We describe how find allMfm-natural operatorsDtransforming torsion free classical linear connections∇onm-manifoldsMintor-th order linear connectionsD

ARCHIVUM MATHEMATICUM (BRNO) Tomus 43 (2007), 285 – 288

HIGHER ORDER LINEAR CONNECTIONS FROM FIRST ORDER ONES

W. M. Mikulski

Abstract. We describe how find allMfm-natural operatorsDtransforming torsion free classical linear connections∇onm-manifoldsMintor-th order linear connectionsD(∇) onM.

Introduction

We study the problem how a torsion free classical linear connection∇on anm- dimensional manifoldM can induce ar-th order linear connectionD(∇) :T M→ J^rT M onM (or equivalently a right invariant connectionD(∇) in the principal bundle L^rM = invJ₀^r(R^m, M)). This problem is related to Mfm-natural oper- atorsD:Q^τ Q^r in the sense of [3]. We describe how find all operatorsD in question.

The category ofm-dimensional manifolds and their embeddings is denoted by Mf^m. All manifolds and maps are assumed to be of classC^∞.

1. Higher order connections on manifolds

Given anm-manifoldM we have the principal bundleL^rM = invJ₀^r(R^m, M) with the standard group G^rm = invJ₀^r(R^m,R^m)0 acting on right by the com- position of jets. Any Mf^m-map ϕ: M → N induces principal bundle map L^rϕ:L^rM →L^rN by composition of jets. The correspondenceL^r:Mf^m→ F M is a natural bundle overm-manifolds, [3].

A principal r-th order connection onM is aG^rm-invariant section Γ : L^rM → J¹L^rM of the first jet prolongationJ¹L^rM →L^rM ofL^rM →M, which can be identified with the corresponding (G^rm-invariant) lifting map Γ :T M×^ML^rM → T L^rM, see [3]. A linearr-th order connection onM is a linear sectionλ: T M→ J^rT M of ther-jet prolongationJ^rT M →T M of the tangent bundleT M →M.

From the introduction of [2] we have

2000Mathematics Subject Classification: 58A20.

Key words and phrases: higher order linear connection, natural operator.

Received February 16, 2007.

286 W. M. MIKULSKI

Fact 1. A linearr-th order connectionλ:T M →J^rT MonM induces a principal r-th order connection Γ^λ:T M ×ML^rM →T L^rM onM by Γ^λ(v, p) =L^rV(p), v ∈ T^xM, p∈L^rxM, x∈ M, whereλ(v) = jx^rV ∈ Jx^rT M and L^rV denotes the flow lifting ofV toL^rM. Conversely, any principal connection Γ :T M×^mL^rM → T L^rM onM induces a linearr-th order connectionλ^Γ:T M →J^rT M onM by λ^Γ(v) =j^rxV, v ∈T^xM,x∈M, where L^rV(p) = Γ(v, p) for some (and then for all)p∈L^rxM. The correspondenceλ→Γ^λ is one to one with the inverse one by Γ→λ^Γ.

Thus a first order linear connectionλ: T M→J¹T M onM is in fact a classical linear connection onM (which can be also defined by its covariant derivative∇).

2. Natural operators

The general concept of natural operators is given in [3]. We need only the following partial definition.

Definition 1. AMf^m-natural operatorD:Q^τ Q^r is aMf^m-invariant family of regular operators (functions)

D:Q^τ(M)→Q^r(M)

for any m-manifold M, where Qτ(M) is the set of torsion free classical linear connections on M and Q^r(M) is the set of all r-th order linear connections on M. The invariance means that if ∇1 ∈Qτ(M1) and ∇2 ∈Qτ(M2) are ϕ-related (by a Mf^m-mapϕ:M1→M2) then D(∇1) and D(∇2) are ϕ-related, too. The regularity means thatDtransforms smoothly parametrized families of connections into smoothly parametrized families.

3. The exponential extension of a classical linear connection The following construction has been (in equivalent way) presented by I. Kol´aˇr [1].

Example 1. Let∇ be a torsion free classical linear connection onM. We define anr-th order linear connection Exp (∇) :T M →J^rT M by

(1) Exp (∇)(v) =jx^r (exp^∇x)^∗˜v ,

where exp^∇x :T^xM →M is the exponent of∇inx(defined on some neighborhood of 0∈T^xM onto some neighborhood ofx) and where ˜v is the constant vector field on the vector spaceT^xM corresponding tov(˜v(w) = [w+tv]). The correspondence Exp:Q^τ Q^ris anMf^m-natural operator.

4. An isomorphism

Example 2. Let∇ be a torsion free classical linear connection on a manifoldM. Define a vector bundle isomorphism

ψ∇:J^rT M → ⊕^rk=0S^kT^∗M⊗T M

HIGHER ORDER LINEAR CONNECTIONS FROM FIRST ORDER ONES 287

(depending canonically on ∇) as follows. Let τ ∈ Jx^rT M, x ∈ M. Let ϕ be a∇-normal coordinate system onM with centerx. We put

(2) ψ∇(τ) =⊕^rk=0S^kT₀^∗ϕ⁻¹⊗T0ϕ⁻¹ I(J^rT ϕ(τ)) ,

where I:J0^rTR^m → ⊕^rk=0S^kT0^∗R^m⊗T0R^m is the usual identification. If ϕ1 is another such∇-normal coordinate system with centerxthenϕ1=A◦ϕnearxfor someA ∈ GL(m). The identification I is GL(m)-equivariant. Then standardly we verify that right hand sides of (2) for ϕ and ϕ1 coincide. That is why the definition ofψ∇(τ) is independent of the choice ofϕ.

5. The main result

Theorem 1. Let D:Q^τ Q^r be an Mf^m-natural operator transforming tor- sion free classical linear connections ∇ on m-manifolds M into r-th order linear connections D(∇) :T M → J^rT M on M. Then there exist uniquely determined Mf^m-natural operators Ak :Qτ T^∗⊗S^kT^∗⊗T for k= 0, . . . , r transforming torsion free classical linear connections ∇ on m-manifolds M into tensor fields Ak(∇)of typeT^∗⊗S^kT^∗⊗T on M such that A0= 0and

(3) D(∇)(v) = Exp (∇)(v) + (ψ^∇)⁻¹ hA0(∇)(x), vi, . . . ,hA^r(∇)(x), vi for any torsion free classical linear connection ∇ on M and any v ∈ T^xM,x ∈ M, where ψ^∇ is the isomorphism from Example 2 and Exp is the operator from Example 1 and the brackets h·,·i denote the obvious contractions ht, vi =t(v,·), v in the first position.

Conversely, given Mf^m-natural operators Ak :Qτ T^∗⊗S^kT^∗⊗T for k= 0, . . . , rwith A0= 0, the formula (3)defines anMf^m-natural operator D:Qτ

Q^r.

Proof. We must define Mf^m-natural operators Ak:Qτ T^∗⊗S^kT^∗⊗T by hA^k(∇)(x), vi^r

k=0 = ψ∇ D(∇)(v)−Exp (∇)(v)

, v ∈ TxM, x ∈ M. Clearly A0= 0 and we have (3).

Remark 1. Theorem 1 together with the result of Section 33.4 in [3] gives a com- plete description of allMf^m-natural operatorsD:Q^τ Q^r. In fact, each covari- ant derivative of the curvatureR(∇)∈C^∞(T M⊗T^∗M ⊗ ∧²T^∗M) of a classical linear connection ∇ is an (Mf^m-)natural tensor. Further every tensor multipli- cation of two natural tensors and every contraction on one covariant and one contravariant entry of a natural tensor give new natural tensor. Finally, we can tensor any natural tensor with a connection independent natural tensor, we can permute any number of entries in the tensor prodduct and we can repeat of these steps and take linear combinations. In this way we can obtain any natural tensor of type (p, q) (in particular of type (1, k+ 1)). Then each natural tensor of type T^∗⊗S^kT^∗⊗T (i.e. Mf^m-natural operatorQτ T^∗⊗S^kT^∗⊗T) can be obtained from a natural tensor of type (1, k+ 1) by using the respective symmetrization.

288 W. M. MIKULSKI

6. Natural operators Qτ Q^rτ

By [4], anr-th order linear connection λ∈Q^r(M) onM is called torsion-free if its torsion tensorτ^λ: ∧²T M →J^r−1T M,τ^λ(u, v) ={λ(u), λ(v)},u, v∈T^xM, x ∈ M, where {jx^rX, jx^rY} := jx^r−1([X, Y]), jx^rX, jx^rY ∈ Jx^rT M, vanishes. An equivalent notion of torsion free r-th order linear connections is presented in [1].

The construction of [1] clarify that the exponential prolongation (equivalently defined in Example 1) is a torsion-free connection on L^rM. By Proposition 5 in [1], the difference of torsion-free connections on L^rM over the same connection onL^r−1M is an arbitrary section of the tensor bundleS^r+1T^∗M⊗T M. Now we easily observe that if D: Qτ Q^rτ is an Mf^m-natural operator sending torsion free classical linear connections ∇ ∈ Qτ(M) into torsion-free r-th order linear connectionsD(∇)∈Q^rτ(M) then the defined in the proof of Theorem 1 operators Ak have values in tensor fields of type S^k+1T^∗⊗T, i.e.Ak: Qτ S^k+1T^∗⊗T. Thus we have

Theorem 2. Let D:Qτ Q^rτ be an Mf^m-natural operator transforming tor- sion free classical linear connections ∇ on m-manifolds M into torsion free r-th order linear connectionsD(∇)onM. Then there exist uniquely determinedMf^m- natural operatorsA^k:Q^τ S^k+1T^∗⊗T fork= 0, . . . , rtransforming torsion free classical linear connections∇ onm-manifoldsM into tensor fieldsA^k(∇)of type S^k+1T^∗⊗T onM such thatA0= 0and we have (3)for any torsion free classical linear connection ∇ onM and any v∈T^xM,x∈M.

References

[1] Kol´aˇr, I.,Torsion-free connections on higher order frame bundles, in New Development in Differential Geometry, Proceedings (Conference in Debrecen), Kluwer 1996, 233–241.

[2] Kol´aˇr, I.,On the torsion-free connections on higher order frame bundles, Publ. Math. De- brecen67(3-4), (2005), 373–379.

[3] Kol´aˇr, I., Michor, P. W., Slov´ak, J.,Natural Operations In Differential Geometry, Springer- Verlag Berlin 1993.

[4] Paluszny, M., Zajtz, A., Foundations of differential geometry of natural bundles, Lecture Notes Univ. Caracas, 1984.

Institute of Mathematics, Jagellonian University Reymonta 4, Krak´ow, Poland

E-mail:[email protected]