mapping T r g : T P T P. Let Ξ be a vector field on P. We say that Ξ is right invariant if Ξpg = T r g Ξp for all p P and g G. In coordinates we have. Ξ = ξ x + Ξ a x B a, where B a is the base of vertical right invariant vector fields on P which are induced by B a. So Ξ are sections of the bundle T P/G M. Example.. Let as recall that the bundle of r-th order frames on a manifold is the principal bundle P r M = P r M, M, π r ; G r m, where P r M = invj0 r R m, M is the bundle of invertible r-jets with the source 0 and the structure group G r m = invj0 r R m, R m 0, [9]. Local coordinates x on M then induce the fibered coordinates x, x µ,..., x µ...µ r on P r M. The coordinates in the structure group G r m are denoted by a µ,..., a µ...µ r. µ...µi The base of the induced right invariant vertical vector fields B, i =,..., r, on P r M can be obtained by the following construction. Let us consider a vector field ξ = ξ x on M. Then we have the flow lift P r ξ given by P r F l t ξ = F l t P r ξ which is a vector field on P r M projectable on ξ. Then, [4],.2 P r ξ = ξ x + r i= i ξ x x µ... x µi Bµ...µ i and, if we compare it with the expression of P r ξ in the canonical coordinates, we obtain the relations between the canonical base of vertical vector fields µ...µi µ...µi and the base of right invariant vertical vector fields B. For instance for r = we have.3 P ξ = ξ + ξ x ρ xρ µ µ, which implies that B ρ = xρ µ µ. For r = 2 we have P 2 ξ = ξ + ξ x ρ xρ µ µ 2 + ξ.4 x ρ x σ xρ µ x σ ν + ξ x ρ xρ µν µν = ξ + ξ x ρ x ρ µ µ + xρ µν µν + 2 ξ x ρ x σ xρ µ x σ ν µν. Then we obtain the right invariant vector fields on P 2 M given by.5 Bρ = xρ µ µ + xρ µν µν, Bρσ = xρ µ x σ ν µν. Definition.. For each principal bundle P = P, M, π; G and for each integer r we can define the principal bundle W r P = P r M M J r P W r P, M, p; W r mg. The structure group is the semidirect product W r mg = G r m T r mg. See [9].

3 PRINCIPAL CONNECTIONS NATURALLY INDUCED ON W 2 P E 537 The group W r mg is the group of r-jets at 0, e of all automorphisms ϕ: R m G R m G with ϕ0 = 0, where the multiplication µ is defined by the composition of jets,.6 µ j r ϕ0, e, j r ψ0, e = j r ψ ϕ0, e. If we replace holonomic jets by nonholonomic or semiholonomic ones, we obtain the nonholonomic or semiholonomic principal prolongations W r P and W r P, respectively. Analogously we have and W r P = P r M M J r P, W r P = P r M M J r P W r mg = G r m T r mg W r mg = Ḡr m T r mg. Remark.. Moreover, W r : PB m G PB m W r mg defines a functor. Let ϕ be a principal morphism in the category PB m G over a base diffeomorphism ϕ then W r ϕ = P r ϕ, J r ϕ. W r is then a gauge-natural bundle functor of order r and plays a fundamental role in the theory of gauge-natural bundles. Really, any r-th order G-gauge-natural bundle is a fiber bundle associated with W r P. See [3, 9]. Remark.2. On W r P we have the induced natural local fiber coordinates x, x µ,..., x µ...µ r, z a, zν a,..., zν a...ν r. The base of the induced right invariant ν...νj vector fields, B a, i =,..., r, j = 0,..., r, on W r P can be obtained µ...µi B by the following construction, [4]. Let us consider a right invariant vector field Ξ on M given by.. Then we have the flow lift W r Ξ which is a vector field on W r P obtained as.7 W r Ξ = P r ξ ξ J r Ξ, where P r ξ is the flow lift of the vector field ξ on P r M and J r Ξ is the r-th order jet lift of Ξ on J r P E. Then r.8 W r Ξ = ξ i ξ x r x +...µ i j Ξ a x Bµ x µ... x µi + Bν...ν j x ν... x νj a. i= Comparing.8 with.7 expressed in the canonical coordinates on W r P we obtain relations between the canonical base of vertical vector fields µ...µi, a ν...νj µ...µi ν...νj and the base of right invariant vertical vector fields B, B a. We have two canonical group homomorphisms.9 π r r : W r mg W r m G, p : W r mg G r m, which induce homomorphisms of Lie algebras j=0.0 π r r : w r mg w r m g, p : w r mg g r m, principle bundle morphisms, over the identity of M,. π r r : W r P W r P, p : W r P P r M, and homomorphisms of associated vector bundles.2 π r r : adw r P adw r P, p : adw r P adp r M.

4 538 J. VONDRA.2. Principal connection on P and on W r P. A principal connection on P is defined as a lifting linear mapping In coordinates Γ: T M T P/G..3 Γ = d + Γ a x B a, where Γ a x are functions on M. If we identify Γ with the functions Γ a x then Γ can be considered as a section of the bundle QP M of principal connections on P. Moreover, we have [9]. Proposition.. Let P M be a principal bundle and QP M is the bundle of principal connections on P. Then QP is a -order G-gauge-natural affine bundle associated with the vector bundle adp T M M. This implies that the standard fiber of the functor Q is g R m. Remark.3. Let us recall that every principal connection Γ on P induce the linear adjoint connection AdΓ on adp given in coordinates by AdΓ = d + c a deγ d u e a. Example.2. Let us consider the r-order frame bundle P r M of M see Example.. A principal connection Λ r on P r M has a coordinate expression r.4 Λ r = d +. i= Λ ν µ...µ i B µ...µi ν We remark that principal connections on P M are in the bijection with classical connections on M linear connections on T M. Let Γ r be a principal connection on W r P given in coordinates by r r.5 Γ r = d µ...µi + B ν + Γ a κ...κ j i= Λ ν µ...µ i j=0 B κ...κj a Remark.4. We remark that coefficients Λ ν µ...µ i, respective Γa κ...κ j, satisfy the symmetry condition with respect to indices µ... µ i, respective κ... κ j. The projections. of W r P and the functor Q induce the projections.6 π r r : QW r P QW r P, p : QW r P QP r M, so any principal connection Γ r on W r P projects on the principal connection Λ r on P r M, see.4, and on the principal connection Γ r on W r P. Remark.5. By Proposition. QW r P M is the affine bundle modeled over the vector bundle adw r P T M M. Now, if we consider two principal connections Γ r and Γ r on W r P such that they are over the same Λ r on P r M and Γ r on W r P, then the difference Γ r Γ r is in the intersection of the kernels of projections π r r id T M : adw r P T M adw r P T M and p id T M : adw r P T M adp r M T M..

5 PRINCIPAL CONNECTIONS NATURALLY INDUCED ON W 2 P E 539 Let ξ be a vector field on M, Γ be a principal connection on P and Λ be a principal connection on P M such that the horizontal lift h Λ ξ = P ξ. Let h Γ ξ denote the horizontal lift of ξ with respect to Γ. Let us denote by Fl t h Γ ξ the flow of h Γ ξ. Then the expression W r Fl t h Γ ξ = P r Fl t ξ, J r Fl t h Γ ξ = Fl t h WrΓ ξ gives a principal connection W r Γ on W r P which depends on Γ in order r and on Λ in order r. So W r Γ is a natural operator called the flow prolongation of Γ. W r Γ: J r QP M M J r QP QW r P Remark.6. There is another construction of a prolongation of the pair Λ, Γ of principal connections to principal connections on W r P. In literature this construction is usually denoted by pλ, Γ and in order one was described in [0] and in any order r in []. Consider general problem of the classification of all principal connections on W r P which are naturally given by a principal connection Γ on P and by a classical connection Λ on the base M. In general this problem is still open. There are only some partial results. Namely, the classification of principal connections on W P for a torsion free Λ and any Γ is given in [0] and the full classification for the linear gauge group GLn and order one is given in [8]. Clearly, the solution of this problem in any order depends essentially on the structure group of principal bundles. In this paper we give the full classification of principal connections for the linear gauge group GLn and for the order r = Connections on frame bundle P E. Let E M be a vector bundle with m-dimensional base and n-dimensional fibers. Let us denote by x, y i local linear fiber coordinate charts on E. Let P E M be the frame bundle of E, i.e. P E is the principal bundle with the structure group GLn and the induced fiber coordinates x, x i j. Principal connections on P E are in bijection with general linear connections on E. The coordinate expression of a linear connection K on E is of the type K = d + K i j xy j i and, if we consider K as a principal connection on P E, K = d + K i pxx p j j i = d + K i j x B j i. Applying the functor W r on P E we obtain the principal prolongation W r P E with the structure group Wm r GLn = G r m Tm r GLn. For example, in order 2 we have the structure group Wm 2 GLn with coordinates a µ, a µν, a i j, ai jµ, ai jµν. If we denote by Xµ, Xµν, Xj i, Xi jµ, Xi jµν the induced coordinates on the Lie algebra w 2 mgln we can compute easily the adjoint action of the group Wm 2 GLn on

6 540 J. VONDRA w 2 mgln. In coordinates if X = AdgX we have X µ = a ϱx ϱ τ ã τ µ, X µν = a ϱx ϱ αβãα µã β ν + pola, X, X j i = a i pxq p ã q j, Xi jϱ = a i pxqαã p α ϱ ã q j + pola, X, X jϱσ i = a i qτ Xαβã τ α ϱ ã β σã q j + ai px p qαβãα ϱ ã β σã q j + pola, X, where pola, X is a polynomial on W 2 m GLn w mgln such that any monomial contains exactly one coordinate of π 2 X of orders less then the leading terms. Proposition.2.. The restriction of the adjoint action of the group W 2 m Gln on the kernel of the projection π 2 : w 2 mgln w mgln has the form X µν = a ϱx ϱ αβãα µã β ν, X i jϱσ = a i qτ X τ αβã α ϱ ã β σã q j + ai px p qαβãα ϱ ã β σã q j. 2. The restriction of the adjoint action of the group Wm 2 Gln on the kernel of the projection p : w 2 mgln g 2 m has the form X j i = a i pxq p ã q j, Xi jϱ = a i pxqαã p α ϱ ã q j + pola, X, X jϱσ i = a i px p qαβãα ϱ ã β σã q j + pola, X, where polx, a is a polynomial on W 2 m GLn t mgln such that any monomial contains exactly one coordinate of orders less then the leading terms. Proof.. The kernel of the projection π 2 : w 2 mgln w mgln is given by Xµ = 0, Xj i = 0 and Xi jµ = 0, i.e. all pola, X = The kernel of the projection p : w 2 mgln g 2 m is given by Xµ = 0 and Xµ µ 2 = 0. Now, as a direct consequence of Proposition.2, we have. Theorem.. The restriction of the adjoint action of the group W 2 m Gln on the intersection of kernels of the projections π 2 : w 2 mgln w mgln and p : w 2 mgln g 2 m is given by X i jϱσ = a i px p qαβãα ϱ ã β σã q j, i.e. Ker π 2 Ker p is gln S 2 R m with the action of the group G m GLn given as the tensor product of the adjoint action of GLn on its Lie algebra gln and the tensor action of the group G m on S 2 R m. Corollary.. The intersection of kernels of the projections π 2 : ad W 2 mp E ad W mp E and p : ad W 2 mp E ad P 2 M is the vector bundle adp E S 2 T M M. Next we will describe the flow lift see Section.2 of a principal connection K on P E and a classical connection Λ on M on the principal bundle W 2 P E. We have the induced fibered coordinates on W 2 P E denoted by x µ, x µν, x i j, xi jµ, xi jµν. A right invariant vector field on P E has the coordinate expression Ξ = ξ x + Ξ i jx B j i = ξ x + Ξ i pxx p j j i.

7 PRINCIPAL CONNECTIONS NATURALLY INDUCED ON W 2 P E 54 The flow lift of Ξ on W 2 P E is then W 2 Ξ = P 2 ξ ξ J 2 Ξ, where P 2 ξ is the flow lift of the vector field ξ on P 2 M given by.4 and J 2 Ξ is the 2nd order jet lift of Ξ on J 2 P E, see [9, ]. We have.7 J 2 Ξ = ξ + Ξ i pg p j j i + Ξ i pg p j + Ξi pg p j ξ µ g i jµ j i + µ Ξ i pg p j + Ξ i pg p jµ + µξ i pg p j + Ξi pg p jµ µ ξ ϱ g i jϱ ξ ϱ g i jϱµ µ ξ ϱ g i jϱ jµ i and from.7 and.4 we obtain the relations between the right invariant vertical vector fields on W 2 P E and the canonical vertical vector fields in the form.8 B ϱ = xϱ µ µ + xϱ µν µν gi j jϱ i B ϱσ = xϱ µx σ ν µν gi j jϱσ i, B p i = gp j j i + gp j j i B p i = g p j j i + g p jµ jµ i, g i jµ jϱµ i, + g p jµ jµ i, Bpµ i = g p j jµ i. Now, let us assume a principle connection on Γ 2 on W 2 P E given in coordinates by see.5 Γ 2 = d + Λ ν µ Bµ ν + Λ ν µµ2 µ µ 2 B ν + Kp i B p i + Γi pµ pµ B i + Γ i pµµ2 pµ µ 2 B i and, in the canonical base, by using the relations.8,.9 Γ 2 = d + Λ ν µx µ ϱ ν ϱ + Λ ν µ µ 2x µ ϱ x µ2 σ + Λ ν µx µ ϱσ ν ϱσ + K i px p j j i + Γi pµx p j + Ki px p jµ Λν µx i jν jµ i + Γ i pµ µ 2x p j + Γi pµ x p jµ 2 Λ ν µ µ 2x i jν + Kpx i p jµ µ 2 Λ ν µ x i jνµ 2 jµµ2 i. Now, if we assume Ξ to be the horizontal lift of ξ given by the connection K and P ξ to be the horizontal lift of ξ given by Λ, i.e. Ξ i j = K j i ξ and µ ξ = Λ µρξ ρ, we have the following coordinates expression W 2 Ξ = ξ [ + Λ ν µx µ ϱ ν ϱ + µ Λ ν µ 2 + Λ ν µ αλ α µ 2 x µ ϱ x µ2 σ + Λ ν αx α σϱ ϱσ ν + Kpx i p j j i + µ Kp i + KpαΛ i α µx p j + Ki px p jµ Λϱ µx i jϱ jµ + µµ 2 K i p + µ K i pϱλ ϱ µ 2 + K j pϱ µ Λ ϱ µ 2 + K j pϱλ ϱ µ σλ σ µ 2 x p j + µ Kp i + KpϱΛ i ϱ µ x p jµ 2 µ Λ ϱ µ 2 + Λ ϱ µ σλ σ µ 2 x p jϱ + Kpx i p jµ µ 2 Λ ϱ µ x i jϱµ 2 jµ µ 2 ] i. So, the vector field W 2 Ξ can be considered as the horizontal lift of ξ with respect to the connection.9 on W 2 P E where Γ 2 is given by connections Λ and K. Especially, Γ 2 = W 2 KΛ, K. From the symmetry of ν ϱσ in ϱ, σ and jµµ2 i in µ, µ 2 we obtain the coordinate expression of W 2 K given by i

8 542 J. VONDRA Λ ν µ µ 2 = 2 µ Λ ν µ 2 + µ2 Λ ν µ + Λ ν µ αλ α µ 2 + Λ ν µ 2 αλ α µ, Γ i pµ = µ K i p + K i pϱλ ϱ µ, Γ i pµ µ 2 = µµ 2 K i p + µ K i pϱλ ϱ µ 2 + µ2 K i pϱλ ϱ µ + 2 Kj pϱ µ Λ ϱ µ 2 + µ2 Λ ϱ µ + Λ ϱ µ σλ σ µ 2 + Λ ϱ µ 2 σλ σ µ. Remark.7. The flow prolongation W KΛ, K we obtain by the projection π 2 of W 2 KΛ, K on W P E, so it is given by.2 and projects on Λ and K. Remark.8. The second prolongation pλ, K on W P E from Remark.6 is over the same connections Λ and K and has the coordinate expresion Γ i pµ = K i pµ + K i j µk j p K i j K j pµ + K i pϱλ ϱ µ. The difference pk, Λ W ΓK, Λ is a section M adp E 2 T M given by.23 K i pµ µ K i p + K i j µk j p K i j K j pµ, where Λ denotes the classical conjugate connection to Λ obtain by the exchange of indices. It is easy to see that.23 is exactly the curvature tensor of the connection K. 2. Classification of naturally induced connections on W 2 P E Let us recall that any principal connection Γ 2 on W 2 P E projects by.6 on principal connections Λ 2 on P 2 M and Γ on W P E. So first we will classify principal connections on P 2 M naturally given by a classical connection Λ and a general linear connection K considered as a principal connection on the corresponding frame bundle. 2.. Natural connections on the 2nd order frame bundle. The connection W 2 KΛ, K described in Section.3 projects on the natural principal connection Λ 2 Λ, K on P 2 M given by.20, i.e. it is independent of K. Λ 2 Λ, K can be constructed by an other construction. Let as remark that P 2 M P 2 M W P M are natural reductions with respect to the subgroups G 2 m Ḡ2 m WmG m, [9, p. 53]. Here P 2 M = inv J 0 2 R m, M and Ḡ2 m = inv J 0 2 R m, R m 0, where J 2 denotes the semiholonomic 2-jet functor. Then we can consider the flow lift of a principal connection Λ on P M with respect to the same principal connection Λ and obtain the principal connection W Λ = W ΛΛ, Λ on W P M. This connection is reducible to the principal connection on P 2 M. Now, if Λ 2 is a principal connection on the semiholonomic 2-order frame bundle, then it can be considered as Ḡ2 m-invariant section Λ 2 : P 2 M J P 2 M and the composition iλ 2 = J i Λ 2 is the principal connection on P 2 M conjugated with Λ 2. Here i: P 2 M P 2 M is the canonical natural involution. Let us remark, that if Λ 2 is given by coefficients Λ ν µ, Λν µ, then iλ µ 2 2 is given by coefficients Λ ν µ, Λν µ. 2µ Then the connection 2 W Λ + iw Λ is reducible to the principal connection on

9 PRINCIPAL CONNECTIONS NATURALLY INDUCED ON W 2 P E 543 P 2 M and we obtain the principal connection on P 2 M which is exactly the above connection Λ 2 Λ, K. In what follows we shall denote this connection by P 2 Λ and call it the flow prolongation of Λ. Any natural principal connection on P 2 M given by Λ is projectable on a natural connection on P M given by Λ. So, we have to classify first all natural connections Λ on P M given by Λ. This result is well known and we have, [9, p. 220]. Proposition 2.. All natural operators transforming a principal connection Λ on P M into principal connections Λ Λ on P M is of zero order and form a 3-parameter family Λ Λ = Λ + Φ Λ, where Φ Λ is a natural, 2-tensor field of the form Φ = a T + a 2 Id T M ˆT + a 3 ˆT IdT M, a i R, where T denote the torsion tensor of Λ and ˆT denote the contraction. Corollary 2.. Any connection Λ 2 Λ on P 2 M naturally given by Λ is projectable on a natural connection in the family Λ Λ. Theorem 2.. Let Λ 2 Λ on P 2 M be a natural connection given by Λ projectable on Λ Λ then the difference Λ 2 Λ P 2 Λ Λ: M T M T M S 2 T M. Proof. As a corollary of Theorem. we obtain that the kernel of the projection π 2 : g 2 m g m is g m R m which implies that the kernel of the projection adp 2 M adp M is the vector bundle adp M T M. The difference of two principle connections on P 2 M over the same Λ Λ on P M is then in the kernel kerπ 2 id T M = adp M T M T M. Moreover, adp M = T M T M and from the symmetry property we obtain Theorem 2.. Lemma 2.. Any natural tensor field Φ 2 Λ: M T M T M S 2 T M naturally given by Λ is of maximal order one and all Φ 2 Λ form 6-parameter family. Proof. A non-symmetric connection Λ can be decomposed as the sum of the classical symmetric connection Λ obtained by the symmetrization of Λ and the torsion tensor T. From the reduction theorems for classical non-symmetric connections, [7], we obtain that Φ 2 Λ is given as a zero order operator of the curvature tensor R[ Λ] of Λ and its covariant differentials R[ Λ] with respect to Λ and the torsion tensor T and its covariant differentials T with respect to Λ. From the homogeneous function theorem, [9, p. 23], it follows, that a natural, 3-tensor field can be constructed only from R[ Λ], T and T. So the maximal order of Φ 2 Λ is one and Φ 2 Λ = ϱr[ Λ] + τt + σ T, where ϱ, τ and σ are given by tensorial operations from the indicated tensor fields. Let X, Y, Z be vector fields on M. Then from the first Bianchi identity and the symmetry of Φ 2 Λ in the last two vector variables we obtain that ϱ is given by the

10 544 J. VONDRA tensor product of R[ Λ] with the identity of T M and contractions. Such operators form a 4-parameter family ϱx, Y, Z = b RX, ZY + RX, Y Z + b2 X c 2RY, Z + c 2RZ, Y + b 3 Y c RX, Z + Zc RX, Y + b 4 Y c 2 RX, Z + Zc 2RX, Y, where b i R. In what follows c i j denotes the contraction on the indicated indices. Similarly τ is given by the tensor product of T two times, the identity of T M and contractions. Such operators form a 7-parameter family τx, Y, Z = d T T X, Z, Y + T T X, Y, Z + d 2 T X, Y c T Z + T X, Y c T Z + d 3 Xc 2 3T T Y, Z + d 4 Xc 2 3T T Y, Z + d 5 Y c 2 2T T X, Z + Zc 2 2T T X, Y + d 6 Y c 2 3T T X, Z + Zc 2 3T T X, Y + d 7 Y c 2 3T T X, Z + Zc 2 3T T X, Y, d i R. Finally, σ is given by T, the tensor product with the identity of T M and contractions. We have a 5-parameter family σx, Y, Z = e Z T X, Y + Y T X, Z + e 2 X c Z T Y + c Y T Z + e 3 Y c Z T X + Zc Y T X + e 4 Y c X T Z + Zc X T Y + e 5 Y c 3 T X, Z + Zc 3 T X, Y, e i R. Theorem 2.2. All natural principal connections Λ 2 Λ on P 2 M given by a classical connection Λ on M form a 9-parameter family. Proof. Every natural connection Λ 2 Λ on P 2 M has form P 2 Λ Λ+Φ 2 Λ, which is over Λ Λ = Λ + Φ Λ. So this theorem is the corollary of Proposition 2. and Lemma 2.. Remark 2.. Let as remark that without using the reduction theorem natural principal connections on the semi-holonomic frame bundle P 2 M naturally given by Λ was studied in [, p. 85]. Corollary 2.2. All natural principal connections on P 2 M given by a classical connection Λ on M and by a principal connection K on P E form a 20-parameter family. Proof. It is sufficient to consider that Φ and Φ 2 are natural tensor fields given by Λ and K. By the reduction theorem for general linear connections, [6], and the homogeneous function theorem we have that Φ Λ, K = Φ Λ given by

11 PRINCIPAL CONNECTIONS NATURALLY INDUCED ON W 2 P E 545 Proposition 2.. On the other hand for Φ 2 Λ, K we obtain that Φ 2 Λ, K include also the tensor product of the contracted curvature tensor R[K] of K and the identity of T M. By using the the symmetry condition we have Φ 2 Λ, KX, Y, Z = Φ 2 ΛX, Y, Z + ky c R[K]X, Z + Zc R[K]X, Y, k R. Corollary 2.3. All natural connections on P 2 M given by a classical symmetric connection Λ on M and by a principal connection K on P E form a 5-parameter family Classification of natural connections on W 2 P E. We consider all connections and tensor fields naturally given by Λ and K and we will not denote this fact explicitly. Proposition 2.2. Any natural principal connection Γ 2 on W 2 P E is in a bijective correspondence with the triplet Γ 2 Λ 2, Γ, Ψ 2, where Λ 2 is a natural principal connection on P 2 M, Γ is a natural principal connection on W P E and Ψ 2 is natural tensor field M E E T M S 2 T M. Moreover, Λ 2 and Γ are over the same Λ on P M. Proof. Let us assume a natural connection Γ 2 projectable on Γ and Λ 2 over the same Λ. The difference of Γ 2 with any other natural connection Γ 2 over the same natural connections Γ and Λ 2 is by Corollary. natural tensor field Ψ 2. From Proposition 2.2 it follows that for the classification of Γ 2 we need to classify all natural Λ 2 see Corollary 2.2, all natural Γ and all natural tensor fields Ψ 2. First, we have to classify all natural principal connections Γ on W P E. Let as recall that we have the correspondence Γ Λ, Γ 0, Ψ, [9]. Λ is described in Proposition 2.. Lemma 2.2. All natural operators transforming a classical connection Λ on M and a principal connection K on P E into principal connections Γ 0 on P E are of zero order and form a -parameter family Γ 0 = K + Ψ 0 = K + m Id E ˆT, m R. Proof. If we consider any natural principal connection Γ 0 on P E, then the difference Γ 0 K is a section Ψ 0 : M E E T M. So to classify all natural Γ 0 it is sufficient to classify natural Ψ 0. Then from the reduction theorem and the homogeneous function theorem we obtain that Ψ 0 is of zero order and is obtained as a multiple of the tensor product of the identity of E and 0, -tensor field given by the contraction of the torsion of Λ. Proposition 2.3. All natural operators transforming a classical connection Λ on M and a principal connection K on P E into principal connections Γ on W P E are of the maximal order and form a 4-parameter family.

12 546 J. VONDRA Proof. We have the 3-parameter family of connections Λ given by Proposition 2., the -parameter family of connections Γ 0 given by Lemma 2.2. First, we have to prove that there exists a natural connection Γ projectable on Λ and Γ 0. But it is exactly the flow lift W Γ 0 with respect to Λ. Then the difference Γ Γ is a natural tensor field Ψ : M E E T M T M. From the reduction theorems, [6], and the homogeneous function theorem, [9], follows, that Ψ is given by tensorial operations on from R[K], R[ Λ], T, T and the identity of E. So the maximal order of Ψ is and from the Bianchi identity and the symmetry condition we obtain 0-parameter family. Remark 2.2. Let us remark that in [8] Proposition 2.3 was proved without using the reduction theorem, so the base of 0-parameter family Ψ is different. Lemma 2.3. All natural tensor fields ω : M T M S 2 T M of type 0, 3 given by Λ and K are of the maximal order two and form a 3-parameter family. Proof. From the reduction theorems for classical non-symmetric connections and general linear connections, [6], and the homogeneous function theorem we obtain that ω is given by the tensorial operations from R[ Λ], R[ Λ], T, T, 2 T, R[K] and R[K] is assumed with respect to K and Λ, see [5]. So natural ω is of the maximal order two. The first part of ω has order two and is given by the contracted R[K], R[ Λ] and 2 T. From the Bianchi and the Ricci identities and the symmetry condition we obtain a 6-parameter family. The second part of ω has order one and is given by R[K], R[ Λ] and T multiplied tensorially with T and two contractions. In this way we obtain a 8-parameter family. The last part of ω has order zero and is given by the tensor product of T three times only and three contractions. In this way we obtain a 7-parameter family. Lemma 2.4. All natural tensor fields Ψ 2 : M E E T M S 2 T M given by Λ and K are of the maximal order two and form a 34-parameter family. Proof. As in Lemma 2.3 we have that all natural Ψ 2 are given by tensorial operations from R[ Λ], R[ Λ], T, T, 2 T, R[K] and R[K]. Then Ψ 2 has two parts. The first one is given by R[K] or by a composition of R[K] and T. By using the symmetry property and the Bianchi identity we obtain 3-parameter family. Exactly we have n Z R[K]X, Y + Y R[K]X, Z + n 2 R[K]X, Y c T Z + R[K]X, Zc T Y + n 3 R[K]T X, Z, Y + R[K]T X, Y, Z, where X, Y, Z are vector fields on M. The second part is of the form Id E ω, where ω is the natural tensor of type 0, 3 described in Lemma 2.3. All together we have a 34-parameter family.

13 PRINCIPAL CONNECTIONS NATURALLY INDUCED ON W 2 P E 547 Theorem 2.3. Natural principal connections on W 2 P E given by a classical connection Λ on M and by a principal connection K on P E form a 65-parameter family. Proof. By Proposition 2.2 and by [9] we have Γ 2 Λ 2, Γ, Ψ 2 Λ, Φ 2, Γ 0, Ψ, Ψ 2 Λ, Φ, Φ 2, K, Ψ 0, Ψ, Ψ 2. By Proposition 2. Φ form a 3-parameter family, by proof of Corollary 2.2 Φ 2 form a 7-parameter family, by Lemma 2.2 Ψ 0 form a -parameter family, by [8] Ψ form a 0-parameter family, and, finally, by Lemma 2.4 Ψ 2 form a 34-parameter family. All together we obtain a 65-parameter family of natural connections. Corollary 2.4. All natural connections on W 2 P E given by a classical symmetric connection Λ on M and by a principal connection K on P E form a 3-parameter family. References [] Doupovec, M., Mikulski, W. M., Reduction theorems for principal and classical connections, to appear. [2] Doupovec, M., Mikulski, W. M., Holonomic extensions of connections and symmetrization of jets, Rep. Math. Phys , [3] Eck, D. J., Gauge-natural bundles and generalized gauge theories, Mem. Amer. Math. Soc , 48p. [4] Fatibene, L., Francaviglia, M., Natural and Gauge Natural Formalism for Classical Field Theories, Kluwer Academic Publishers, Dordrecht-Boston-London, [5] Janyška, J., On the curvature of tensor product connections and covariant differentials, Rend. Circ. Mat. Palermo 2 Suppl , [6] Janyška, J., Reduction theorems for general linear connections, Differential Geom. Appl , [7] Janyška, J., Higher order valued reduction theorems for classical connections, Cent. Eur. J. Math , [8] Kolář, I., Some natural operators in differential geometry, Differential Geom. Appl., D. Reidel, 987, pp [9] Kolář, I., Michor, P. W., Slovák, J., Natural Operations in Differential Geometry, Springer Verlag, 993. [0] Kolář, I., Virsik, G., Connections in first principal prolongations, Rend. Circ. Mat. Palermo 2 Suppl , [] Krupka, D., Janyška, J., Lectures on Differential Invariants, Folia Fac. Sci. Natur. Univ. Purkynian. Brun. Math., 990. [2] Nijenhuis, A., Natural bundles and their general properties, Differential Geom. 972, , In honour of K. Yano, Kinokuniya, Tokyo. [3] Terng, C. L., Natural vector bundles and natural differential operators, Amer. J. Math , Department of Mathematics and Statistics, Masaryk University Kotlářská 2, 6 37 Brno, Czech Republic vondra@math.muni.cz