The Diffiety Institute Preprint Series

Preprint DIPS–1/2007 March 22, 2007

Available via INTERNET: http://diffiety.ac.ru; http://diffiety.org

The Diffiety Institute

Iterated Differential Forms V: C–Spectral Sequence on J^{∞}(π)

by

A. M. Vinogradov and L. Vitagliano

DIPS-1/2007

                                                 arXiv:math.DG/0703661

Iterated Differential Forms V: C–Spectral Sequence on J^{∞}(π)

A. M. VINOGRADOV AND L. VITAGLIANO

Abstract. In the preceding note [1] the Λ_{k−1}C–spectral sequence, whose first term is composed of secondary iterated differential forms, was constructed for a generic diffiety. In this note the zero and first terms of this spectral sequence are explicitly computed for infinite jet spaces. In particular, this gives an explicit description of secondary covariant tensors on these spaces and some basic operations with them. On the basis of these results a description of the Λ_{k−1}C–spectral sequence for infinitely prolonged PDE’s will be given in the subsequent note.

Introduced in [1] secondary iterated differential forms on a generic diffiety (O, C) are elements of the first term of the Λ_{k−1}C– spectral sequence associated with it. In the present note we give an explicit description of the zeroth and first terms of this spectral sequence for the infinite jet space of a vector bundle O = J^{∞}(π), π : E ⟶ M. Moreover, analogues of basic operations of tensor analysis in secondary calculus on these spaces are constructed. This goal is got by a due generalization of the approach developed in [2] taking into account some improvements proposed in [3].

1. Notations and conventions

In what concerns geometry of infinite jet manifolds and general theory of iterated differential forms (shortly, IDFs), we follow [4] and [5], respectively. The notation is slightly simplified against [4, 5]. Namely, when the context allows we omit the reference to arguments “π” and “J^{∞}(π)”. For instance, we use F and Λ for algebras of smooth functions and differential forms on J^{∞}(π), respectively, instead of standard F(π) and Λ(J^{∞}(π)), etc. Also the algebra of horizontal differential forms on J^{∞}(π) is denoted by HΛ (rather than Λ_{0}) and the algebra of vertical differential forms by C^{•}Λ. Accordingly, d^{h} and d^{v} stand for horizontal and vertical differentials, respectively. As usually, {…, x^{μ}, …, u^{j}, …}, μ = 1, …, n, j = 1, …, m, n = dim M, m = dim E− n, denote an adapted to π local chart on E, assuming that {…, x^{μ}, …} is a local chart on M. We follow the notation of [1] in what concerns the theory of secondary IDF’s.

2000 Mathematics Subject Classification. 58A10, 58A20, 58J10.

Key words and phrases. Differential Forms, Jet Spaces, C–Spectral Sequence, Secondary Calculus.

The wedge product of IDFs is denoted, according to [1], by the “dot”. Finally, the fact that an isomorphism is canonical is stressed using the symbol ≃.

2. IDFs on J^{∞}(π)

Recall that, due to the fibered structure of J^{∞}(π), the de Rham differential d splits into vertical and horizontal parts: d = d^{v} + d^{h}. (Λ, d^{v}, d^{h}) is a bi–complex, for the first time introduced in [6] (see also [7, 8, 2]) and later called the variational bicomplex (see [3, 4], see also [9]). A variational bi–complex exists locally for any diffiety and may be understood as a local description of the C–spectral sequence. Moreover, Λ^{1} splits naturally as Λ^{1} = CΛ^{1} ⊕ HΛ^{1}. Accordingly, Λ is factorized as Λ ≃ C^{•}Λ ⊗_{F} HΛ (C^{•}Λ ≡ ⋀^{•} CΛ^{1}, HΛ ≡ ⋀^{•} HΛ^{1}). In a dual manner, D splits as D = CD ⊕ D^{v} (see [4]). Note that d^{v} and d^{h} extends as derivations to Λ_{k}. Abusing the notation we continue to denote these extensions by d^{v} and d^{h}. Put d_{m}^{v} = κ_{1m} ∘ d^{v} ∘ κ_{1m} : Λ_{k} ⟶ Λ_{k}, d_{m}^{h} = κ_{1m} ∘d^{h} ∘κ_{1m} : Λ_{k} ⟶ Λ_{k}, κ_{1m} : Λ_{k} ⟶ Λ_{k} being the involution that interchanges d_{1} and d_{m}, m ≤ k and also d_{K}^{v} = d_{k_{1}}^{v} ∘ ⋯ ∘ d_{k_{s}}^{v}, K = {k_{1}, …, k_{s}} ⊂ {1, …, k}. (Λ_{k}, (d_{1}^{v}, d_{1}^{h}), …, (d_{k}^{v}, d_{k}^{h})), where d_{1}^{v} = d^{v} and d_{1}^{h} = d^{h}, is a multiple bi–complex. Define HΛ_{k}^{1} ⊂ Λ_{k}^{1} as the Λ_{k−1}–submodule generated by π_{∞}^{∗}(Λ_{k}^{1}(M)), π_{∞} : J^{∞}(π) ⟶ M being a natural projection. CΛ_{k}^{1} ⊂ Λ_{k}^{1} (see [1]) is a locally free Λ_{k−1}–submodule generated by d_{k}^{v}(Λ_{k−1}). Elements

CΛ_{k}^{1} ∋ d_{k}^{v}d_{K}^{v}u_{σ}^{j}, K ⊂ {1, …, k − 1}, j = 1, …, m, σ a lenght n multi–index (1)

form a local basis of it and must be understood as iterated Cartan forms. Similarly, HΛ_{k}^{1} ⊂ Λ_{k}^{1} is a locally free Λ_{k−1}–submodule generated by d_{k}^{h}(Λ_{k−1}) and elements

HΛ_{k}^{1} ∋ d_{k}d_{K}x^{μ}, K ⊂ {1, …, k − 1}, μ = 1, …, n  (2)

form an its local basis. Denote by C^{•}Λ_{k} ⊂ Λ_{k} (resp. HΛ_{k} ⊂ Λ_{k}) the Λ_{k−1}–subalgebra generated by 1 and CΛ_{k}^{1} (resp. HΛ_{k}^{1}). Clearly, d_{k}^{v}(C^{•}Λ_{k}) ⊂ C^{•}Λ_{k} and d_{k}^{h}(HΛ_{k}) ⊂ HΛ_{k}. Moreover, Λ_{k}^{1} = CΛ_{k}^{1} ⊕ HΛ_{k}^{1} and Λ_{k} ≃ C^{•}Λ_{k} ⊗_{Λ_{k−1}} HΛ_{k}. Put D^{v}(Λ_{k−1}) =^{def} {X ∈ D(Λ_{k−1}, Λ_{k−1}) | X ∘ π_{∞}^{∗} = 0}. Elements of D^{v}(Λ_{k−1}) are called vertical derivations of Λ_{k−1}. D(Λ_{k−1}, Λ_{k−1}) splits as D(Λ_{k−1}, Λ_{k−1}) ≃ CD(Λ_{k−1}) ⊕ D^{v}(Λ_{k−1}) (see [1]). Moreover, CD(Λ_{k−1}) ≃ Hom_{Λ_{k−1}}(HΛ_{k}^{1}, Λ_{k−1}) and D^{v}(Λ_{k−1}) ≃ Hom_{Λ_{k−1}}(CΛ_{k}^{1}, Λ_{k−1}). Denote by V_{j}^{σ,K} ∈ D^{v}(Λ_{k−1}) and D_{μ}^{K} ∈ CD(Λ_{k−1}) with K ⊂ {1, …, k−1}, j = 1, …, m, μ = 1, …, n and σ being a lenght n multi–index, elements of the dual local bases of (1) and (2), respectively. The following two sub–algebras of Λ_{k−1} will be of a special interest in the sequel.

• C_{⋆}Λ_{k−1}: the F–subalgebra generated by elements of the form d_{K}^{v}f, f ∈ F,

K ⊂ {1, …, k − 1} (note that C_{⋆}Λ_{0} ≡ C_{⋆}Λ = C^{•}Λ).

• C_{∘}Λ_{k−1}: the C^{∞}(E)–subalgebra generated by elements of the form d_{K}^{v}g, g ∈

C^{∞}(E), K ⊂ {1, …, k − 1}.

ITERATED DIFFERENTIAL FORMS V: C–SPECTRAL SEQUENCE ON J^{∞}(π) 3

Clearly, C_{∘}Λ_{k−1} ⊂ C_{⋆}Λ_{k−1}. Moreover, V_{j}^{σ,K}(C_{∘}Λ_{k−1}) ⊂ C_{∘}Λ_{k−1} with K ⊂ {1, …, k − 1}, j = 1, …, m and σ being a lenght n multi–index. Denote by Λ_{k−1}κ the C_{⋆}Λ_{k−1}– module of C_{⋆}Λ_{k−1}–valued derivations of C_{∘}Λ_{k−1}. Λ_{k−1}κ is locally generated by derivations V_{j}^{K} ≡ V_{j}^{(0,...,0),K}, K ⊂ {1, …, k − 1}, j = 1, …, m. k–times IDFs–valued symmetries (see [1]) of J^{∞}(π) are easily described in terms of vertical derivations of Λ_{k−1}. Indeed, let χ = [X] ∈ Λ_{k−1}Sym, X ∈ D_{C}(Λ_{k−1}). There is a unique vertical representative of χ denoted by �_{χ}. �_{χ} is called an evolutionary derivation of Λ_{k−1}. �_{χ}(C_{⋆}Λ_{k−1}) ⊂ C_{⋆}Λ_{k−1}, since �_{χ} ∈ D_{C}(Λ_{k−1}) ∩ D^{v}(Λ_{k−1}).

Proposition 1. The correspondence Λ_{k−1}Sym ∋ χ ⟼ �_{χ}|_{C_{∘}Λ_{k−1}} ∈ Λ_{k−1}κ is an isomorphism of vector spaces, whose inverse looks locally as Λ_{k−1}κ ∋ χ_{K}^{j}V_{j}^{K} ⟼ (D_{σ}χ_{K}^{j})V_{j}^{σ,K} ∈ Λ_{k−1}Sym with χ_{K}^{j} ∈ C_{⋆}Λ_{k−1}, D_{σ} = (D_{1}^{(0,...,0)})^{σ_{1}} ∘ ⋯ ∘ (D_{n}^{(0,...,0)})^{σ_{n}}, K ⊂ {1, …, k − 1}, j = 1, …, m and σ = (σ_{1}, …, σ_{n}) being a lenght n multi–index.

As a consequence Λ_{k−1}Sym inherits the structure of a C_{⋆}Λ_{k−1}–module. On the other hand Λ_{k−1}κ inherits the structure of a graded Lie–algebra denoted by (Λ_{k−1}κ, {·, ·}). In the following we identify Λ_{k−1}Sym and Λ_{k−1}κ. Finally, note that d_{m}^{v} is an evolutionary derivation of Λ_{k−1} for any m < k. Denote by U_{m} the corresponding element in Λ_{k−1}κ, i.e., U_{m} = d_{m}^{v}|_{C_{∘}Λ_{k−1}}.

3. Adjoint graded C–differential operators and horizontal modules

In what follows the algebra C_{⋆}Λ_{k−1} is considered to be the ground algebra. In particular, all differential operators are operators over this algebra. Obviously, D_{σ}(C_{⋆}Λ_{k−1}) ⊂ C_{⋆}Λ_{k−1} for any lenght n multi–index σ. Let P, Q be locally free graded C_{⋆}Λ_{k−1}–modules of finite rank.

Definition 2. A linear differential operator □ : P ⟶ Q is called C–differential if for any local basis {e_{1}, …, e_{r}} of P, □ is locally of the form □(p) = (−1)^{|p|·|α|}□_{α}^{σ}D_{σ}p^{α}, where □_{α}^{σ} ∈ Q and p = p^{α}e_{α} ∈ P, |α| ≡ |e_{α}|, p^{α} ∈ C_{⋆}Λ_{k−1}, α = 1, …, r.

The totality of all C–differential operators □ : P ⟶ Q has a natural C_{⋆}Λ_{k−1}–module structure. This module is denoted by CDiff(P, Q). Similarly, denote by CDiff_{(p)}^{alt}(P, Q) the C_{⋆}Λ_{k−1}–module of Q–valued, skew–symmetric, multi–C–differential operators over P with p ≥ 0 entries (CDiff_{(0)}^{alt}(P, Q) ≡ Q). HΛ_{k} is a C_{⋆}Λ_{k−1}–module and d_{k}^{h} : HΛ_{k} ⟶ HΛ_{k} is a C–differential operator. The theory of adjoint C–differential operators in the category of C_{⋆}Λ_{k−1}–modules may be developed almost literally as in the standard non–graded case (see [4]). Here we limit ourselves to those elements of this theory that are necessary for the proposed below description of the Λ_{k−1}C–spectral sequence terms . Let P be a graded C_{⋆}Λ_{k−1}–module. Consider the map w_{k}^{P} : CDiff(P, HΛ_{k}) ⟶ CDiff(P, HΛ_{k}), given by w_{k}^{P} (□) = d_{k}^{h} ∘ □, □ ∈ CDiff(P, HΛ_{k}). w_{k}^{P} is a differential, i.e., w_{k}^{P} ∘ w_{k}^{P} = 0. The cohomology space of w_{k}^{P} carries a natural C_{⋆}Λ_{k−1}–module structure defined by ω · [□] =^{def} (−1)^{|ω|·|□|}[□ ∘ ω], ω ∈ C_{⋆}Λ_{k−1}, □ ∈ CDiff(P, HΛ_{k}), d_{k}^{h} ∘ □ = 0.

The corresponding C_{⋆}Λ_{k−1}–module is denoted by P̂ and called the adjoint to P module.

Put Λ_{k−1}B ≡ { \atop C_{⋆}Λ_{k−1}}.

Proposition 3. Λ_{k−1}B ≃ C_{⋆}Λ_{k−1} ⊗_{F} HΛ_{1}^{n}. Moreover, if P is a locally free C_{⋆}Λ_{k−1}– module of finite rank, then P̂ ≃ Hom_{C_{⋆}Λ_{k−1}}(P, Λ_{k−1}B).

In particular, proposition 3 tells that Λ_{k−1}B is a locally free C_{⋆}Λ_{k−1}–module of rank

one. Thus, if P is a locally free C_{⋆}Λ_{k−1}–module of finite rank, then ˆ P̂ ≃ P. Further on our exposition is based on identifications of proposition 3. Note that Λ_{k−1}B is embedded in ker d_{k}^{h} ⊂ HΛ_{k} in a natural way by means of the correspondence Λ_{k−1}B ≃ C_{⋆}Λ_{k−1} ⊗_{F} HΛ_{1}^{n} ∋ ω ⊗ σ ↦ ω · κ_{1k}(σ) ∈ HΛ_{k}, ω ∈ C_{⋆}Λ_{k−1}, σ ∈ HΛ_{1}^{n}. If P, Q are C_{⋆}Λ_{k−1}–modules and □ ∈ CDiff(P, Q), then a differential operator ˆ□ ∈ CDiff(Q̂, P̂) is well–defined by ˆ□[Δ] = (−1)^{|Δ|·|□|}[Δ∘□], Δ ∈ CDiff(Q, HΛ_{k}), d_{k}^{h}∘Δ = 0. ˆ□ is called the adjoint to □ operator. Let P be as above, ξ_{1}, …, ξ_{p−2} ∈ P and □ ∈ CDiff_{(p−1)}^{alt}(P, P̂). Define a C–differential

operator □_{ξ_{1},...,ξ_{p−2}} : P ⟶ P̂ by putting □_{ξ_{1},...,ξ_{p−2}}(ξ) =^{def} □(ξ_{1}, …, ξ_{p−2}, ξ) ∈ P̂. Also

put

L_{p}^{(k)}(P) =^{def} \{□ ∈ CDiff_{(p−1)}^{alt}(P, P̂) | { \atop □_{ξ_{1},...,ξ_{p−2}}} = −□_{ξ_{1},...,ξ_{p−2}}, ∀ξ_{1}, …, ξ_{p−2} ∈ P \}.

A C_{⋆}Λ_{k−1}–module P is called horizontal if it is of the form P ≃ C_{⋆}Λ_{k−1} ⊗_{C^{∞}(M)} P_{0} for a (possibly graded) C^{∞}(M)–module P_{0}. Evolutionary derivations of Λ_{k−1} act naturally on horizontal modules. Namely, let χ ∈ Λ_{k−1}Sym and �_{χ} be the corresponding evolutionary derivation of Λ_{k−1}. For ξ = ω ⊗ ξ_{0} ∈ P, ω ∈ C_{⋆}Λ_{k−1} and ξ_{0} ∈ P_{0} we put

�_{χ}ξ =^{def} �_{χ}ω ⊗ ξ_{0} ∈ P.  (3)

Since �_{χ} is a vertical derivation, definition (3) extends unambiguously to the whole P. Now, fix an element ξ ∈ P and define an operator ℓ_{ξ}^{{k}} : Λ_{k−1}κ ⟶ P by putting

ℓ_{ξ}^{{k}}(χ) =^{def} (−1)^{|ξ|·|χ|}�_{χ}ξ, χ ∈ Λ_{k−1}κ. ℓ_{ξ}^{{k}} is a C–differential operator called the universal Λ_{k−1}–linearization of ξ.

Example 4. HΛ_{k} is a horizontal module. Indeed, HΛ_{k} ≃ C_{⋆}Λ_{k−1} ⊗_{C^{∞}(M)} Λ_{k}(M).

Example 5. Λ_{k−1}κ and { \atop Λ_{k−1}κ} are horizontal modules. Indeed, let W_{k−1} ⊂ Λ_{k−1}κ be the C^{∞}(M)–submodule of derivations that are locally of the form χ_{K}^{j}V_{j}^{K} with χ_{K}^{j} ∈ C^{∞}(M), K ⊂ {1, …, k − 1}, j = 1, …, m. W_{k−1} is well defined. Then Λ_{k−1}κ ≃

C_{⋆}Λ_{k−1} ⊗_{C^{∞}(M)} W_{k−1} and { \atop Λ_{k−1}κ} ≃ C_{⋆}Λ_{k−1} ⊗_{C^{∞}(M)} Hom_{C^{∞}(M)}(W_{k−1}, Λ^{n}(M)).

4. Secondary IDFs on J^{∞}(π)

The complexes (Λ_{k−1}CE_{0}^{0,•} = Λ_{k}/CΛ_{k}, d_{k,0}^{0,•}) and (HΛ_{k}, d_{k}^{h}) are isomorphic in a natural way and further on we shall identify Λ_{k−1}CE_{0}^{0,•} with HΛ_{k}.

Remark 6. For any p > 0, there exists a natural isomorphism of complexes

[figure: Λ_{k−1}CE_{0}^{p,•}; d_{0,k}^{p,•}; η_{k−1}; ; Λ_{k−1}CE_{0}^{p,•}; η_{k−1}; ; CDiff_{(p)}^{alt}(Λ_{k−1}κ, HΛ_{k}); w_{k}^{p}; CDiff_{(p)}^{alt}(Λ_{k−1}κ, HΛ_{k})] ,

where w_{k}^{p}(□) =^{def} (−1)^{p}d^{h} ∘ □, □ ∈ CDiff_{(p)}^{alt}(Λ_{k−1}κ, HΛ_{k}). The isomorphism η_{k−1} is defined by

η_{k−1}([ω]_{C^{p+1}Λ_{k}})(χ_{1}, …, χ_{p}) = (−1)^{|ω|·(|χ_{1}|+···+|χ_{p}|)+p(p−1)/2}[(i_{�_{χ_{1}}}^{{k}} ∘⋯∘i_{�_{χp}}^{{k}})(ω)]_{CΛ_{k}} ∈ HΛ_{k}.

The following two theorems generalize the results of [6, 2] concerning the first term of the C–spectral sequence to that of the Λ_{k−1}C–spectral one. In particular, they give an explicit description of secondary IDFs on J^{∞}(π) and basic operations over them.

Theorem 7 (One Line Theorem).

• Λ_{k}CE_{1}^{p,q} = 0 if either q > n, or p > 0 and q < n;

• Λ_{k−1}CE_{1}^{0,•} ≃ H(HΛ_{k}, d_{k}^{h}) ≃ Λ_{k−2}CE_{1}^{•,•};
• Λ_{k−1}CE_{1}^{p,n} ≃ L_{p}^{(k)}(Λ_{k−1}κ), if p > 0.

Corollary 8. Λ_{k−1}CE stabilizes at the second term and Λ_{k−1}CE_{2}^{0,q} ≃ H^{q}(Λ_{k}, d_{k}) ≃ H^{q}(E), q ≤ n, and Λ_{k−1}CE_{2}^{p,n} = H^{p+n}(E), p ≥ 0.

There is a distinguished element 𝔲 in { \atop HΛ_{k}} ≃ Hom_{C_{⋆}Λ_{k−1}}(HΛ_{k}, Λ_{k−1}B) whose local expression is

𝔲(Ω_{μ_{1}···μ_{r}}^{J_{1}···J_{r}}d_{J_{1}}x^{μ_{1}} ⋯ d_{J_{r}}x^{μ_{r}}) =^{def}Ω_{μ_{1}···μ_{r}}^{J_{1}···J_{r}} ⊗ ((κ_{1k} ∘ ν)(d_{J_{1}}x^{μ_{1}} ⋯ d_{J_{r}}x^{μ_{r}}))
∈ C_{⋆}Λ_{k−1} ⊗_{F} HΛ_{1}^{n} ≃ Λ_{k−1}B,

where Ω_{μ_{1}···μ_{r}}^{J_{1}···J_{r}} ∈ C_{⋆}Λ_{k−1} and ν : HΛ_{k} ⟶ HΛ_{k}^{n} is the projection onto the homogeneous component of multi–degree (0, …, 0, n) ∈ ℤ^{k}.

Theorem 9.

• The differential d_{k,1}^{0,n} : H^{n}(HΛ_{k}, d_{k}^{h}) ⟶ { \atop Λ_{k−1}κ} is given by

d_{k,1}^{0,n}[Ω]_{im d_{k}^{h}} = ℓ̂_{Ω}^{{k}}(𝔲), Ω ∈ HΛ_{k}^{n};

and its local description is

d_{k,1}^{0,n}([Ω]_{im d_{k}^{h}})(χ) = (−1)^{σ+|χ|·|Ω|}χ_{L}^{j}(D_{σ} ∘ V_{j}^{σL})(A) ⊗ dx^{1} ⋯ dx^{n}
∈ C_{⋆}Λ_{k−1} ⊗_{F} HΛ_{1}^{n} ≃ Λ_{k−1}B,

assuming that Ω = A d_{k}x^{1} ⋯ d_{k}x^{n} ∈ HΛ_{k}^{n}, A ∈ C_{⋆}Λ_{k−1} and χ = χ_{L}^{j}V_{j}^{L} ∈ Λ_{k−1}κ; • the differential d_{k,1}^{p,n} : L_{p}^{(k)}(Λ_{k−1}κ) ⟶ L_{p+1}^{(k)}(Λ_{k−1}κ) acting on secondary iterated

p–forms, p > 0, is given by

d_{k,1}^{p,n}(Θ)(χ_{1}, …, χ_{p}) = ∑_{i=1}^{p}(−1)^{a(i)+i+1}(�_{χ_{i}})(Θ(χ_{1}, …, χ̂_{i}, …, χ_{p}))

+ ∑_{i<j}(−1)^{c(i,j)+i+j}Θ({χ_{i}, χ_{j}}, χ_{1}, …, χ̂_{i}, …, χ̂_{j}, …, χ_{p})

+ \frac{1}{p} ∑_{i=1}^{p}(−1)^{i+1}[(−1)^{a(i)}(p − 1)ℓ̂_{χ_{i}}^{\{k+1\}}(Θ(χ_{1}, …, χ̂_{i}, …, χ_{p}))

− (−1)^{b(i)}ℓ̂_{Θ(χ_{1},...,χ̂_{i},...,χ_{p})}^{{k+1}}(χ_{i})],

Θ ∈ L_{p}^{(k)}(Λ_{k−1}κ), where a(i) = |χ_{i}| · (|Θ| + |χ_{1}| + ⋯ + |χ_{i−1}|), b(i) = |χ_{i}| · (|χ_{i+1}| + ⋯ + |χ_{p}|), c(i, j) = a(i) + a(j) − |Θ| · (|χ_{i}| + |χ_{j}|) − |χ_{i}| · |χ_{j}|;

• the action L_{χ}^{{k}} : L_{p}^{(k)}(Λ_{k−1}κ) ⟶ L_{p}^{(k)}(Λ_{k−1}κ) of χ ∈ Λ_{k−1}Sym on secondary iterated p–forms, p > 0, is given by

(L_{χ}^{{k}}Θ)(χ_{1}, …, χ_{p−1})(χ_{p}) = (�_{χ} ∘ Θ)(χ_{1}, …, χ_{p})

− ∑_{i=1}^{p}(−1)^{|χ|·(|χ_{1}|+···+|χ_{i−1}|+|Θ|)}Θ(χ_{1}, …, {χ, χ_{i}}, …, χ_{p}),

Θ ∈ L_{p}^{(k)}(Λ_{k−1}κ).

• the insertion i_{χ}^{\{k\}} : { \atop Λ_{k−1}κ} ⟶ H(HΛ_{k}, d_{k}^{h}) of χ ∈ Λ_{k−1}Sym in secondary iterated 1–forms is given by

i_{χ}^{\{k\}}Θ = (−1)^{|χ|·|Θ|}[Θ(χ)]_{im d_{k}^{h}}, Θ ∈ { \atop Λ_{k−1}κ}, Θ(χ) ∈ Λ_{k−1}B ↪ ker d_{k}^{h}.

• the insertion i_{χ}^{{k}} : L_{p}^{(k)}(Λ_{k−1}κ) ⟶ L_{p−1}^{(k)}(Λ_{k−1}κ) of χ ∈ Λ_{k−1}Sym in secondary iterated p–forms, p > 1, is given by

(i_{χ}^{{k}}Θ)(χ_{1}, …, χ_{p−2}) = (−1)^{|χ|·|Θ|}Θ(χ, χ_{1}, …, χ_{p−2}).

5. Secondary covariant tensors

Recall that τ ∈ Λ_{k−1}CE_{1}^{(1,...,1),•} is a secondary covariant k–tensor iff L_{I_{m}^{K′}}^{{k}}τ = i_{I_{m}^{K}}^{{k}}τ = 0 for any m < k and K = {k_{1}, …, k_{s}}, K′ = {k′_{1}, …, k′_{s′}} ⊂ {1, …, k − 1}, s ≥ 2, s′ ≥ 1 (see [1]). In what follows we characterize such τ’s. First of all, there is a well defined F–linear surjective map 𝔭_{k−1} : Λ_{k−1}κ ⟶ Λ_{0}κ ≡ κ locally given by

𝔭_{k−1}(χ_{K}^{j}V_{j}^{K}) =^{def} p_{0}(χ^{j})\frac{∂}{∂u^{j}} , χ_{K}^{j} ∈ C_{⋆}Λ_{k−1}, K ⊂ \{1, …, k − 1\}, j = 1, …, m,

p_{0} : C_{⋆}Λ_{k−1} ⟶ F being the projection onto the homogeneous component of multi– degree (0, …, 0) ∈ ℤ^{k−1}. Then there is an injective morphism of F–modules 𝔦_{k} :

(CΛ^{1})^{⊗k−1} ⊗_{F} κ̂ ⟶ Λ_{k}CE_{1}^{(1,...,1),•} ⊂ { \atop Λ_{k−1}κ} given by

𝔦_{k}(ω_{1} ⊗ ⋯ ⊗ ω_{k−1} ⊗ ψ)(χ) =^{def}(ω_{1} · κ_{12}(ω_{2}) · · · · · κ_{1k−1}(ω_{k−1})) ⊗ ((ψ ∘ 𝔭_{k−1})(χ))
∈ C_{⋆}Λ_{k−1} ⊗_{F} HΛ_{1}^{n} ≃ Λ_{k−1}B,

ω_{1}, …, ω_{k−1} ∈ CΛ^{1}, ψ ∈ κ̂, χ ∈ Λ_{k−1}κ. Now we are able to characterize secondary IDFs on J^{∞}(π) in the same manner as it was done for ordinary IDFs in [5]

Proposition 10. τ ∈ Λ_{k−1}CE_{1}^{(1,...,1),•} is a secondary covariant k–tensor iff τ ∈ im 𝔦_{k}.

6. Conclusions

The above given description of secondary IDFs on J^{∞}(π) was not conceptual in all its aspects just because the necessary for this purpose constructions can not be put in frames of a short note. This will be done in the subsequent detailed exposition together with basic elements of secondary tensor calculus and some applications to mechanics and field theory.

References

[1] A. M. Vinogradov, L. Vitagliano, Dokl. Math., to appear in, see also The Diffiety Inst. Preprint

   Series, DIPS 5/06, http://diffiety.ac.ru/preprint/2006/05-06abs.htm or arXiv:math.
   DG/0610917 .

[2] A. M. Vinogradov, J. Math. Anal. Appl. 100 (1984) 1.

[3] I. S. Krasil’shchik, A. M. Verbovetsky, Homological Methods in Equation of Mathematical Physics Open Education (Opava) 1998. See also Diffiety Inst. Preprint Series, DIPS 7/98, http://

   diffiety.ac.ru/preprint/2006/07_06abs.htm .

[4] A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor’kova, I. S. Krasil’shchik, A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky, and A. M. Vinogradov, Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, AMS “Translation of Mathematical Monographs” 182 (1999).

[5] A. M. Vinogradov, L. Vitagliano, Dokl. Math. 73, n^{∘} 2 (2006) 169, see also The Diffiety

   Inst. Preprint Series, DIPS 1/06, http://diffiety.ac.ru/preprint/2006/01-06abs.htm or
   arXiv:math.DG/0605113 .

[6] A. M. Vinogradov, Soviet Math. Dokl. 19 (1978) 144.

[7] W. M. Tulczyjew, in “Lecture Notes in Mathematics” 836, Springer-Verlag, New York (1980) 22.

[8] T. Tsujishita, Osaka J. Math. 19 (1982) 311.

[9] I. M. Anderson, Contemp. Math. 132 (1992) 51.

Alexandre Vinogradov, DMI, Facoltà di Scienze MM. FF. NN. Università degli Studi di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA) and INFN, sez. Napoli–Salerno, Italy

 E-mail address: vinograd@unisa.it

Luca Vitagliano, DMI, Facoltà di Scienze MM. FF. NN. Università degli Studi di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA) and INFN, sez. Napoli–Salerno, Italy

E-mail address: luca vitagliano@fastwebnet.it