Reduction Theorems for General Linear Connections ✩

Differential Geometry and its Applications 20 (2004) 177–196 www.elsevier.com/locate/difgeo

Reduction theorems for general linear connections ✩

Josef Janyška

Department of Mathematics, Masaryk University, Janáˇckovo nám. 2a, 662 95 Brno, Czech Republic Received 26 February 2003 Communicated by I. Kolár˘

Abstract It is well known that natural operators of linear symmetric connections on manifolds and of classical tensor fields can be factorized through the curvature tensors, the tensor fields and their covariant differentials. We generalize this result for general linear connections on vector bundles. In this gauge-natural situation we need an auxiliary linear symmetric connection on the base manifold. We prove that natural operators defined on the spaces of general linear connections on vector bundles, on the spaces of linear symmetric connections on base manifolds and on certain tensor bundles can be factorized through the curvature tensors of linear and classical connections, the tensor fields and their covariant differentials with respect to both connections.  2003 Elsevier B.V. All rights reserved.

MSC: 53C05; 58A20

Keywords: Gauge-natural bundle; Natural operator; Linear connection; Classical connection; Reduction theorem

Introduction

The reduction (or replacement) theorems are classical theorems in the theory of classical, i.e., linear symmetric, connections on manifolds. The reduction theorems for classical connections can be formulated as follows.

The ﬁrst reduction theorem. All natural differential operators with values in a natural bundle of order one of a classical connection are natural differential operators of the curvature tensor and its covariant differentials.

✩ This paper has been supported by the Ministry of Education of the Czech Republic under the Project MSM 143100009. E-mail address: [email protected] (J. Janyška).

The second reduction theorem. All natural differential operators with values in a natural bundle of 1 m order one of a classical connection and ﬁelds Φ,...,Φ of geometrical objects of order one are natural 1 m differential operators of the curvature tensor, Φ,...,Φ and their covariant differentials.

The history of the first reduction theorem comes back to the paper by Christoffel, [1], and the history of the second reduction theorem comes back to the paper by Ricci and Levi Civita, [12]. For further references see [10,13,15]. In [13] the proof for algebraic concomitants is given. In [7] the first and the second reduction theorems are proved for all natural differential operators by using the modern approach of natural bundles and natural differential operators, [9,11,14]. The situation with natural differential operators of general linear connections is completely different. Eck, [2], has proved that first order operators of any principal connection with values in volume forms of the base manifold factorize through the curvature, which is the generalized Utiyama theorem. In [3] it was proved that the first order operators of any linear connection on a vector bundle with values in a (1, 0)-order gauge-natural bundle factorize through the curvature and in [7] this result is generalized for any principal connection. In [5,6] operators of classical connections of a spacetime (phase fields), of linear connections of a quantum bundle (quantum connections) and of sections of the quantum bundle were studied. By using the methods of gauge-natural geometry, [7], it was proved that such second order operators can be factorized through curvature tensors of both connections and an operator which can be considered as the second order “covariant differential” of sections of the quantum bundle with respect to both connections. This result was the motivation of the present paper. We generalize the above situation considering linear connections K on vector bundles, classical connections Γ on the base manifolds and fields of geometrical objects Φ of order (1, 0). We introduce the notion of the covariant differential ∇(K,Γ )Φ of Φ with respect to the pairs of connections (K, Γ ) and we prove the first and the second reduction theorems. The paper is organized as follows. In Section 1 we recall basic definitions and properties of gauge- natural bundles and natural operators in the sense of [2,7]. Basic properties of linear connections on vector bundles and of classical connections are recalled in Section 2 and some properties of tensor product connections and covariant differentials with respect to a pair of linear and classical connections are described in Section 3, [4]. The first reduction theorem for linear and classical connections is proved in Section 4 and the second reduction theorem is proved in Section 5. All manifolds and maps are assumed to be smooth. The sheaf of (local) sections of a fibered manifold p : Y → X is denoted by C∞(Y ), C∞(Y , R) denotes the sheaf of (local) functions.

1. Gauge-natural bundles

∞ Let Mm be the category of m-dimensional C -manifolds and smooth embeddings. Let FMm be the category of smooth ﬁbered manifolds over m-dimensional bases and smooth ﬁber manifold maps over embeddings of bases and PBm(G) be the category of smooth principal G-bundles with m-dimensional bases and smooth G-bundle maps (ϕ, f ), where the map f ∈ Mor Mm.

Deﬁnition 1.1. A gauge-natural bundle is a covariant functor F from the category PBm(G) to the category FMm satisfying J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196 179

(i) for each π : P → M in PBm(G), πP : F P → M is a ﬁbered manifold over M, (ii) for each map (ϕ, f ) in PBm(G), F ϕ = F(ϕ,f)is a ﬁbered manifold morphism covering f , (iii) for any open subset U ⊆ M, the immersion ι : π −1(U)→ P is transformed into the immersion −1 → Fι: πP (U) F P .

→ ∈ PB r r Rm × → Let (π : P M) Ob m(G) and W P be the space of all r-jets J(0,e)ϕ,whereϕ : G P m r is in Mor PBm(G), 0 ∈ R and e is the unit in G. The space W P is a principal fiber bundle over the r r manifold M with the structure group WmG of all r-jets J(0,e)Ψ of principal fiber bundle isomorphisms Rm × → Rm × Rm → Rm = r Ψ : G G covering the diffeomorphisms ψ : such that ψ(0) 0. The group WmG r = r Rm Rm r = r Rm is the semidirect product of Gm inv J0 ( , )0 and TmG J0 ( ,G) with respect to the action of r r r = r r → ¯ ∈ PB Gm on TmG given by the jet composition, i.e., WmG Gm TmG.If(ϕ : P P ) Mor m(G), then we can define the principal bundle morphism W r ϕ : W r P → W r P¯ by the jet composition. The ∈ PB r ∈ PB r ∈ PB rule transforming any P Ob m(G) into W P Ob m(WmG) and any ϕ Mor m(G) into r ∈ PB r W ϕ Mor m(WmG) is a gauge-natural bundle. The gauge-natural bundle functor W r plays a fundamental role in the theory of gauge-natural bundles. We have, [2,7],

Theorem 1.2. Every gauge-natural bundle is a ﬁber bundle associated to the bundle W r for a certain order r.

The number r from Theorem 1.2 is called the order of the gauge-natural bundle.SoifF is an r-order gauge-natural bundle, then = r = r F P W P ,SF ,Fϕ W ϕ,idSF , r where SF is a WmG-manifold called the standard fiber of F . λ a i λ i If (x ,z ) is a local fiber coordinate chart on P and (y ) a coordinate chart on SF ,then(x ,y ) is the fiber coordinate chart on F P which is said to be adapted. Let F be a gauge-natural bundle of order s and let r s be a minimal number such that the action of s = s s s s → r WmG Gm TmG on SF can be factorized through the canonical projection πr : TmG TmG. Then r is (s,r) = s r called the gauge-order of F and we say that F is of order (s, r). We shall denote by Wm G Gm TmG the Lie group acting on the standard fiber of an (s, r)-order gauge-natural bundle. Then there is a one- (s,r) to-one, up to equivalence, correspondence of smooth left Wm G-manifolds and gauge-natural bundles of order (s, r),[2].Soany(s, r)-order gauge-natural bundle can be represented by its standard fiber with (s,r) an action of the group Wm G. If F is an (s, r)-order gauge-natural bundle, then J kF is an (s + k,r + k)-order gauge-natural bundle k = k Rm with the standard fiber TmSF J0 ( ,SF ). The class of gauge-natural bundles contains the class of natural bundles in the sense of [7,9,11,14]. Namely, if F is an r-order natural bundle, then F is the (r, 0)-order gauge-natural bundle with trivial gauge structure. ¯ Let F be a gauge-natural bundle and (ϕ, f ) : P → P be in the category PBn(G).Letσ be a section ∗ = ◦ ◦ −1 ¯ of F P . Then we define the section ϕF σ Fϕ σ f of F P .LetH be another gauge-natural bundle.

Deﬁnition 1.3. A natural differential operator from F to H is a collection D ={D(P ), P ∈ Ob PBn(G)} of differential operators from C∞(F P ) to C∞(H P ) satisfying D(P¯ ) ◦ ϕ∗ = ϕ∗ ◦ D(P ) for each map ¯ F H (ϕ, f ) ∈ Mor PBn(G), ϕ : P → P . 180 J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196

D is of order k if all D(P ) are of order k.LetD be a natural differential operator of order k from k F to H .ForanyP ∈ Ob PBn(G) we have the associated map D(P ) : J F P → H P , over M,deﬁned D k = ∈ → by (P )(jx σ) D(P )σ (x) for all x M and any section σ : M F P . From the naturality of D k it follows that D ={D(P ), P ∈ ObPBn(G)} is a natural transformation of J F to H . The following theorem is due to Eck, [2].

Theorem 1.4. Let F and H be gauge-natural bundles of order (s, r), s r. Then we have a one- s+k,r+k to-one correspondence between natural differential operators of order k from F to H and Wn G- k equivariant maps from TmSF to SH .

So according to Theorem 1.4 a classification of natural operators between gauge-natural bundles is equivalent to the classification of equivariant maps between standard fibers. Very important tool in classifications of equivariant maps is the orbit reduction theorem, [7–9]. Let p : G → H be a Lie group epimorphism with the kernel K, M be a left G-space, Q bealeftH -space and π : M → Q be a p- equivariant surjective submersion, i.e., π(gx) = p(g)π(x) for all x ∈ M, g ∈ G.Havingp, we can consider every left H -space N asaleftG-space by gy = p(g)y, g ∈ G, y ∈ N.

Theorem 1.5. If each π −1(q), q ∈ Q is a K-orbit in M, then there is a bijection between the G-maps f : M → N and the H -maps ϕ : Q → N given by f = ϕ ◦ π.

2. Linear connections on vector bundles

In what follows let G = GL(n, R) be the group of linear automorphisms of Rn with coordinates i VB (aj ). Let us consider the category m,n of vector bundles with m-dimensional bases, n-dimensional fibers and local fibered linear diffeomorphisms. Then any vector bundle (p : E → M) ∈ Ob VBm,n can be considered as a zero order gauge-natural functor PBm(G) → VBm,n. Local linear fiber coordinate charts on E will be denoted by (xλ,yi ). The corresponding base of local ∗ i sections of E or E will be denoted by bi or b , respectively. The induced coordinate charts on T M or ∗ λ λ λ ∗ T M will be denoted by (x , x˙ ) or (x , x˙λ) and the induced local bases of sections of T M or T M will λ be denoted by (∂λ) or (d ), respectively.

Deﬁnition 2.1. We deﬁne a linear connection on E to be a linear splitting K : E → J 1E.

Proposition 2.2. Considering the contact morphism J 1E → T ∗M ⊗ T E over the identity of T M,a linear connection can be regarded as a T E-valued 1-form K : E → T ∗M ⊗ T E projecting into the identity of T M. The coordinate expression of a linear connection K is of the type λ i j i ∞ K = d ⊗ ∂λ + Kj λy ∂i , with Kj λ ∈ C (M, R). J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196 181

Linear connections can be regarded as sections of a (1, 1)-order G-gauge-natural bundle Lin E → M, [2,7]. The standard ﬁber of the functor Lin will be denoted by R = Rn∗ ⊗ Rn ⊗ Rm∗,elementsofR will be said to be formal linear connections, the induced coordinates on R will be said to be formal symbols i (1,1) × → of formal linear connections and will be denoted by (Kj λ). The action β : Wm G R R of the group (1,1) = 1 1 Wm G Gm TmG on the standard ﬁber R is given in coordinates by i ◦ = i p ˜ q ˜ρ −˜p Kj λ β ap Kq ρ aj aλ ajλ , λ i i (1,1) ˜ where (aµ,aj ,ajλ) are coordinates on Wm G and denotes the inverse element.

Note 2.3. Let us note that the action β gives, in a natural way, the action r (r+1,r+1) × r → r β : Wm G TmR TmR determined by the jet prolongation of the action β.

r+1,r+1 (r+1,r+1) → (r,r) Remark 2.4. Let us consider the group epimorphism πr,r : Wm G Wm G and its kernel Br+1,r+1G := Ker π r+1,r+1.OnBr+1,r+1G we have the induced coordinates (aλ ,ai ).Then r,r r,r r,r µ1···µr+1 jµ1···µr+1 ¯r r r+1,r+1 the restriction β of the action β to Br,r G has the following coordinate expression i i ¯r i i i i K ,...,K ··· ◦ β = K ,...,K ··· ,K ··· −˜a , (2.1) j µ1 j µ1,µ2 µr+1 j µ1 j µ1,µ2 µr j µ1,µ2 µr+1 jµ1···µr+1 i i i r where (Kj µ1 ,Kj µ1,µ2 ,...,Kj µ1,µ2···µr+1 ) are the induced jet coordinates on TmR.

The curvature of a linear connection K on E turns out to be the vertical valued 2-form R[K]=−[K,K] : E → V E ⊗∧2T ∗M, where [ , ] is the Froehlicher–Nijenhuis bracket. The coordinate expression is i j λ µ i p i j λ µ R[K]=R[K]j λµy ∂i ⊗ d ∧ d =−2 ∂λKj µ + Kj λKp µ y ∂i ⊗ d ∧ d . If we consider the identiﬁcation V E = E ×E and linearity of R[K], the curvature R[K] can be M considered as the curvature tensor ﬁeld R[K] : M → E∗ ⊗ E ⊗∧2T ∗M and R[K] : C∞(Lin E) → C∞ E∗ ⊗ E ⊗∧2T ∗M

(2,2) is a natural operator which is of order one, i.e., we have the associated Wm G-equivariant map, called the formal curvature map of formal linear connections, R 1 → L : TmR U with the coordinate expression i i i p i p i uj λµ ◦ RL = Kj λ,µ − Kj µ,λ + Kj µKp λ − Kj λKp µ, (2.2) i n∗ n 2 m∗ ∗ where (uj λµ) are the induced coordinates on the standard ﬁber U := R ⊗ R ⊗∧ R of E ⊗ E ⊗ ∧2T ∗M. If K∗ is the connection on E∗ dual to K,thenR[K∗]=−R[K]. 182 J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196

We deﬁne a classical connection on M to be a linear symmetric connection on the tangent vector bundle pM : T M → M with the coordinate expression λ µ ν ˙ λ ∞ λ λ Γ = d ⊗ ∂λ + Γν λx˙ ∂µ ,Γµ ν ∈ C (M, R), Γµ ν = Γν µ. Classical connections can be regarded as sections of a 2nd order natural bundle Cla M → M,[7].The standard ﬁber of the functor Cla will be denoted by Q = Rm ⊗ S2Rm∗,elementsofQ willbesaidtobe formal classical connections, the induced coordinates on Q will be said to be formal Christoffel symbols λ 2 × → of formal classical connections and will be denoted by (Γµ ν ). The action α : Gm Q Q of the group 2 Gm on Q is given in coordinates by λ ◦ = λ ρ ˜σ ˜ τ −˜ρ Γµ ν α aρ Γσ τ aµaν aµν .

Note 2.5. Let us note that the action α gives, in a natural way, the action r r+2 × r → r α : Gm TmQ TmQ determined by the jet prolongation of the action α.

r+2 r+2 → r+1 r+2 := Remark 2.6. Let us consider the group epimorphism πr+1 : Gm Gm and its kernel Br+1 Ker π r+2. We have the induced coordinates (aλ ) on Br+2. Then the restriction α¯ r of the action r+1 µ1···µr+2 r+1 r r+2 α to Br+1 has the following coordinate expression λ λ r Γ ,...,Γ ··· ◦¯α µ1 µ2 µ1 µ2,µ3 µr+2 λ λ λ λ = Γ ,...,Γ ··· ,Γ ··· −˜a , (2.3) µ1 µ2 µ1 µ2,µ3 µr+1 µ1 µ2,µ3 µr+2 µ1···µr+2 λ λ λ r where (Γµ1 µ2 ,Γµ1 µ2,µ3 ,...,Γµ1 µ2,µ3···µr+2 ) are the induced jet coordinates on TmQ.

The curvature tensor of a classical connection is a natural operator R[Γ ] : C∞(ClaM) → C∞ T ∗M ⊗ T M ⊗∧2T ∗M

3 which is of order one, i.e., we have the associated Gm-equivariant map, called the formal curvature map of formal classical connections,

R 1 → ∗ ∗ C : TmQ ST ⊗T ⊗∧2T with the coordinate expression ρ ρ ρ σ ρ σ ρ wν λµ ◦ RC = Γν λ,µ − Γν µ,λ + Γν µΓσ λ − Γν λΓσ µ, ρ := = Rm∗ ⊗Rm ⊗∧2Rm∗ where (wν λµ) are the induced coordinates on the standard ﬁber W ST ∗⊗T ⊗∧2T ∗ .

3. Tensor product connections and covariant differentials

p,r := ⊗p ⊗⊗q ∗ ⊗⊗r ⊗⊗s ∗ Let us denote by Eq,s E E T M T M the tensor product over M and recall p,r that Eq,s is a vector bundle which is a G-gauge-natural bundle of order (1, 0). J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196 183

Proposition 3.1. A classical connection Γ on M and a linear connection K on E induce the linear p ⊗ r := ⊗p ⊗⊗q ∗ ⊗⊗r ⊗⊗s ∗ p,r tensor product connection Kq Γs K K Γ Γ on Eq,s p ⊗ r p,r → ∗ ⊗ p,r Kq Γs : Eq,s T M T Eq,s M with the coordinate expression ki ···i λ ···λ i ···i − kλ ···λ Kp ⊗ Γ r = dν ⊗ ∂ + K i1 y 2 p 1 r +···+K ip y 1 p 1 1 r q s ν k ν j1···jq µ1···µs k ν j1···jq µ1···µs ··· ··· ··· ··· − K k yi1 ipλ1 λr −···−K k yi1 ipλ1 λr j1 ν kj2···jq µ1···µs jq ν j1···jq−1kµ1···µs i ···i ρλ ···λ i ···i λ ···λ − ρ + Γ λ1 y 1 p 2 r +···+Γ λr y 1 p 1 r 1 ρ ν j1···jq µ1···µs ρ ν j1···jq µ1···µs ··· ··· ··· ··· ··· ··· − Γ ρ yi1 ipλ1 λr −···−Γ ρ yi1 ipλ1 λr ∂j1 jq µ1 µs , µ1 ν j1···jq ρµ2···µs µs ν j1···jq µ1···µs−1ρ i1···ipλ1···λr i ···i λ ···λ where (xλ,y 1 p 1 r ) are the induced linear ﬁber coordinates on Ep,r. j1···jq µ1···µs q,s

[ p ⊗ r p ⊗ r ] The Froehlicher–Nijenhuis bracket Kq Γs ,Kq Γs has values in the vertical tangent bundle p,r p,r = p,r × p,r V Eq,s . By considering V Eq,s Eq,s Eq,s we have, [4], M

Proposition 3.2. The curvature p ⊗ r := − p ⊗ r p ⊗ r p,r → p,r ⊗∧2 ∗ R Kq Γs Kq Γs ,Kq Γs : Eq,s Eq,s T M is determined by the curvatures R[K] and R[Γ ]. We have the coordinate expression ki ···i λ ···λ i ···i − kλ ···λ R Kp ⊗ Γ r = R[K] i1 y 2 p 1 r +···+R[K] ip y 1 p 1 1 r q s k ν1ν2 j1···jq µ1···µs k ν1ν2 j1···jq µ1···µs ··· ··· ··· ··· − R[K] k yi1 ipλ1 λr −···−R[K] k yi1 ipλ1 λr j1 ν1ν2 kj2···jq µ1···µs jq ν1ν2 j1···jq−1kµ1···µs i ···i ρλ ···λ i ···i λ ···λ − ρ + R[Γ ] λ1 y 1 p 2 r +···+R[Γ ] λr y 1 p 1 r 1 ρ ν1ν2 j1···jq µ1···µs ρ ν1ν2 j1···jq µ1···µs ··· ··· ··· ··· − R[Γ ] ρ yi1 ipλ1 λr −···−R[Γ ] ρ yi1 ipλ1 λr µ1 ν1ν2 j1···jq ρµ2···µs µs ν1ν2 j1···jq µ1···µs−1ρ ··· ··· × ⊗ j1 jq ⊗ ⊗ µ1 µs ⊗ ν1 ∧ ν2 bi1···ip b ∂λ1···λr d d d , ··· = ⊗ ··· ⊗ j1 jq = j1 ⊗ ··· ⊗ jq = ⊗ ··· ⊗ where we have put bi1···ip bi1 bip , b b b , ∂λ1···λr ∂λ1 ∂λr , ··· dµ1 µs = dµ1 ⊗···⊗dµs .

p ⊗ r Let us note that the tensor product connection Kq Γs can be considered as a linear splitting p ⊗ r p,r → 1 p,r Kq Γs : Eq,s J Eq,s .

∈ ∞ p,r Definition 3.3. Let Φ C (Eq,s ).Wedefinethecovariant differential of Φ with respect to the pair of p,r ⊗ ∗ connections (K, Γ ) as a section of Eq,s T M defined by ∇(K,Γ ) = 1 − p ⊗ r ◦ Φ j Φ Kq Γs Φ. 184 J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196

··· ··· ∞ p,r i1 ipλ1 λr j ···j µ ···µ Proposition 3.4. Let Φ ∈ C (E ), Φ = φ b ··· ⊗ b 1 q ⊗ ∂ ··· ⊗ d 1 s . Then we have q,s j1···jq µ1···µs i1 ip λ1 λr the coordinate expression

i ···i λ ···λ ··· ··· ∇(K,Γ ) =∇(K,Γ ) 1 p 1 r ⊗ j1 jq ⊗ ⊗ µ1 µs ⊗ ν Φ ν φj ···j µ ···µ bi1···ip b ∂λ1···λr d d 1 q 1 s i ···i λ ···λ ki ···i λ ···λ i ···i − kλ ···λ = ∂ φ 1 p 1 r − K i1 φ 2 p 1 r −···−K ip φ 1 p 1 1 r ν j1···jq µ1···µs k ν j1···jq µ1···µs k ν j1···jq µ1···µs ··· ··· ··· ··· + K k φi1 ipλ1 λr +···+K k φi1 ipλ1 λr j1 ν kj2···jq µ1···µs jq ν j1···jq−1kµ1···µs i ···i ρλ ···λ i ···i λ ···λ − ρ − Γ λ1 φ 1 p 2 r −···−Γ λr φ 1 p 1 r 1 ρ ν j1···jq µ1···µs ρ ν j1···jq µ1···µs ··· ··· ··· ··· + Γ ρ φi1 ipλ1 λr +···+Γ ρ φi1 ipλ1 λr µ1 ν j1···jq ρµ2···µs µs ν j1···jq µ1···µs−1ρ ··· ··· × ⊗ j1 jq ⊗ ⊗ µ1 µs ⊗ ν bi1···ip b ∂λ1···λr d d .

Remark 3.5. Eck, [2], has deﬁned the so-called gauge-covariant differential ∇K Φ of Φ with respect to K. But the gauge-covariant differential ∇K Φ is a section of a (2, 0)-order G-gauge-natural bundle and so its order is different from the order of Φ. This is why the gauge-covariant differentials can not be used in reduction theorems for linear connections and why we have to use auxiliary classical connections on base manifolds.

i ···i λ ···λ i ···i λ ···λ In what follows we set ∇=∇(K,Γ ) and φ 1 p 1 r =∇ φ 1 p 1 r . j1···jq µ1···µs ;ν ν j1···jq µ1···µs We have the following relations between the covariant differentials and the curvatures, [4].

Theorem 3.6 (The generalized Bianchi identity). We have

i i i R[K]j λµ;ν + R[K]j µν;λ + R[K]j νλ;µ = 0.

∈ ∞ p,r Theorem 3.7. Let Φ C (Eq,s ). Then we have 1 Alt∇2Φ =− R Γ p ⊗ Kr ◦ Φ ∈ C∞ Ep,r ⊗∧2T ∗M , 2 q s q,s where Alt is the antisymmetrization.

[ p ⊗ r ] Remark 3.8. From the above Theorem 3.7 and the coordinate expression of R Kq Γs it follows, that ∇2 p,r [ ] [ ] Alt Φ is a Eq,s -valued 2-form which is a quadratic polynomial in R K ,R Γ ,Φ.

Example 3.9. We have

Alt∇2R[K] : M → E∗ ⊗ E ⊗∧2T ∗M ⊗∧2T ∗M, given in coordinates by 1 ∇2 [ ]=− [ ] i [ ] p − [ ] p [ ] i Alt R K R K p ν1ν2 R K j λµ R K j ν1ν2 R K p λµ 2 − [ ] ω [ ] i − [ ] ω [ ] i j ⊗ ⊗ λ ∧ µ ⊗ ν1 ∧ ν2 R Γ λ ν1ν2 R K j ωµ R Γ µ ν1ν2 R K j λω b bi d d d d . J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196 185

4. The ﬁrst reduction theorem for linear and classical connections

Let us introduce the following notations. ∗ 2 ∗ i ∗ r Let WM = W0M := T M ⊗ T M ⊗∧ T M, Wi M = WM ⊗⊗T M, i 0. Let us put W M = r W0M ×···×Wr M.ThenWi M and W M are natural bundles of order one and the corresponding M M r m∗ m 2 m∗ i m∗ standard ﬁbers will be denoted by Wi and W ,whereW0 := W = R ⊗R ⊗∧ R , Wi = W ⊗⊗ R , r i 0, and W = W0 ×···×Wr . We denote by R i+1 → C,i : Tm Q Wi i+3 the Gm -equivariant map associated with the ith covariant differential of curvature tensors of classical connections i ∞ ∞ ∇ R[Γ ] : C (Cla M) → C (WiM).

The map RC,i issaidtobetheformal curvature map of order i of classical connections. Let CC,i ⊂ Wi be a subset given by identities of the ith covariant differentials of the curvature tensors r = ×···× of classical connections. Let us put CC CC,0 CC,r and Rr := R R r+1 → r C ( C,0,..., C,r) : Tm Q W (4.1) r which has values in CC . Similarly let ∗ 2 ∗ i ∗ UE = U0E := E ⊗ E ⊗∧ T M,UiE = UE ⊗⊗T M,i 0, r U E = U0E × ···×Ur E. M M r Then UiE and U E are G-gauge-natural bundles of order (1, 0) and the corresponding standard ﬁbers r n∗ n 2 m∗ i m∗ will be denoted by Ui and U ,whereU0 := U = R ⊗ R ⊗∧ R , Ui = U ⊗⊗R , i 0, and r U = U0 ×···×Ur . We denote by R i−1 × i+1 → L,i : Tm Q Tm R Ui (i+2,i+2) the Wm G-equivariant map associated with the ith covariant differential of curvature tensors of linear connections i ∞ ∞ ∇ R[K] : C ClaM × LinE → C (UiE). M

The map RL,i is said to be the formal curvature map of order i of linear connections. Let CL,i ⊂ Ui be a subset given by identities of the ith covariant differentials of the curvature tensors r = ×···× of linear connections. Let us put CL CL,0 CL,r and Rr := R R r−1 × r+1 → r L ( K,0,..., L,r) : Tm Q Tm R U r which has values in CL. r r r In [7] it was proved that CC is a submanifold in W and the restriction of (4.1) to CC isasurjective submersion. Now we shall prove 186 J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196

− s × r s × r s × r = Lemma 4.1. If s r 2, r 0,thenCC CL is a submanifold of W U . It holds (CC CL) Rs Rr s+1 × r+1 ( C, L)(Tm Q Tm R) and the restricted map Rs Rr s+1 × r+1 → s × r C, L : Tm Q Tm R CC CL is a surjective submersion.

r r Proof. First we prove CL is a submanifold in U . = 0 = 0 = = We prove it by induction. For r 0wehaveCL U U.Forr 1wehaveCL,1 a linear subspace of U1 given by the solution of the formal generalized Bianchi identity (see Theorem 3.6)

i i i uj λµρ + uj µρλ + uj ρµλ = 0. (4.2) 1 0 × Hence CL is a vector subbundle of CL U1. r−1 ⊂ r−1 r−1 × Assume CL U is a submanifold and consider the product bundle CL Ur . Equations deﬁning CL,r consist of the formal covariant differentials of Eq. (4.2) and of formal expressions of alternations of second order covariant differentials and their covariant differentials. By Theorem 3.7 and Example 3.9 we have the following two systems of equations

i u ··· = 0, (4.3) j (λµρ1)ρ2 ρr i + r−2 × r−2 = uj λµρ1···[ρs−1ρs ]···ρr pol CC CL 0, (4.4) = r−2 × r−2 s 2,...,r,where(...) denotes the cyclic permutation and pol(CC CL ) are some polynomials r−2 × r−2 on CC CL . The map defined by the left-hand sides of (4.3) and (4.4) represents an affine bundle r−1 × → r−1 × RN = morphism CL Ur CL of constant rank, N the number of Eqs. (4.3) and (4.4). Its kernel r r−1 × CL is a subbundle of CL Ur . Rs Rr Rr s+1 − To prove ( C, L) is a surjective submersion it is sufficient to prove that the map L(j0 γ, ) : r+1 → r s+1 ∈ s+1 Tm R CL is a surjective submersion for any j0 γ Tm Q. We shall prove it by induction. For = R0 s+1 − ≡ R R 1 = 0 = r 0wehave L(j0 γ, ) L and we shall prove L(TmR) CL U. The coordinate expression R 1 1 → of L is (2.2). This is an affine bundle morphism of affine bundle π0 : TmR R into U of constant rank. We know that the values of RL lie in U, so that it suffices to prove that the image is the whole U at one ∈ R¯ R 1 −1 ∈ point 0 R. Consider the restriction L of L to the fiber (π0 ) (0),0 R.Then

i ¯ i i uj λµ ◦ RL = Kj λ,µ − Kj µ,λ. ¯ RL is a surjection if and only if = 1 −1 − R¯ dim U dim π0 (0) dim Ker L. = 1 2 − 1 −1 = 2 2 R¯ = 1 2 + But we have dim U n m(m 1),dim(π0 ) (0) n m and dim Ker L n m(m 1). This implies ¯ 2 2 that RL is surjective and hence RL is a surjective submersion. Assume by induction Rr−1 s+1 − r → r−1 L j0 γ, : TmR CL J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196 187 is a surjective submersion. So we have the commutative diagram

+ Rr (js 1γ,−) r+1 L 0 r Tm R CL

r+1 πr Rr−1 s+1 − r L (j0 γ, ) r−1 TmR CL Rr s+1 − where the bottom arrow is by assumption the surjective submersion. L(j0 γ, ) is affine (with respect to affine structures of both projections) and of constant rank on all fibers. Hence if fibers of the projection r+1 r+1 → r r → r−1 πr : Tm R TmR are mapped surjectively on fibers of the projection CL CL , the top arrow is a R¯ r s+1 − Rr s+1 − surjective submersion. Let us denote by L(j0 γ, ) the restriction of L(j0 γ, ) to the fiber over ∈ r R¯ r s+1 − = R¯ s+1 − r+1 −1 → 0 TmR.Then L(j0 γ, ) L,r(j0 γ, ) : (πr ) (0) (CL,r)0,where(CL,r)0 is the fiber of r → r−1 r−1 the projection CL CL over the zero in CL . R¯ s+1 − r+1 −1 → We shall prove L,r(j γ, ) : (πr ) (0) (CL,r)0 is surjective. In coordinates 0 i ◦ R¯ s+1 − = i − i uj λµρ1···ρr L,r j0 γ, Kj λ,µρ1···ρr Kj µ,λρ1···ρr . r m∗ r−1 m∗ The fiber (CL,r)0 = (CL,0 ⊗ S R ) ∩ (CL,1 ⊗ S R ), which follows from (4.3) and (4.4). r+1 −1 = Rn∗ ⊗ Rn ⊗ Rm∗ ⊗ r+1Rm∗ Let us note that (πr ) (0) S . Consider an element r m∗ r−1 m∗ X ∈ CL,0 ⊗ S R ∩ CL,1 ⊗ S R = (CL,r)0. By the induction assumption the map R¯ s+1 − Rn∗ ⊗ Rn ⊗ Rm∗ ⊗ 2Rm∗ → L,1 j0 γ, : S (CL,1)0 is a surjection and hence s+1 ∗ ∗ ∗ − ∗ − ∗ R¯ − ⊗ − ∗ Rn ⊗ Rn ⊗ Rm ⊗ 2Rm ⊗ r 1Rm → ⊗ r 1Rm L,1 j0 γ, idSr 1Rm : S S (CL,1)0 S ∈ Rn∗ ⊗ Rn ⊗ Rm∗ ⊗ 2Rm∗ ⊗ r−1Rm∗ R¯ s+1 − ⊗ is also a surjection. So we have Y S S such that ( L,1(j0 γ, ) = idSr−1Rm∗ )(Y ) X, i.e., in coordinates i = i − i Xj λµρ1···ρr Yj λµρ1···ρr Yj µλρ1···ρr . The symmetrization gives ¯ i = i ∈ Rn∗ ⊗ Rn ⊗ Rm∗ ⊗ r+1Rm∗ = r+1 −1 Yj λµρ1ρ2ρ3···ρr Yj λµ(ρ1ρ2)ρ3···ρr S πr (0) R¯ s+1 ¯ = R¯ s+1 − such that L,r(j0 γ,Y) X and hence L,r(j0 γ, ) is a surjection. It implies that the top arrow in the above diagram is a surjective submersion. ✷

s × r s × r In the above Lemma 4.1 we have proved CC CL is a submanifold of W U . It is easy to see that s × r (1,0) = 1 × R CC CL is closed with respect to the action of the group Wm G Gm GL(n, ) and, according to the general theory of gauge-natural bundles, we have the (1, 0)-order G-gauge-natural vector subbundle s × r s × r CC M CLE of the (1, 0)-order G-gauge-natural vector bundle W M U E called the (s, r)-order M M curvature bundle of linear and classical connections. (1,0) Let F be a G-gauge-natural bundle of order (1, 0), i.e., SF is a Wm G-manifold. 188 J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196

− (s+2,r+1) Theorem 4.2. Let s r 2, r 0. For every Wm G-equivariant map s × r → f : TmQ TmR SF (1,0) s−1 × r−1 → there exists a unique Wm G-equivariant map g : CC CL SF satisfying = ◦ Rs−1 Rr−1 f g C , L .

Proof. Let us consider the space m s m∗ n∗ n r m∗ SC,s := R ⊗ S R or SL,r := R ⊗ R ⊗ S R λ i s with coordinates (s µ1µ2···µs ) or (sj µ1···µr ), respectively. Let us consider the action of Gm on SC,s and the (r,r) action of Wm G on SL,r given by λ λ λ i i i s¯ ··· = s ··· −ã , s¯ ··· = s ··· −ã . (4.5) µ1µ2 µs µ1µ2 µs µ1···µs j µ1 µr j µ1 µr jµ1···µr From (2.1), (2.3) and (4.5) it is easy to see that the symmetrization maps s → r → σC,s : TmQ SC,s+2,σL,r : TmR SL,r+1 given by λ ◦ = λ i ◦ = i s µ1µ2···µr+1 σC,s Γ(µ1 µ2,µ3···µs+2), sj µ1···µr+1 σL,r Kj (µ1,µ2···µr+1) are equivariant. s+2 We have the Gm -equivariant map := s R s → × s−1 × ϕC,s σC,s,πs−1, C,s−1 : TmQ SC,s+2 Tm Q Ws−1. s+2 On the other hand we define the Gm -equivariant map × s−1 × → s ψC,s : SC,s+2 Tm Q Wr−1 TmQ s−1 over the identity of Tm Q by the following coordinate expression λ = λ + λ − s−1 Γµ ν,ρ1···ρs s µνρ1···ρs lin wµ νρ1···ρs pol Tm Q , (4.6) where lin denotes the linear combination with real coefficients which arises in the following way. We recall that RC,s−1 gives the coordinate expression λ − λ = λ − s−1 Γµ ν,ρ1···ρs Γµ ρ1,νρ2···ρs wµ νρ1···ρs pol Tm Q . (4.7) We can write λ = λ + λ − λ Γµ ν,ρ1···ρs s µνρ1···ρs Γµ ν,ρ1···ρs Γ(µ ν,ρ1···ρs ) . Then the term in brackets can be written as a linear combination of terms of the type λ − λ Γµ ν,ρi ρ1···ρi−1ρi+1···ρs Γµ ρi ,νρ1···ρi−1ρi+1···ρs , i = 1,...,s, and from (4.7) we get (4.6). Moreover,

◦ = s ψC,s ϕC,s idTmQ . J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196 189

(r+1,r+1) Similarly we have the Wm G-equivariant map r r−2 r r−2 r−1 ϕ := σ , id r−2 × π , R − : T Q × T R → S + × T Q × T R × U − L,r L,r Tm Q r−1 L,r 1 m m L,r 1 m m r 1 (r+1,r+1) and we deﬁne the Wm G-equivariant map × r−2 × r−1 × → r−2 × r ψL,r : SL,r+1 Tm Q Tm R Ur−1 Tm Q TmR r−2 × r−1 over the identity of Tm Q Tm R by the following coordinate expression i = i + i − r−2 × r−1 Kj λ,ρ1···ρr sj λρ1···ρr lin uj λρ1···ρr pol Tm Q Tm R , (4.8) where lin denotes the linear combination with real coefﬁcients which arises in the following way. We recall that RL,r−1 gives the coordinate expression i − i = i − r−2 × r−1 Kj λ,ρ1···ρr Kj ρ1,λρ2···ρr uj λρ1···ρr pol Tm Q Tm R . (4.9) We can write i = i + i − i Kj λ,ρ1···ρr sj λρ1···ρr Kj λ,ρ1···ρr Kj (λ,ρ1···ρr ) . Then the term in brackets can be written as a linear combination of terms of the type i − i Kj λ,ρi ρ1···ρi−1ρi+1···ρr Kj ρi ,λρ1···ρi−1ρi+1···ρr , i = 1,...,r, and from (4.9) we get (4.8). Moreover,

ψL,r ◦ ϕL,r = id r−2 × r . Tm Q TmR Now we have to distinguish three possibilities. = − r+1 (r+1,r+1) r−1 (A) Let s r 1. We have the same orders of groups Gm and Wm G acting on Tm Q and r TmR, respectively. Let us denote by r := r−2 × r−1 × × A Tm Q Tm R Wr−2 Ur−1. r Then the map f ◦(ψC,r−1,ψL,r) : SC,r+1 ×SL,r+1 ×A → SF satisﬁes the conditions of the orbit reduction r+1,r+1 (r+1,r+1) → (r,r) Theorem 1.5 for the group epimorphism πr,r : Wm G Wm G and the surjective submersion × × r → r × r+1,r+1 pr3 : SC,r+1 SL,r+1 A A . Indeed, the space SC,r+1 SL,r+1 is a Br,r G-orbit. Moreover, (4.5) r+1,r+1 × implies that the action of Br,r G on SC,r+1 SL,r+1 is simply transitive. Hence there exists a unique (r,r) Wm G-equivariant map r = r−2 × r−1 × × → gr : A Tm Q Tm R Wr−2 Ur−1 SF such that the following diagram

× × r (ψC,r−1,ψL,r ) r−1 × r f SC,r+1 SL,r+1 A Tm Q TmR SF

− pr r 1× r R R idS 3 (πr−2 πr−1, C,r−2, L,r−1) F

r idAr r gr A A SF 190 J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196

◦ = ◦ commutes. So f (ψC,r−1,ψL,r) gr pr3 and if we compose both sides with (ϕC,r−1,ϕL,r),by r−1 r considering pr ◦(ϕC,r−1,ϕL,r) = (π − × π − , RC,r−2, RL,r−1), we obtain 3 r 2 r 1 = ◦ r−1 × r R R f gr πr−2 πr−1, C,r−2, L,r−1 .

In the second step we consider the same construction for the map gr and obtain the commutative diagram

(ψ − ,ψ − ,id × ) r−1 C,r 2 L,r 1 Wr−2 Ur−1 r gr SC,r × SL,r × A × Wr−2 × Ur−1 A SF

r−2 r−1 pr × R − R − × 3,4,5 (πr−3 πr−2 , C,r 3, L,r 2,idWr−2 Ur−1 ) idSF

id r−1× × r−1 A Wr−2 Ur−1 r−1 gr−1 A × Wr−2 × Ur−1 A × Wr−2 × Ur−1 SF (r−1,r−1) r−1 × × → So that there exists a unique Wm G-equivariant map gr−1 : A Wr−2 Ur−1 SF such that = ◦ r−2 × r−1 R R gr gr−1 (π − π − , C,r−3, L,r−2, idW − ×U − ), i.e., r 3 r 2 r 2 r 1 = ◦ r−1 × r R R R R f gr−1 πr−3 πr−2, C,r−3, C,r−2, L,r−2, L,r−1 . (1,1) × r−2 × Proceeding in this way we get in the last step a unique Wm G-equivariant map g1 : R W U r−1 → S such that F = ◦ r ◦ Rr−2 Rr−1 f g1 π0 pr2, C , L . 1,1 (1,1) → (1,0) Finally, g1 satisﬁes the orbit reduction theorem for the group epimorphism π1,0 : Wm G Wm G × r−2 × r−1 → r−2 × r−1 1,1 and the surjective submersion pr2,3 : R W U W U . Indeed, R is a B1,0 G-orbit (1,0) r−2 × r−1 → (it follows from (2.1)). So there is a unique Wm G-equivariant map g : W U SF such that g1 = g ◦ pr and hence 2,3 = ◦ Rr−2 Rr−1 f g C , L . = − r r−2 (B) Let s r 2. We have the action of the group Gm on Tm Q and the action of the group (r+1,r+1) r Wm G on TmR. r−2 r−1 Then the map f ◦ (id r−2 ,ψ ) : S + × T Q × T R × U − → S satisﬁes the conditions of Tm Q L,r L,r 1 m m r 1 F r+1,r+1 (r+1,r+1) → (r,r) the orbit reduction Theorem 1.5 for the group epimorphism πr,r : Wm G Wm G and the × r−2 × r−1 × → r−2 × r−1 × surjective submersion pr2,3,4 : SL,r+1 Tm Q Tm R Ur−1 Tm Q Tm R Ur−1. Indeed, the r+1,r+1 r+1,r+1 space SL,r+1 is a Br,r G-orbit. Let us note that the action of Br,r G on SL,r+1 is transitive, but not (r,r) r−2 × r−1 × → simple transitive. Hence there exists a unique Wm G-equivariant map gr : Tm Q Tm R Ur−1 SF such that the following diagram

(id r−2 ,ψL,r ) × r−2 × r−1 × Tm Q r−2 × r f SL,r+1 Tm Q Tm R Ur−1 Tm Q TmR SF

× r R pr2,3,4 (id r−2 π − , L,r−1) idS Tm Q r 1 F

id r−2 × r−1 × r−2 × r−1 × Tm Q Tm R Ur−1 r−2 × r−1 × gr Tm Q Tm R Ur−1 Tm Q Tm R Ur−1 SF

◦ − = ◦ − commutes. So f (idT r 2Q,ψL,r) gr pr2,3,4 and if we compose both sides with (idT r 2Q,ϕL,r),by m r m considering pr ◦(id r−2 ,ϕL,r) = (id r−2 ×π − , RL,r−1), we obtain 2,3,4 Tm Q Tm Q r 1 r f = g ◦ id r−2 ×π , R − . r Tm Q r−1 L,r 1 J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196 191

(1,0) Further we proceed as in the second step in (A) and we get a unique Wm G-equivariant map r−3 r−1 g : W × U → SF such that = ◦ Rr−3 Rr−1 f g C , L . − (s+2,r+1) s × r (C) Let s>r 1. We have the action of the group Wm G on TmQ TmR. s−1 r Then the map f ◦ (ψ , id r−2 ) : S + × T Q × T R × W − → S satisﬁes the conditions of C,s Tm Q C,s 2 m m s 1 F s+2,r+1 (s+2,r+1) → (s+1,r+1) the orbit reduction Theorem 1.5 for the group epimorphism πs+1,r+1 : Wm G Wm G and × s−1 × r × → s−1 × r × the surjective submersion pr2,3,4 : SC,s+2 Tm Q TmR Ws−1 Tm Q TmR Ws−1. Indeed, s+2,r+1 s+2,r+1 the space SC,s+2 is a Bs+1,r+1 G-orbit. Moreover, the action of Bs+1,r+1 G on SC,s+2 is simply transitive. (s+1,r+1) Hence there exists a unique Wm G-equivariant map s−1 × r × → gs+1 : Tm Q TmR Ws−1 SF such that the following diagram

(ψ ,id r ) × s−1 × r × C,s TmR s × r f SC,s+2 Tm Q TmR Ws−1 TmQ TmR SF

pr (πs ×id r ,R − ) 2,3,4 s−1 TmR C,s 1 idSF

id s−1 × r × s−1 × r × Tm Q TmR Ws−1 s−1 × r × gs+1 Tm Q TmR Ws−1 Tm Q TmR Ws−1 SF

◦ r = + ◦ r commutes. So f (ψC,s, idTmR) gs 1 pr2,3,4 and if we compose both sides with (ϕC,s, idTmR,),by s ◦ r = × r R − considering pr2,3,4 (ϕC,s, idTmR) (πs−1 idTmR, C,s 1), we obtain s = + ◦ × r R − f gs 1 πs−1 idTmR, C,s 1 .

In the second step we consider the same construction for the map gs+1 and we obtain that there (s,r+1) s−2 × r × × → = exists a unique Wm G-equivariant map gs : Tm Q TmR Ws−2 Ws−1 SF such that gs+1 s−1 ◦ × r R − R − gs (πs−2 idTmR, C,s 2, C,s 1), i.e., s = ◦ × r R − R − f gs πs−2 idTmR, C,s 2, C,s 1 . (r+1,r+1) Proceeding in this way we get a Wm G-equivariant map r−1 × r × ×···× → gr+1 : Tm Q TmR Wr−1 Ws−1 SF such that s = + ◦ × r R − R − f gr 1 πr−1 idTmR, C,r 1,..., C,s 1 . (1,0) But the map gr+1 satisﬁes the conditions of the case (A) and so we obtain a unique Wm G-equivariant morphism

s−1 r−1 g : W × U → SF = ◦ Rr−2 Rr−1 such that gr+1 g ( C , L , idWr−1×···×Ws−1 ), i.e., = ◦ Rs−1 Rr−1 f g C , L . 192 J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196

Summarizing all cases we have = ◦ Rs−1 Rr−1 f g C , L − r−2 × r−1 for any s r 2 and the restriction of g to CC CL is uniquely determined map we wished to ﬁnd. ✷

In the above Theorem 4.2 we have ﬁnd a map g which factorizes f , but we did not prove, that Rs−1 Rr−1 s × r → s−1 × r−1 ( C , L ) : TmQ TmR CC CL satisfy the orbit conditions, namely we did not prove that Rs−1 Rr−1 −1 s+2,r+1 ∈ s−1 × r−1 ( C , L ) (rC ,rL) is a B1,0 G-orbit for any (rC ,rL) CC CL . Now we shall prove it.

s r s ´ r ´ ∈ s × r Lemma 4.3. If (j0 γ,j0 κ),(j0 γ,j0 κ) TmQ TmR satisfy Rs−1 Rr−1 s r = Rs−1 Rr−1 s ´ r ´ C , L j0 γ,j0 κ C , L j0 λ, j0 κ , ∈ s+2,r+1 s ´ r ´ = s r then there is an element h B1,0 G such that h(j0 γ,j0 κ) (j0 γ,j0 κ).

s × r s+2,r+1 (1,0) Proof. Consider the orbit set (TmQ TmR)/B1,0 G.ThisisaWm G-set. Clearly the factor projection s × r → s × r s+2,r+1 p : TmQ TmR TmQ TmR /B1,0 G (s+2,r+1) (1,0) is a Wm G-map. By Theorem 4.2 there is a Wm G-equivariant map s−1 × r−1 → s × r s+2,r+1 g : CC CL TmQ TmR /B1,0 G = ◦ Rs−1 Rr−1 Rs−1 Rr−1 s r = Rs−1 Rr−1 s ´ r ´ = satisfying p g ( C , L ).If( C , L )(j0 γ,j0 κ) ( C , L )(j0 λ, j0 κ) (rC ,rL),then s r = s ´ r ´ = ✷ p(j0 γ,j0 κ) p(j0 γ,j0 κ) g(rC,rL). This proves Lemma 4.3.

(k) 0 1 k Setting ∇ = (∇ = idV , ∇ ,...,∇ ), then, as a direct consequence of Theorem 4.2, we obtain the ﬁrst reduction theorem for linear and classical connections.

Theorem 4.4. Let s r − 2, r 0. All natural differential operators f : C∞ Cla M × Lin E → C∞(F E) M which are of order s with respect to classical connections and of order r with respect to linear connections are of the form f(Γ,K)= g ∇(s−1)R[Γ ], ∇(r−1)R[K] , where g is a zero order natural operator ∞ s−1 × r−1 → ∞ g : C CC M CL E C (F E). M

Remark 4.5. From the proof of Theorem 4.2 it follows that the operator g is the restriction of a zero order operator deﬁned on the gauge-natural tensor bundle W s−1M × U r−1E. M J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196 193

5. The second reduction theorem for linear and classical connections

∗ Write (Ep1,p2 ) := Ep1,p2 ⊗⊗iT M, i 0, and set q1,q2 i q1,q2 k p1,p2 = p1,p2 × ···× p1,p2 Eq ,q Eq ,q Eq ,q . 1 2 1 2 0 M M 1 2 k The kth order covariant differential of sections of Ep1,p2 with respect to (K, Γ ) is a natural operator q1,q2 ∞ ∞ ∇k × × p1,p2 → p1,p2 : C Cla M LinE Eq ,q C Eq ,q M M 1 2 1 2 k which is of order (k − 1) with respect to classical and linear connections and of order k with respect to sections of Ep1,p2 . Let us note that Ep1,p2 is a (1, 0)-order gauge-natural bundle and let us denote by q1,q2 q1,q2 ··· ··· p n q n∗ p m q m∗ A i1 ip1 λ1 λp2 V := ⊗ 1 R ⊗⊗ 1 R ⊗ 2 R ⊗⊗ 2 R its standard fiber with coordinates (v ) = (v ··· ··· ).By j1 jq1 µ1 µq2 V or V k := V ×···×V we denote the standard fibers of (Ep1,p2 ) or (Ep1,p2 )k, respectively. i 0 k q1,q2 i q1,q2 (k+1,k+1) Hence we have the associated Wm G-equivariant map, denoted by the same symbol, ∇k k−1 × k−1 × k → : Tm Q Tm R TmV Vk. A A A k If (v ,v λ,...,v λ1···λk ) are the induced jet coordinates on TmV (symmetric in all subscripts) and A ∇k (V λ1···λk ) are the canonical coordinates on Vk,then is of the form A ◦∇k = A + k−1 × k−1 × k−1 V λ1···λk v λ1···λk pol Tm Q Tm R Tm V , (5.1) k−1 × k−1 × k−1 where pol is a quadratic homogeneous polynomial on Tm Q Tm R Tm V . We define the kth order formal Ricci equations, k 2, as follows. For k = 2wehavebyRemark3.8 A − 0 × 0 × = V [λµ] pol CC CL V 0. (E2)

For k>2, (Ek) is obtained by the formal covariant differentiating of (Ek−1) and antisymmetrization of the last two formal covariant differentials. They are of the form A − k−2 × k−2 × k−2 = V λ1···[λi λi+1]···λk pol CC CL V 0, (Ek) i = 1,...,k− 1.

k ⊂ k−2 × k−2 × k Deﬁnition 5.1. The kth order formal Ricci subspace Z CC CL V is deﬁned by equations 0 1 1 (E2),...,(Ek), k 2. For k = 0, 1 we set Z = V and Z = V .

k k−2 × k−2 × k Lemma 5.2. Z is a submanifold of CC CL V , it holds k = Rk−2 Rk−2 ∇(k) k−1 × k−1 × k Z C , L , Tm Q Tm R TmV and the restricted morphism Rk−2 Rk−2 ∇(k) k−1 × k−1 × k → k C , L , : Tm Q Tm R TmV Z is a surjective submersion. 194 J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196

= 0 = ∇0 = = ∇(1) × × 1 → 1 = 1 Proof. For k 0wehaveZ V and idV .Fork 1, : Q R TmV V Z is of the form A A A A V = v ,Vλ = v λ + pol(Q × R × V) so that our claim is trivial. k−1 k−3 × k−3 × Assume by induction Z is a submanifold and the restriction of the projection pr1,2 : CC CL V k−1 → Ck−3 × Ck−3 to Zk−1 is a surjective submersion. Consider the fiber product C L k−2 × k−2 × k−1 CC CL Z k−3× k−3 CC CL and the fiber product k−2 × k−2 × k−1 × CC CL Z Vk. k−3× k−3 CC CL k k By (Ek)Z is defined by affine equations of constant rank. This proves Z is a subbundle and k → k−2 × k−2 Z CC CL is a surjective submersion. Assume by induction Rk−3 Rk−3 ∇(k−1) k−2 × k−2 × k−1 → k−1 C , L , : Tm Q Tm R Tm V Z is a surjective submersion. We have k = k−1 × ⊗ kRm∗ TmV Tm V V S .

By (5.1) and (Ek) Rk−2, Rk−2, ∇(k) : T k−1Q × T k−1R × T k−1V × V ⊗ SkRm∗ C L m m m → k → k−2 × k−2 × k−1 Z CC CL Z k−3× k−3 CC CL is bijective on each ﬁber. This proves Lemma 5.2. ✷

k k−2 × k−2 × k In the above Lemma 5.2 we have proved Z is a submanifold of CC CL V . It is easy to see k (1,0) = 1 × R that Z is closed with respect to the action of the group Wm G Gm GL(n, ) and, according to the general theory of gauge-natural bundles, we have the (1, 0)-order G-gauge-natural vector subbundle − − k k 2 × k 2 × p1,p2 k Z E of the (1, 0)-order G-gauge-natural vector bundle CC M CL E (Eq ,q ) called the kth order M M 1 2 Ricci subbundle.

− − (s+2,r+1) Theorem 5.3. Let s,r k 1, s r 2. For every Wm G-equivariant map s × r × k → f : TmQ TmR TmV SF there exists a unique W (1,0)G-equivariant map m s−1 × r−1 × k → g : CC CL Z SF k−2× k−2 CC CL satisfying = ◦ Rs−1 Rr−1 ∇(k) f g C , L , . J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196 195

Proof. Consider the map (k) s r k s r k s × r ∇ × × → × × idTmQ TmR, : TmQ TmR TmV TmQ TmR V ˜ k ⊂ s × r × k and denote by V TmQ TmR V its image. By (5.1), the restricted morphism ∇˜ (k) s × r × k → ˜ k : TmQ TmR TmV V is bijective for every (j sγ,jr κ) ∈ T s Q × T r R,sothat∇˜ (k) is an equivariant diffeomorphism. Deﬁne 0 0 m m R˜ s−1, R˜ r−1 : V˜ k → Cs−1 × Cr−1 × Zk, C L C L − − Ck 2×Ck 2 C L R˜ s−1 R˜ r−1 s r = Rs−1 Rr−1 s r C , L j0 γ,j0 κ,v C , L j0 γ,j0 κ ,v , s r ∈ ˜ k R˜ s−1 R˜ r−1 (j0 γ,j0 κ,v) V . By Lemma 4.1 ( C , L ) is a surjective submersion. R˜ s−1 R˜ r−1 Thus, Lemmas 4.1 and 4.3 imply that ( C , L ) satisﬁes the orbit conditions for the group + + epimorphism π s 2,r 1 : W s+2,r+1G → W 1,0G and there exists a unique W (1,0)G-equivariant map 1,0 m m m s−1 × r−1 × k → g : CC CL Z SF k−2× k−2 CC CL such that the diagram

(∇˜ (k))−1 f ˜ k s × r × k V TmQ TmR T V SF

R˜ s−1 R˜ r−1 Rs−1 Rr−1 ∇(k) ( C , L ) ( C , L , ) idSF

s−1 × r−1 × k id s−1 × r−1 × k g (CC CL ) Z (CC CL ) Z SF k−2× k−2 k−2× k−2 CC CL CC CL ◦ ∇˜ (k) −1 = ◦ R˜ s−1 R˜ r−1 ∇˜ (k) commutes. Hence f ( ) g ( C , L ). Composing both sides with , by considering (R˜ s−1, R˜ r−1) ◦ ∇˜ (k) = (Rs−1, Rr−1, ∇(k)),weget C L C L = ◦ Rs−1 Rr−1 ∇(k) ✷ f g C , L , . Now as a direct consequence of Theorem 5.3 we obtain the second reduction theorem for linear and classical connections.

Theorem 5.4. Let s,r k − 1, s r − 2. All natural differential operators ∞ ∞ × × p1,p2 → f : C Cla M Lin E Eq ,q C (F E) M M 1 2 which are of order s with respect to classical connections, of order r with respect to linear connections p1,p2 and of order k with respect to sections of Eq ,q are of the form 1 2 f(Γ,K,Φ)= g ∇(s−1)R[Γ ], ∇(r−1)R[K], ∇(k)Φ where g is a zero order natural operator ∞ s−1 × r−1 × k → ∞ g : C CC M CL E Z E C (F E). M k−2 × k−2 CC M CL E M 196 J. Janyška / Differential Geometry and its Applications 20 (2004) 177–196

Remark 5.5. It is easy to see that the second reduction theorem can be generalized for any number of 1 m 1 m ﬁelds Φ,...,Φ of order (1, 0) and that any ﬁnite order operator f(Γ,K,Φ,...,Φ) factorizes through 1 m R[Γ ], R[K], Φ,...,Φ and their covariant differentials.

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