The category of 2-fibred manifold (pairs of surjective submersionsY →X→M between manifolds) and their 2-fibred maps is denoted by 2-F M

ARCHIVUM MATHEMATICUM (BRNO) Tomus 44 (2008), 17–21

THE JET PROLONGATIONS OF 2-FIBRED MANIFOLDS AND THE FLOW OPERATOR

Włodzimierz M. Mikulski

Abstract. Letr,s,m,n,qbe natural numbers such thats≥r. We prove that any 2-F Mm,n,q-natural operator A:T2-proj T J^(s,r) transforming 2-projectable vector fieldsV on (m, n, q)-dimensional 2-fibred manifoldsY → X→M into vector fieldsA(V) on the (s, r)-jet prolongation bundleJ^(s,r)Y is a constant multiple of the flow operatorJ^(s,r).

All manifolds and maps are assumed to be of classC^∞. Manifolds are assumed to be finite dimensional and without boundaries.

The category of all manifolds and maps is denoted byMf. The category of all fibred manifolds (surjective submersionsX →M between manifolds) and fibred maps is denoted by F M. The category of all fibred manifolds withm-dimensional bases and n-dimensional fibres and their fibred embeddings is denoted byF Mm,n. The category of 2-fibred manifold (pairs of surjective submersionsY →X→M between manifolds) and their 2-fibred maps is denoted by 2-F M. The category of all fibred manifoldsY →X →M such thatX→M is anF M_m,n-object and their 2-fibred maps coveringF Mm,n-maps is denoted by 2-F Mm,n. The category of all fibred manifoldsY →X →M such thatX →M is anF Mm,n-object andY →X is anF Mm+n,q-object and their 2-fibred embeddings is denoted by 2-F Mm,n,q. The standard 2-F Mm,n,q-object is denoted by R^m,n,q = (R^m×Rⁿ ×R^q → R^m×Rⁿ →R^m). The usual coordinates onR^m,n,q are denoted byx¹, . . . , x^m, y¹, . . . , yⁿ,z¹, . . . , z^q.

Taking into consideration some idea from [1] one can generalize the concept of jets as follows. Let randsbe integers such that s≥r. LetY →X →M be a 2-F Mm,n-object. Sections σ1, σ2: X →Y of Y →X have the same (s, r)-jet jx^(s,r)σ1=jx^(s,r)σ2 atx∈X iff

j_x^s−r J^rσ₁|X_p0(x)

=j_x^s−r J^rσ₂|X_p0(x) ,

whereJ^rσ_i:X →J^rY is the r-jet mapJ^rσ_i(x) =j_x^rσ_i,x∈X, andX_p0(x) is the fibre ofX →M throughx. Equivalentlyjx^(s,r)σ1=jx^(s,r)σ2 iff (in some and then in every 2-F Mm,n-coordinates)D(α,β)σ1(x) =D(α,β)σ2(x) for allα∈ N∪ {0}m

and β ∈ N∪ {0}ⁿ

with |α| ≤ r and |α|+|β| ≤ s, where D_(α,β) denotes the

2000Mathematics Subject Classification:Primary: 58A20.

Key words and phrases:(s, r)-jet, bundle functor, natural operator, flow operator, 2-fibred manifold, 2-projectable vector field.

Received March 13, 2007. Editor I. Kolář.

iterated partial derivative corresponding to (α, β). Thus we have the so called (s, r)-jets prolongation bundle

J^(s,r)Y =

j_x^(s,r)σ |σ:X →Y is a section of Y →X, x∈X .

Given a 2-F Mm,n-mapf:Y₁ → Y₂ of two 2-F Mm,n-objects covering F Mm,n- -map f:X1 → X2 we have the induced map J^(s,r)f: J^(s,r)Y1 → J^(s,r)Y2 given by J^(s,r)f(j^(s,r)x σ) = j_f(x)^(s,r)(f ◦σ◦f⁻¹), jx^(s,r)σ ∈ J^(s,r)Y1. The correspondence J^(s,r): 2-F Mm,n→ F Mis a (fiber product preserving) bundle functor.

Let Y → X → M be an 2-F Mm,n,q-object. A vector field V on Y is called 2-projectable if there exist (unique) vector fieldsV₁onX andV₀onM such thatV is related with V₁ andV₁is related withV₀ (with respect to the 2-fibred manifold projections). Equivalently, the flow ExptV of V is formed by (local) 2-F Mm,n,q- -isomorphisms. Thus we can apply functorJ^(s,r) to ExptV and obtain new flow J^(s,r)(ExptV) onJ^(s,r)Y. Consequently we obtain vector fieldJ^(s,r)V onJ^(s,r)Y. The corresponding 2-F Mm,n,q-natural operatorJ^(s,r):T2-proj T J^(s,r) is called the flow operator (ofJ^(s,r)).

The main result of the present note is the following classification theorem.

Theorem 1. Letr,s,m,n,qbe natural numbers such thats≥r. Any2-F M_m,n,q- -natural operatorA:T_2-proj T J^(s,r) is a constant multiple of the flow operator

J^(s,r).

Thus Theorem 1 extends the result from [2] on 2-fibred manifolds. More precisely, in [2] it is proved that anyF Mm,n-natural operatorA lifting projectable vector fieldsV from fibred manifolds Y →M to vector fieldsA(V) on J^rY is a constant multiple of the flow operator.

In the proof of Theorem 1 we will use the method from [4] (a Weil algebra technique). We start with the proof of the following lemma. LetA: T2-proj T J^(s,r) be a natural operator in question.

Lemma 1. The natural operator A is determined by the restriction A _∂x^∂1

| J^(s,r)(R^m,n,q)

(0,0), where(0,0)∈R^m×Rⁿ.

Proof. The assertion is an immediate consequence of the naturality and regularity ofAand the fact that any 2-projectable vector field which is not (Y →M)-vertical

is related with _∂x^∂1 by an 2-F Mm,n,q-map.

Now we prove

Lemma 2. LetA be the operator. Let π: J^(s,r)Y →X be the projection. Then there exists the unique real numbercand the uniqueπ-vertical operatorV:T2-proj

T J^(s,r)with V(0) = 0 such that A=cJ^(s,r)+V. Proof. Define C=T π◦A _∂x^∂1

: J^(s,r)(R^m,n,q)

(0,0)→T_(0,0)(R^m×Rⁿ). Using the invariance of Awith respect to 2-F M_m,n,q-maps

(x¹, . . . , x^m, y¹, . . . , yⁿ, τ z¹, . . . , τ z^q)

forτ >0 and puttingt→0 we get thatC j_(0,0)^(s,r)(σ)

=C j_(0,0)^(s,r)(0)

, where 0 is the zero section. Then using the invariance ofA with respect to

(x¹, τ x², . . . , τ x^m, τ y¹, . . . , τ yⁿ, τ z¹, . . . , τ z^q) for τ > 0 and putting t →0 we get thatC j_(0,0)^(s,r)(0)

=c_∂x^∂1|0 for some c ∈R.

We putV=A−cJ^(s,r). Then V is of vertical type because of Lemma 1. Clearly, A=cJ^(s,r)+V.

It remains to show thatV(0) = 0. Clearly, the flow ofV(0) is a family of natural automorphismsJ^(s,r)→J^(s,r). Since the 2-F Mm,n,q-orbit of j_(0,0)^(s,r)(0) is the whole

J^(s,r)(R^m,n,q)

(0,0) (any elementj_(0,0)^(s,r)σ∈ J^(s,r)(R^m,n,q)

(0,0)is transformed by 2-F Mm,n,q-map

x, y, z−σ(x, y)

intoj_(0,0)^(s,r)(0)), then any natural automorphism E:J^(s,r)→J^(s,r) is determined by E j^(s,r)_(0,0)(0)

. Then using the invariance ofE with respect to (τ x¹, . . . , τ x^m, τ y¹, . . . , τ yⁿ, τ z¹, . . . , τ z^q) forτ >0 and puttingτ→0 we getE j_(0,0)^(s,r)(0)

=j_(0,0)^(s,r)(0). ThenE= id and then

V(0) = 0.

Define a bundle functorF:Mf → F M by F N = J^(s,r)(R^m×Rⁿ×N)

(0,0), F f = J^(s,r)(id_Rm×id_Rn×f)

(0,0). Lemma 3. The bundle functor F:Mf → F M is product preserving.

Proof. It is clear.

LetB=FRbe the Weil algebra corresponding toF.

Lemma 4. We have B = D^s_m+n/B, where D^s_m+n = J_(0,0)^s (R^m+n,R) and B = j^s_(0,0)(x¹), . . . , j^s_(0,0)(x^m)r+1

is the (r+ 1)-power of the ideal

j_(0,0)^s (x¹), . . . , j_(0,0)^s (x^m)

, generated by the elements as indicate.

Proof. It is a simple observation.

We have the obvious actionH:G^s_m,n×B→B, H j_(0,0)^s ψ,[j_(0,0)^s γ]

=

j_(0,0)^s (γ◦ψ⁻¹) for anyF M,m,n-mapψ: R^m×Rⁿ,(0,0)

→ R^m×Rⁿ,(0,0)

andγ: R^m+n→ R. This action is by algebra automorphisms.

Lemma 5. For any derivation D∈Der(B)we have the implication: if H j_(0,0)^s (τid)

◦D◦H j^s_(0,0)(τ⁻¹id)

→0 as τ→0 then D= 0. Proof. LetD∈Der(B) be such that

H j^s_(0,0)(τid)

◦D◦H j_(0,0)^s (τ⁻¹id)

→0 as τ→0.

Fori= 1, . . . , mandj= 1, . . . , nwriteD [j_(0,0)^s (xⁱ)]

=Paⁱ_αβ

j_(0,0)^s (x^αy^β) and D [j_(0,0)^s (y^j)]

=Pb^j_αβ

j_(0,0)^s (x^αy^β)

for some (unique) real numbersaⁱ_αβandb^j_αβ, where the sums are over allα∈ N∪ {0}^m

andβ∈ N∪ {0}ⁿ

with|α| ≤rand

|α|+|β| ≤s. We have H j_(0,0)^s (τid)

◦D◦H j^s_(0,0)(τ⁻¹id)

[j_(0,0)^s (xⁱ)]

=X

aⁱ_αβ 1 τ^{|α|+|β|−1}

j_(0,0)^s (x^αy^β) . Then from the assumption on D it follows that aⁱ_αβ = 0 if (α, β) 6= (0),(0)

. Similarly,b^j_αβ= 0 if (α, β)6= (0),(0)

. ThenD [j_(0,0)^s (xⁱ)]

=aⁱ₍₀₎₍₀₎

j^s_(0,0)(1) and D [j_(0,0)^s (y^j)]

= b^j₍₀₎₍₀₎

j_(0,0)^s (1)

for i= 1, . . . , m andj = 1, . . . , n. Then (since j_(0,0)^s ((xⁱ)^r+1)

= 0 andD is a differentiation) we have 0 =D [j_(0,0)^s ((xⁱ)^r+1)]

= (r+ 1)

j_(0,0)^s ((xⁱ)^r)

D [j_(0,0)^s (xⁱ)]

= (r+ 1)aⁱ₍₀₎₍₀₎

j^s_(0,0)((xⁱ)^r) . Thenaⁱ₍₀₎₍₀₎= 0 as

j_(0,0)^s ((xⁱ)^r)

6= 0. Similarly,b^j₍₀₎₍₀₎= 0. ThenD= 0 because the

j_(0,0)^s (xⁱ)

and [j_(0,0)^s (y^j)] generate the algebraB.

Proof of Theorem 1. OperatorV from Lemma 2 defines (by the restriction) Mfq-natural vector fields ˜Vt=V t_∂x^∂₁

|F N onF N for any t∈R. Clearly, V is determined by ˜V₁. By Lemma 2, ˜V₀= 0. By [2], ˜V_t= op(D_t) for someD_t∈Der(B).

Then using the invariance ofV with respect to

(τ x¹, . . . , τ x^m, τ y¹, . . . , τ yⁿ, z¹, . . . , z^q) forτ 6= 0 and puttingτ→0 we obtain that

H j_(0,0)^s (τid)

◦Dt◦H j_(0,0)^s (τ⁻¹id)

→0 as τ→0.

ThenDt= 0 because of Lemma 5. ThenV = 0, and thenA=cJ^(s,r) as well.

Remark 1. There is another (non-equivalent) generalization of jets. Lets≥r. Let Y →X →M be a 2-fibred manifold. By [2], sections σ₁, σ₂:X →Y of Y →X have the samer, s-jetsj_x^r,sσ₁=j_x^r,sσ₂ atx∈X iff

j^r_xσ1=j_x^rσ2 and j_x^s σ1|Xp_o(x)

=j_x^s σ2|Xp_o(x)

,

whereX_po(x)is the fiber ofX →M throughx. Consequently we have the correspon- ding bundleJ^r,sY and the corresponding (fiber product preserving) bundle functor J^r,s: 2-F Mm,n→ F M. In [3], we proved that any 2-F Mm,n,q-natural operator A:T2-proj T J^r,s is a constant multiple of the flow operatorJ^r,s corresponding toJ^r,s (we used quite different method than the one in [4] or in the present note).

References

[1] Cabras, A., Janyška, J., Kolář, I.,On the geometry of variational calculus on some functional bundles, Note Mat.26(2) (2006), 51–57.

[2] Kolář, I., Michor, P.W., Slovák, J., Natural Operations in Differential Geometry, Springer-Verlag Berlin, 1993.

[3] Mikulski, W. M.,The jet prolongations of fibered manifolds and the flow operator, Publ. Math.

Debrecen59(2001), 441–458.

[4] Mikulski, W. M.,The natural operators lifting projectable vector fields to some fiber product preserving bundles, Ann. Polon. Math.81(3) (2003), 261–271.

Institute of Mathematics, Jagellonian University Reymonta 4, Kraków, Poland

E-mail:[email protected]