summary:For natural numbers $r$ and $n\ge 2$ a complete classification of natural affinors on the natural bundle $(J^rT^*)^*$ dual to $r$-jet prolongation $J^rT^*$ of the cotangent bundle over $n$-manifolds is given

Mikulski, Włodzimierz M.

English

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumWłodzimierz M. MikulskiThe natural affinors on (JrT ∗)∗Archivum Mathematicum, Vol. 36 (2000), No. 4, 261--267Persistent URL: http://dml.cz/dmlcz/107740Terms of use:© Masaryk University, 2000Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVUM MATHEMATICUM (BRNO)Tomus 36 (2000), 261 – 267THE NATURAL AFFINORS ON (JrT ∗)∗WLODZIMIERZ M. MIKULSKIFor natural numbers r and n ≥ 2 a complete classification of natu-ral affinors on the natural bundle (J rT ∗)∗ dual to r-jet prolongation J rT ∗ of thecotangent bundle over n-manifolds is given.0. The r-jet prolongation JrT ∗M of the cotangent bundle T∗M of an n-manifoldM is the space of all r-jets of 1-forms on M , i.e.JrT ∗M = {jrxω | ω is a 1-form on M , x ∈M} .It is a vector bundle overM with respect to the source projection. Let (JrT ∗)∗M =(JrT ∗M )∗ be the dual bundle and let π : (JrT ∗)∗M → M be its projection.Clearly, every embedding ϕ : M → N of two n-manifolds induces functorially (inobvious way) a vector bundle mapping (JrT ∗)∗ϕ : (JrT ∗)∗M → (JrT ∗)∗N overϕ, and we obtain a natural vector bundle (JrT ∗)∗ :Mn → VB ⊂ FM.In general, a natural affinor A on a natural bundle F :Mn → FM is a systemof affinorsA : TFM → TFM(i.e. tensor fields of type (1,1) on FM ) for any n-manifold M which is invariantwith respect to local embeddings between n-manifolds.For example, the family id = idTFM : TFM → TFM for any n-manifold M isa natural affinor on F .Another example of a natural affinor on (JrT ∗)∗ is the familyδ : T (JrT ∗)∗M → (JrT ∗)∗M ×M TM ⊂ (JrT ∗)∗M ×M (JrT ∗)∗M=̃V (JrT ∗)∗M ,where the arrow is the system (πT , Tπ) : T (JrT ∗)∗M → (JrT ∗)∗M ×M TM ,πT : T (JrT ∗)∗M → (JrT ∗)∗M is the tangent bundle projection, the inclusion ⊂is induced by the bundle map dual to the target projection JrT ∗M → T ∗M , =̃ isthe standard canonical identification E×M E=̃V E for any vector bundle E →M ,V E ⊂ TE is the vertical bundle of E.The main result of this note is the following classification theorem.1999 Mathematics Subject Classification : 58A20, 53A55.Key words and phrases : bundle functors, natural transformations, natural affinors.Received September 20, 1999.262 W. M. MIKULSKITheorem 1. If n ≥ 2 and r are natural numbers, then any natural affinor A on(JrT ∗)∗ over n-manifolds is a linear combination (over R) of id and δ.In [7], we proved that if r and n ≥ 2 are natural numbers, then any naturalaffinor A on JrT , the r-jet prolongation of the tangent bundle T , is proportionalto the identity affinor. Then (as a corollary of Theorem 1) the natural bundlesJrT and (JrT ∗)∗ are not naturally isomorphic for r and n as above.Natural affinors on F play a very importrant role in the differential geometry.For example, they can be used to define torsions of a connection on F , see [5].That is why classifications of natural affinors on some natural bundles has beenstudied in many papers, see e.g. [1]-[3] and [6]-[8].Throughout this note the usual coordinates on Rn are denoted by x1, . . . , xnand ∂i = ∂∂xi , i = 1, . . . , n.All manifolds and maps are assumed to be of class C∞.1. We have a linear natural transformation δ̃ : T (JrT ∗)∗ → (JrT ∗)∗ given byδ̃ : T (JrT ∗)∗M → TM ⊂ (JrT ∗)∗Mfor any n-manifold M , where the arrow is Tπ : T (JrT ∗)∗M → TM and theinclusion TM ⊂ (JrT ∗)∗M is defined in Item 0. The linearity of δ̃ means that δ̃induces a linear map Ty(JrT ∗)∗M → (JrT ∗)∗π(y)M for any y ∈ (JrT ∗)∗M .The crucial point in the proof of Theorem 1 is the following proposition.Proposition 1. If n ≥ 2 and r are natural numbers, then any linear naturaltransformation A : T (JrT ∗)∗ → (JrT ∗)∗ over n-manifolds is proportional (by areal number) to δ̃.Proof. Clearly, any element from the fibre (JrT ∗)∗0Rn is a linear combination ofthe (jr0(xαdxi))∗ for all α ∈ (N ∪ {0})n with |α| ≤ r and i = 1, . . . , n, where the(jr0(xαdxi))∗ form the basis dual to the jr0(xαdxi) ∈ (JrT ∗)0Rn for α and i asbeside.Any natural transformation A as in the proposition is uniquely determined bythe values 〈A(u), jr0(xαdxi)〉 ∈ R for u ∈ (T (JrT ∗)∗Rn)0=̃Rn × (V (JrT ∗)∗Rn)0=̃ Rn × (JrT ∗)∗0Rn × (JrT ∗)∗0Rn, α ∈ (N ∪ {0})n with |α| ≤ r and i = 1, . . . , n,where =̃ is the standard trivialization and the canonical identification.Since A is invariant with respect to the coordinate permutations, it is uniquelydetermined by the 〈A(u), jr0(xαdx1)〉 for any u and α as above.If |α| ≥ 1, then the local diffeomorphisms ϕα = (x1, x2 + xα, x3, . . . , xn)−1sends jr0(x2dx1) into jr0(x2dx1)+jr0 (xαdx1). Then (using the invariancy of A withrespect to the ϕ’s) A is uniquely determined by the 〈A(u), jr0(x2dx1)〉 ∈ R and the〈A(u), jr0(dx1)〉 ∈ R for any u ∈ (T (JrT ∗)∗Rn)0=̃Rn× (JrT ∗)∗0Rn× (JrT ∗)∗0Rn.At first we study the 〈A(u), jr0(dx1)〉 ∈ R for u as above.By the naturality of A with respect to the homotheties at = (t1x1, . . . , tnxn) fort = (t1, . . . , tn) ∈ Rn+, 〈A(T (JrT ∗)∗(at)(u)), jr0(dx1)〉 = t1〈A(u), jr0(dx1)〉 for anyTHE NATURAL AFFINORS ON (JrT ∗)∗ 263t = (t1, . . . , tn) ∈ Rn+. For any t ∈ Rn, any i = 1, . . . , n and any α ∈ (N∪{0})n wehave T (JrT ∗)∗(at)((jr0 (xαdxi))∗) = tα+ei(jr0(xαdxi))∗. Then by the homogeneousfunction theorem, see [4], we have(1.1) 〈A(u), jr0(dx1)〉 = λu11 + µu2,(0),1 + νu3,(0),1for some λ, µ, ν ∈ R, where u = (u1, u2, u3), u1 = (u11, . . . , un1 ) ∈ Rn, u2,α,i isthe coefficient of u2 ∈ (JrT ∗)∗0Rn corresponding to (jr0(xαdxi))∗, and u3,α,i is thecoefficient of u3 ∈ (JrT ∗)∗0Rn on (jr0(xαdxi))∗, (0) = (0, . . . , 0) ∈ (N ∪ {0})n.Since A is linear, 〈A(u1, u2, u3), jr0(dx1)〉 is linear in (u1, u3) for any u2. Thenµ = 0. Replacing A by A− λδ̃, we can assume that λ = 0. Then(1.2) 〈A(∂C1 |w), jr0(dx1)〉 = 〈A(e1, w, 0), jr0(dx1)〉 = 0for w ∈ (JrT ∗)∗0Rn, where ( )C is the complete lift of vector fields to (JrT ∗)∗.We prove that ν = 0.It is sufficient to show that 〈A(0, 0, (jr0(dx1))∗), jr0(dx1)〉 = 0.For showing the last equality we prove(1.3)0 = 〈A(((x1)r+1∂1)C|w), jr0(dx1)〉= (r + 1)〈A(0, w, (jr0(dx1))∗), jr0(dx1)〉= (r + 1)〈A(0, 0, (jr0(dx1))∗), jr0(dx1)〉 ,where w = (jr0((x1)rdx1))∗.The third equality of (1.3) is clear as in the formula (1.1) λ and µ are 0.We can prove the first equality of (1.3) as follows. Vector fields ∂1 + (x1)r+1∂1and ∂1 have the same r-jets at 0. Then, by the result of Zajtz [9], there exists adiffeomorphism ϕ : Rn → Rn such that jr+10 ϕ = id and ϕ∗∂1 = ∂1 + (x1)r+1∂1near 0. Clearly, ϕ preserves jr0(dx1) because of the jet argument. Then, us-ing the naturality of A with respect to ϕ, from (1.2) it follows that 〈A((∂1 +(x1)r+1∂1)C |w), jr0(dx1)〉 = 0 for any w ∈ (JrT ∗)∗0Rn. Now, applying the linearityof A, we end the proof of the first equality.It remains to prove the second equality of (1.3). Let ϕt be the flow of (x1)r+1∂1.For any α ∈ (N ∪ {0})n with |α| ≤ r we have〈((x1)r+1∂1)C|w, jr0(xαdxi)〉= 〈 ddt |t=0(JrT ∗)∗0(ϕt)(w), jr0(xαdxi)〉=ddt |t=0〈(JrT ∗)∗0(ϕt)(w), jr0(xαdxi)〉=ddt |t=0〈w, jr0((ϕ−t)∗(xαdxi))〉= 〈w, jr0(ddt |t=0(ϕ−t)∗(xαdxi))〉= 〈w, jr0(L(x1)r+1∂1(xαdxi))〉= 〈w, jr0(α1(x1)rxαdxi + (r + 1)δi1(x1)rxαdx1)〉 .264 W. M. MIKULSKIBecause of the definition of w, the last term is equal to r + 1 if jr0(xαdxi) =jr0(dx1) and it is equal to 0 in the other cases. Then ((x1)r+1∂1)C|w = (r +1)(jr0(dx1))∗ under the isomorphism Vw((JrT ∗)∗Rn)=̃(JrT ∗)∗0Rn. It implies thesecond equality of (1.3).To end the proof of the proposition it remains to show 〈A(u), jr0(x2dx1)〉 = 0for any u ∈ (T (JrT ∗)∗Rn)0=̃Rn × (JrT ∗)∗0Rn × (JrT ∗)∗0Rn.To prove this we use similar procedure as in the case of 〈A(u), jr0(dx1)〉.By the naturality of A with respect to the homotheties at = (t1x1, . . . , tnxn)for t = (t1, . . . , tn) ∈ Rn+, the homogeneous function theorem and the linearity ofA, one can easily deduce(1.4)〈A(u), jr0(x2dx1)〉 = λu3,e2,1 + µu3,e1,2 + νu11u2,(0),2+ ρu21u2,(0),1 + σu2,(0),1u3,(0),2 + κu2,(0),2u3,(0),1for some λ, µ, ν, ρ, σ, κ ∈ R, where u = (u1, u2, u3), u1 = (u11, . . . , un1) ∈ Rn,u2, u3 ∈ (JrT ∗)∗0Rn, and uτ,α,i is the coefficient of uτ on (jr0(xαdxi))∗, τ ∈ {2, 3},ei = (0, . . . , 1, 0, . . ., 0) ∈ (N ∪ {0})n, 1 in i-position.Then(1.5) 〈A(∂C1 |w), jr0(x2dx1)〉 = 0for any linear combination w of the (jr0(xαdxi))∗ ∈ (JrT ∗)∗0Rn for |α| ≥ 1 andi = 1, . . . , n.At first we prove that λ = µ = 0, i.e. 〈A(0, 0, (jr0(x2dx1))∗), jr0(x2dx1)〉 = 0 and〈A(0, 0, (jr0(x1dx2))∗), jr0(x2dx1)〉 = 0.For showing the equality 〈A(0, 0, (jr0(x2dx1))∗), jr0(x2dx1)〉 = 0 we prove(1.6)0 = 〈A(((x1)r∂1)C|w), jr0(x2dx1)〉= r〈A(0, w, (jr0(x2dx1))∗), jr0(x2dx1)〉= r〈A(0, 0, (jr0(x2dx1))∗), jr0(x2dx1)〉 ,where w = (jr0(x2(x1)r−1dx1))∗ ∈ (JrT ∗)∗0Rn.The third equality of (1.6) is clear, see (1.4).We can prove the first equality of (1.6) as follows. Vector fields ∂1 + (x1)r∂1and ∂1 have the same r − 1-jets at 0. Then, by the result of Zajtz, there existsa diffeomorphism ϕ = ϕ1 × idRn−1 : Rn = R × Rn−1 → Rn = R × Rn−1 suchthat ϕ1 : R → R, jr0ϕ = id and ϕ∗∂1 = ∂1 + (x1)r∂1 near 0. Let ϕ−1 sendw into w̃. Then w̃ is the linear combination of the (jr0(xαdxi))∗ ∈ (JrT ∗)∗0Rnfor |α| ≥ 1 and i = 1, . . . , n. (For, 〈w̃, j r0(dxj)〉 = 〈w, jr0(d(xj ◦ ϕ−1))〉 = 0for j = 1, . . . , n.) Consequently 〈A(∂C1 |w̃), jr0(x2dx1)〉 = 0. Clearly, ϕ preservesjr0(x2dx1). Then, using the naturality of A with respect to ϕ it follows that〈A((∂1 + (x1)r∂1)C |w), jr0(x2dx1)〉 = 0. Now, applying the linearity of A, we endthe proof of the first equality.THE NATURAL AFFINORS ON (JrT ∗)∗ 265It remains to prove the second equality of (1.6). Using the flow argument onecan easily compute〈((x1)r∂1)C|w, jr0(xαdxi)〉 = 〈w, jr0(α1(x1)r−1xαdxi + rδi1(x1)r−1xαdx1)〉 .Because of the definition of w, the last term is equal to r if jr0(xαdxi) =jr0(x2dx1) and it is equal to 0 in the other cases. Then ((x1)r∂1)C|w = r(jr0(x2dx1))∗under the isomorphism Vw((JrT ∗)∗Rn)=̃(JrT ∗)∗0Rn. It implies the second equal-ity of (1.6).For showing the equality 〈A(0, 0, (jr0(x1dx2))∗), jr0(x2dx1)〉 = 0 we prove(1.7)0 = 〈A(((x1)r∂1)C|w), jr0(x2dx1)〉= 〈A(0, w, (jr0(x1dx2))∗), jr0(x2dx1)〉= 〈A(0, 0, (jr0(x1dx2))∗), jr0(x2dx1)〉 ,where w = (jr0((x1)rdx2))∗.The third equality of (1.7) is clear, see (1.4).The first equality of (1.7) has similar proof as the first equality of (1.6).It remains to prove the second equality of (1.7). Using the flow argument onecan easily compute ((x1)r∂1)C|w = (jr0(x1dx2))∗ under the obvious isomorphism. Itimplies the second equality of (1.7).We have proved that µ = λ = 0.Now, we prove that σ = κ = 0.Local diffeomorphism ψ = (x1 + 12(x1)2, x21+x1 , x3, . . . , xn) preserves jr0(x2dx1),it sends (jr0(x1dx2))∗ into (jr0(x1dx2))∗ − (jr0(dx2))∗, and it preserves (jr0(dx1))∗.Now, using (1.4) with λ = µ = 0 and the naturality of A with respect to ψ weobtain(1.8)0 = −〈A(0, (jr0(dx1))∗, (jr0(x1dx2))∗), jr0(x2dx1)〉= −〈A(0, (jr0(dx1))∗, (jr0(x1dx2))∗) − (jr0(dx2))∗), jr0(x2dx1)〉= 〈A(0, (jr0(dx1))∗, (jr0(dx2))∗), jr0(x2dx1)〉 .Therefore in (1.4) we have σ = 0.Similarly, starting from 0 = −〈A(0, (j r0(x1dx2))∗, (jr0(dx1))∗), jr0(x2dx1)〉 we ob-tain 〈A(0, (jr0(dx2))∗, (jr0(dx1))∗), jr0(x2dx1)〉 = 0, i.e. κ = 0.Now, we prove that in (1.4) we have ν = 0.The above local diffeomorphism ψ sends the germ at 0 of ∂1 into the germ at 0of ∂1 + . . . , where the dots is some vector field vanishing in 0 ∈ Rn. Now, using(1.4) with λ = µ = σ = κ = 0 and the naturality of A with respect to ψ we obtain(1.9)0 = −〈A(e1, (jr0(x1dx2))∗, 0), jr0(x2dx1)〉= −〈A(e1, (jr0(x1dx2))∗)− (jr0(dx2))∗, u3), jr0(x2dx1)〉= 〈A(e1, (jr0(dx2))∗, 0), jr0(x2dx1)〉266 W. M. MIKULSKIfor some u3 ∈ (JrT ∗)∗0Rn. Therefore in (1.4) we have ν = 0.It remains to prove that in (1.4) we have ρ = 0.From (1.4) with µ = λ = ν = σ = κ = 0 and the invariancy of A with respectto the diffeomorphism permuting x1 and x2 we have(1.10) 〈A(e2, (jr0(dx1))∗, u3), jr0(x1dx2)〉 = 0for any u3 ∈ (JrT ∗)∗0Rn. From (1.1) with λ = µ = ν = 0 we have(1.11) 〈A(e2, (jr0(dx1))∗, 0), jr0(dx1)〉 = 0 .Now, using the invariancy of A with respect to Θ = (x1 +x1x2, x2, . . . , xn)−1 from(1.11) we obtain0 = 〈A(e2, (jr0(dx1))∗, u3), jr0(dx1) + jr0(x1dx2) + jr0(x2dx1)〉= 〈A(e2, (jr0(dx1))∗, 0), jr0(x2dx1)〉for some u3 because of (1.1) with λ = µ = ν = 0, (1.4) with λ = µ = ν = σ = κ = 0and (1.10). Therefore in (1.4) we have ρ = 0.The proof of Proposition 1 is complete. 2. The tangent map Tπ : T (JrT ∗) ∗ M → TM of the bundle projection π :(JrT ∗)∗M →M defines a linear natural transformation Tπ : T (JrT ∗)∗ → T overn-manifolds.Proposition 2. If r and n ≥ 2 are natural numbers, then any linear naturaltransformation A : T (JrT ∗)∗ → T over n-manifolds is proportional (by a realnumber) to Tπ.Proof. Applying the inclusion TM ⊂ (JrT ∗)∗M , we have A : T (JrT ∗)∗M →TM ⊂ (JrT ∗)∗M . Then by Proposition 1, A : T (JrT ∗)∗M → TM ⊂ (JrT ∗)∗Mis proportional to δ̃. Then A : T (JrT ∗)∗M → TM is proportional to Tπ. 3. In Item 0, we defined natural affinor δ : T (JrT ∗)∗M → (JrT ∗)∗M ×M TM ⊂(JrT ∗)∗M ×M (JrT ∗)∗M=̃V (JrT ∗)∗M .Proposition 3. If r and n ≥ 2 are natural numbers, then any natural affinorA : T (JrT ∗)∗M → V (JrT ∗)∗M on (JrT ∗)∗ over n-manifolds is proportional (bya real number) to δ.Proof. Define a linear natural transformation Ã = pr2 ◦ A : T (JrT ∗)∗M →V (JrT ∗)∗M=̃(JrT ∗)∗M ×M (JrT ∗)∗M → (JrT ∗)∗M , where pr2 : (JrT ∗)∗M ×M(JrT ∗)∗M → (JrT ∗)∗M is the projection onto second factor. By Proposition 1,Ã = λδ̃ for some λ ∈ R. Then A = (πT , Ã) = λ(πT , δ̃) = λδ. THE NATURAL AFFINORS ON (JrT ∗)∗ 2674. We are now in position to prove Theorem 1. Let A : T (JrT ∗)∗M → T (JrT ∗)∗Mbe the natural affinor on (JrT ∗)∗ over n-manifolds. Then Tπ ◦A : T (JrT ∗)∗ → Tis a linear natural transformation. By Proposition 2, Tπ ◦ A = λTπ for some λ.Clearly, Tπ ◦ id = Tπ. Then A − λid is an affinor on (JrT ∗)∗ of vertical type.Now, applying Proposition 3 we end the proof. References[1] Doupovec, M., Natural transformations between TTT ∗M and TT ∗TM , Czechoslovak Math.J. 43 (118) 1993, 599–613.[2] Doupovec, M., Kolář, I., Natural affinors on time-dependent Weil bundles, Arch. Math.(Brno) 27 (1991), 205–209.[3] Gancarzewicz, J., Kolář, I., Natural affinors on the extended r-th order tangent bundles,Suppl. Rendiconti Circolo Mat. Palermo, 30 (1993), 95–100.[4] Kolář I., Michor P. W., Slovák J., Natural operations in differential geometry, Springer-Verlag,Berlin 1993.[5] Kolář, I., Modugno, M., Torsions of connections on some natural bundles, Diff. Geom. andAppl. 2(1992), 1–16.[6] Kurek, J., Natural affinors on higher order cotangent bundles, Arch. Math. (Brno) 28 (1992),175–180.[7] Mikulski, W. M., Natural affinors on r-jet prolongation of the tangent bundle, Arch. Math.(Brno) 34(2)(1998), 321–328.[8] Mikulski, W. M., The natural affinors on ⊗kT (r), Note di Matematica, to appear.[9] Zajtz, A., On the order of natural operators and liftings, Ann. Polon. Math. 49(1988),169–178.E-mail Wlodzimierz.Mikulski@im.uj.edu.pl

The natural affinors on $(J^rT^*)^*$

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The natural affinors on $(J^rT^)^$

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The natural affinors on (JrT∗)∗(J^rT^*)^*(JrT∗)∗

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The natural affinors on $(J^rT^)^$