The Diffiety Institute Preprint Series

Preprint DIPS–1/2006 May 2, 2006

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

The Diffiety Institute

Iterated Differential Forms I: Tensors

by

A. M. Vinogradov and L. Vitagliano

DIPS-1/2006

                                            arXiv:math.DG/0605113

Iterated Differential Forms I: Tensors

A. M. VINOGRADOV AND L. VITAGLIANO

Abstract. This note is the first in a series of short communications dedicated to general theory and some applications of iterated differential forms. Both are developed either in the “classical”context, or in the “quantistic”one, i.e., of Secondary Calculus (see [1], [2]). Detailed expositions containing proofs of the announced results will be appearing in due course. With iterated forms we solve the problem of a conceptual foundation of tensor calculus. In particular, we show that covariant tensors are differential forms over a certain graded commutative algebra called the algebra of iterated differential forms. From one side, this interpretation extends noteworthy frames of the traditional tensor calculus and enriches it by numerous new natural operators. On the other side, it allows various generalizations of tensor calculus, the most important of which is that to secondary (“quantized”) calculus. In particular, this leads to an unified solution of the secondarization (“quantization”) problem (see [1], [2]) for arbitrary tensors. In this communication the algebra of iterated (differential) forms over an arbitrary (graded) commutative algebra is defined. It is also shown how tensors on a (smooth) manifold M are naturally interpreted as iterated differential forms over the algebra C^{∞}(M).

According to the original definition a tensor on a manifold M is a system of suitably indexed functions, called components, associated with a local chart, which change according to a certain rule when passing to another chart. The modern definition presents a tensor to be a C^{∞}(M)–multilinear and C^{∞}(M)–valued function in a number of variables that are either vector fields, or differential 1–forms on M. These definitions are manifestly descriptive. For instance, on the basis of any of them it is not possible to understand why the natural exterior differential d is defined on skew–symmetric covariant tensors, i.e., differential forms, but not on symmetric ones. Or, why a natural connection, namely, that of Levi–Civita, is associated with (non–degenerate) symmetric covariant 2–tensors, but not with skew–symmetric ones, etc. According to [3] the meaning of “controvariant”objects of differential calculus, say, vector fields, or general differential operators, is given by functors of differential calculus. For instance, the

2000 Mathematics Subject Classification. 53A55, 58A10, 58A50. Key words and phrases. Differential Calculus, Graded Algebras, Differential Forms, Tensors.

derivation functor D corresponds to vector fields and Diff_{k} to k–th order differential operators. On the other hand, covariant objects are related to representing such functors objects. For instance, differential i–forms and k–jets are elements of modules that represent functors D_{i} and Diff_{k}, respectively. Concerning conceptual definition of a species of covariant tensors the problem is to attribute them to the module representing a certain functor of differential calculus. What makes this problem not very banal is that such a direct attribution is not, generally, possible. This is, for instance, the case of symmetric tensors. On the contrary, skew-symmetric tensors, i.e., differential forms, allow such one and the corresponding functors are D_{i}’s. We overcome this difficulty by looking for the necessary direct attribution not over the original ground commutative algebra, but over another one, naturally associated with the former. Below this idea is realized by passing to the filtered graded commutative algebra A = Λ_{0} ⊂ Λ_{1} ⊂ ⋯ ⊂ Λ_{k} ⊂ ⋯ ⊂ Λ_{∞}, where A is the ground algebra and Λ_{k} is the algebra of differential forms over the algebra Λ_{k−1}. This way we respond the question: what properly are covariant tensors? In the subsequent note the first application of the theory sketched here to Riemannian geometry will be given. In particular, the nature of the Levi–Civita connection will be clarified and answered the question posed above.

1. Differential calculus over graded algebras

A pair G = (G, μ), G being an abelian group and μ : G × G ⟶ ℤ_{2} a symmetric ℤ– bilinear map, is a grading group. Below 𝕜 stands for a field of zero–characteristic and we use g·h for μ(g, h). The category of unitary, G–graded, associative, graded–commutative 𝕜–algebras is denoted by Alg_{𝕜}^{G}. If A ∈ Alg_{𝕜}^{G} and a ∈ A is homogeneous, then |a| ∈ G denotes the degree of a. Recall that graded–commutativity means that ab = (−1)^{|a|·|b|}ba for all homogeneous elements a, b ∈ A. In the sequel we adopt the following convention: in the exponent of (−1) the symbol, say, σ, denoting a homogeneous element is used as a substitute of |σ|. For instance, (−1)^{a·b} means (−1)^{|a|·|b|}. The differential calculus over an algebra A ∈ Alg_{𝕜}^{G} is introduced along the lines of [3, 4, 5, 6] where the reader will find further details. Denote by Mod_{A}^{G} the category of G–graded A–modules and by D_{A} : Mod_{A}^{G} ⟶ Mod_{A}^{G} the functor associating with P ∈ Mod_{A}^{G} the graded A–module D_{A}(P) of P – valued derivations of A. D_{A}(A) is a graded 𝕜–Lie algebra with respect to the graded commutator. Functors D_{k} : Mod_{A}^{G} ⟶ Mod_{A}^{G}, k ∈ ℕ are defined in [3, 4, 5]. Recall that an element ∇ ∈ D_{k}(P) may be viewed as a graded skew–symmetric P –valued multi–derivation of A of multiplicity k. In particular, D_{1} = D_{A}. Denote by Λ^{k}(A) the graded A–module of differential k–forms over A and by Λ^{k}(A)^{g}, g ∈ G, its homogeneous component of grade g. The direct sum Λ(A) = ⨁_{k=0}^{∞} Λ^{k}(A) has a natural structure of an unitary, G ⊕ ℤ–graded, associative, graded–commutative 𝕜–algebra which is called the algebra of differential forms over A. It is naturally isomorphic to the G–exterior algebra ⋀^{∗} Λ^{1}(A).

Thus, any σ ∈ Λ(A) is of the form

σ = ∑ a_{α_{1}···α_{k}}db_{α_{1}} ∧ ⋯ ∧ db_{α_{k}},  (1)

for some a_{α_{1}...α_{k},}b_{α_{1}}, … b_{α_{k}} ∈ A. For any σ ∈ Λ^{k}(A)^{g} put |σ| = (g, k) ∈ G ⊕ ℤ. Then, σ ∧ ρ = (−1)^{σ·ρ}ρ ∧ σ for any homogeneous elements σ, ρ ∈ Λ(A), where (g, k) · (h, l) =^{def} g · h + [kl]_{2} ∈ ℤ_{2}. The exterior differential in Λ(A) will be denoted by d : Λ(A) ⟶ Λ(A). It is a graded Λ(A)–derivation of bi–degree |d| = (0, 1) ∈ G ⊕ ℤ and d^{2} = 0. So, (Λ(A), d) is a G ⊕ ℤ–graded differential algebra. The correspondence A ⟼ (Λ(A), d) is a functor from the category Alg_{𝕜}^{G} to the category dAlg_{𝕜}^{G⊕ℤ} of unitary, G ⊕ ℤ–graded, associative, graded–commutative, differential 𝕜–algebras. In particular, if A, A′ ∈ Alg_{𝕜}^{G}, then a morphism ϕ : A ⟶ A′ is extended to a morphism Λ(ϕ) : Λ(A) ⟶ Λ(A′) compatible with the exterior differential, i.e., d ∘ Λ(ϕ) = Λ(ϕ) ∘ d. If σ ∈ Λ(A) is of the form (1), then

Λ(ϕ)(σ) = ∑ ϕ(a_{α_{1}···α_{k}})d(ϕ(b_{α_{1}})) ∧ ⋯ ∧ d(ϕ(b_{α_{k}})).

The insertion of a derivation X ∈ D_{A}(A) operator will be denoted by i_{X} : Λ(A) ⟶ Λ(A). It is a Λ(A)–derivation of bi–degree (|X|, −1). The Lie derivative along X ∈ D_{A}(A) is defined by L_{X} = [i_{X}, d]. It is a graded Λ(A)–derivation of bi–degree (|X|, 0) and extends X to Λ(A). For any X, Y ∈ D_{A}(A) the following (graded) commutation relations hold:

[i_{X}, i_{Y}] = [L_{X}, d] = 0, [i_{X}, L_{Y}] = i_{[X,Y]}, [L_{X}, L_{Y}] = L_{[X,Y]}.  (2)

2. Iterated differential forms

In what follows we put Λ ≡ Λ(A). So, Λ is an unitary, graded, associative, graded– commutative algebra and all constructions of the previous section are applied to Λ and so on. This leads to the following [7]

Definition 1. Given k ∈ ℕ the algebra Λ_{k} of k–times iterated differential forms over A is defined inductively as Λ_{k} = Λ(Λ_{k−1}), by starting with Λ_{0} = A. The exterior differential d_{k} in Λ_{k} (d_{1} = d) is called k–th iterated exterior differential.

So, (Λ_{k}, d_{k}) is a differential, G ⊕ ℤ^{k}–graded commutative algebra. Natural inclusions Λ_{k−1} ⊂ Λ(Λ_{k−1}) = Λ_{k} define the filtered algebra Λ_{∞}:

Λ_{0} ≡ A ⊂ Λ_{1} ≡ Λ ⊂ Λ_{2} ⊂ ⋯ ⊂ Λ_{k} ⊂ ⋯ ⊂ Λ_{∞},

where Λ_{∞} ≡ ⋃_{k} Λ_{k}. Λ_{∞} is a G ⊕ ℤ^{∞}–graded commutative algebra. If (g, K) ∈ G ⊕ ℤ^{k}, denote by Λ_{k}^{(g,K)} ⊂ Λ_{k} the homogeneous component of grade (g, K). Put also

Λ_{k}^{K} =^{def} ⨁_{g∈G} Λ_{k}^{(g,K)} ⊂ Λ_{k}.

Definition 2. Λ_{∞} is called the algebra of iterated forms over A.

The operator of insertion of ∇ ∈ D_{Λ_{k}}(Λ_{k}) into forms Λ(Λ_{k}) = Λ_{k+1} over Λ_{k} is denoted by i_{∇}^{(k+1)} ∈ D_{Λ_{k+1}}(Λ_{k+1}). The Lie derivative L_{∇}^{(k+1)} = [i_{∇}^{(k+1)}, d_{k+1}] ∈ D_{Λ_{k+1}}(Λ_{k+1}) extends ∇ to Λ_{k+1}. In its turn, L_{∇}^{(k+1)} can be extended to Λ_{k+2} and so on up to Λ_{∞}. Hence any ∇ ∈ D_{Λ_{k}}(Λ_{k}) extends to a derivation of Λ_{∞}. This extension will be denoted by the same symbol ∇. Note that this notation is consistent with the third relation in (2). In particular, the k–th differential d_{k} can be extended to a derivation of Λ_{∞}. Moreover, It follows from commutation relations (2) that d_{j}^{2} = 0 and [d_{i}, d_{j}] = 0 for all i, j. This way (Λ_{∞}, d_{1}, …, d_{k}, …) becomes a multiple complex. Note also that for a given k the correspondence A ⟼ (Λ_{∞}, d_{k}) is a functor from the category Alg_{𝕜}^{G} to the category dAlg_{𝕜}^{G⊕ℤ^{∞}}. Note that Λ_{k+l} = Λ_{l}(Λ_{k}) for any k, l ∈ ℕ. Therefore, Λ_{l}(Λ_{∞}) = Λ_{∞}, l ∈ ℕ. Indeed,

Λ_{l}(Λ_{∞}) = Λ_{l}(_{⋃k}Λ_{k}) = _{⋃k}Λ_{l}(Λ_{k}) = _{⋃k}Λ_{k+l} = Λ_{∞}.

We express this fact by saying accordingly that Λ_{∞} is Λ–closed.

Proposition 3. There is an isomorphism κ_{(12)} of the double complexes (Λ_{2}, d_{1}, d_{2}) and (Λ_{2}, d_{2}, d_{1}).

Proof. The proposition is proved via the following commutative diagram

[figure: A; d_{1}; Λ_{1}^{1}; i_{d_{2}}; Λ_{1}; d_{2}; κ′; ⋀^{∗}(i_{d_{2}}); Λ_{2}^{1}; κ′′; Λ^{1}(κ′); Λ_{2}; κ_{(12)}; ⋀^{∗} κ′′; A; d_{2}; Λ_{2}^{(0,1)}; Λ_{2}^{(0,∗)}; d_{1}; d′′; Λ_{2}^{(1,∗)}; Λ_{2}; ⋀_{A}^{∗}(Λ_{2}^{(0,1)}); _{⋀_{′}}; Λ^{1}(Λ_{2}^{(0,∗)}); i_{d′′}; ⋀_{Λ_{2}^{(0,∗)}}^{∗}(Λ_{2}^{(1,∗)}); _{⋀_{′′′}}]

where the following notations have been adopted. Λ_{2}^{(m,∗)} = ⨁_{l} Λ_{2}^{(m,l)}, m = 0, 1, ∧′ is the product in Λ_{2} of elements belonging to Λ_{2}^{(0,1)} ⊂ Λ_{2} and κ′ = ∧′∘⋀^{∗}(i_{d_{2}}). Note that Λ_{2}^{(0,∗)} is a G ⊕ ℤ–graded, unitary, graded–commutative algebra and d′′ : Λ_{2}^{(0,∗)} ⟶ Λ^{1}(Λ_{2}^{(0,∗)}) is its first de Rham differential. Moreover κ′′ = i_{d′′} ∘ Λ^{1}(κ′). Finally ∧′′′ is the product in Λ_{2} of elements belonging to Λ_{2}^{(1,∗)} ⊂ Λ_{2} and κ_{(12)} = ∧′′′ ∘ ⋀^{∗} κ′′. It is straightforward to see that

κ_{(12)}(d_{1}g_{1} ∧ ⋯ ∧ d_{1}g_{p} ∧ d_{2}h_{1} ∧ ⋯ ∧ d_{2}h_{q} ∧ d_{1}d_{2}ℓ_{1} ∧ ⋯ ∧ d_{1}d_{2}ℓ_{r})
= d_{2}g_{1} ∧ ⋯ ∧ d_{2}g_{p} ∧ d_{1}h_{1} ∧ ⋯ ∧ d_{1}h_{q} ∧ d_{1}d_{2}ℓ_{1} ∧ ⋯ ∧ d_{1}d_{2}ℓ_{r}

for any g_{1}, …, g_{p}, h_{1}, …, h_{q}, ℓ_{1}, …, ℓ_{r} ∈ A. Therefore, κ_{(12)} is an involution of Λ_{2}. □

Let S_{k} be the group of permutations of {1, …, k}.

Corollary 4. For any k ∈ ℕ and σ ∈ S_{k} there is an isomorphism κ_{σ} of the multiple complexes (Λ_{k}, d_{1}, …, d_{k}) and (Λ_{k}, d_{σ(1)}, …, d_{σ(k)}).

Proof. The proof is by induction on k. Obviously, it is sufficient to prove the assertion for a transposition σ. The base of induction, k = 2, is provided by the above proposition. Now, suppose the corollary be true for Λ_{k−1}. In particular, for any i, j < k there is an isomorphism κ_{i,j} of the complexes (Λ_{k−1}, d_{i}) and (Λ_{k−1}, d_{j}) which commutes with all the other differentials. By extending the d_{i}’s, i < k, as derivations to Λ_{k} one finds that the complexes (Λ_{k}, d_{i}) and (Λ_{k}, d_{j}), i, j < k are isomorphic as well. Moreover, there exists an isomorphism of the complexes (Λ_{k}, d_{k−1}) and (Λ_{k}, d_{k}). Therefore (Λ_{k}, d_{i}) and (Λ_{k}, d_{j}) are isomorphic for any i, j ≤ k. By abusing the notation we again denote by κ_{i,j} such isomorphism. Since for any algebra automorphism which commutes with a given derivation X, the corresponding automorphism of the differential form algebra commutes with the Lie derivative along X, it is clear that κ_{i,j} commutes with the differentials d_{l}, l ≠ i, j. □

Corollary 5. For any permutation σ ∈ S_{ℕ} there exists an isomorphism κ_{σ} of the multiple complexes (Λ_{∞}, d_{1}, …, d_{k}, …) and (Λ_{∞}, d_{σ(1)}, …, d_{σ(k)}, …).

Proof. Let σ ∈ S_{ℕ} and Ω ∈ Λ_{k} ⊂ Λ_{∞}. Let l = max{σ(1), …, σ(k)} and σ̃ ∈ S_{l} be such that σ̃(i) = σ(i) for any i ≤ k. Put κ_{σ}(Ω) ≡ κσ̃(Ω). κ_{σ} does not depend on the choice of σ̃ and is therefore well-defined. Moreover, κ_{σ} has inverse κ_{σ^{−1}}. Thus it is an isomorphism of the multiple complexes (Λ_{∞}, d_{1}, …, d_{k}, …) and (Λ_{∞}, d_{σ(1)}, …, d_{σ(k)}, …). □

The correspondence σ ⟼ κ_{σ} defines an action of the group S_{ℕ} on Λ_{∞}. An important fact is that the cohomology of complexes (Λ_{∞}, d_{k}), k ∈ ℕ is “constant”.

Theorem 6. Complexes (Λ_{2}, d_{2}) and (Λ, d) are naturally homotopy equivalent.

Proof. In the proof of proposition 3 a natural isomorphism between the complexes (Λ_{2}^{(0,∗)}, d_{2}) and (Λ, d) was established. Let ι : (Λ_{2}^{(0,∗)}, d_{2}) ⟶ (Λ_{2}, d_{2}) be a natural embedding and π : (Λ_{2}, d_{2}) ⟶ (Λ_{2}^{(0,∗)}, d_{2}) a natural projection. We shall prove that the pair (ι, π) is a homotopy equivalence. Since π ∘ ι = id_{Λ_{2}^{(0,∗)}} it suffices to prove that

ι ∘ π is homotopy equivalent to id_{Λ_{2}}. If Ω ∈ Λ_{2}^{(s,t)}, then

(id_{Λ_{2}} − ι ∘ π)(Ω) = \{ \begin{matrix}Ω & if s ≠ 0 \\ 0 & if s = 0 \end{matrix} .

Denote by d^{0} the 0–degree component of the ordinary de Rham differential. It is a Λ^{1}–valued derivation of A. The corresponding insertion operator C = i_{d^{0}} : Λ ⟶ Λ is a derivation of Λ and we have C(σ) = sσ for σ ∈ Λ^{s}.

Consider the 0–degree map i_{C}^{(2)} : Λ_{2} ⟶ Λ_{2} and define the map H_{2} : Λ_{2} ⟶ Λ_{2} by:

H_{2}(Ω) = \{ \begin{matrix}\frac{1}{s}i_{C}^{(2)}(Ω) & if s ≠ 0 \\ 0 & if s = 0 \end{matrix} ,

for Ω ∈ Λ_{2}^{(s,t)}. Prove that H_{2} is a homotopy connecting the chain maps ι and π. Indeed, if s = 0, then [H_{2}, d_{2}](Ω) = 0 = (id_{Λ_{2}} − ι ∘ π)(Ω). If s ≠ 0, then [H_{2}, d_{2}](Ω) = \frac{1}{s}[i_{C}^{(2)}, d_{2}](Ω) = \frac{1}{s}L_{C}^{(2)}Ω. Show that L_{C}^{(2)}Ω = sΩ. Assume that

Ω = ∑ σ_{α_{1}...α_{l}} ∧ d_{2}σ_{α_{1}} ∧ ⋯ ∧ d_{2}σ_{α_{l}}

for σ_{α_{1}···α_{l}}, σ_{α_{1}}, …, σ_{α_{l}} ∈ Λ. Then

L_{C}Ω = ∑ C(σ_{α_{1}···α_{l}}) ∧ d_{2}σ_{α_{1}} ∧ ⋯ ∧ d_{2}σ_{α_{l}} + ∑ σ_{α_{1}···α_{l}} ∧ d_{2}C(σ_{α_{1}}) ∧ σ_{α_{2}} ∧ ⋯ ∧ d_{2}σ_{α_{l}}
+ ⋯ + ∑ σ_{α_{1}···α_{l}} ∧ d_{2}σ_{α_{1}} ∧ ⋯ ∧ d_{2}σ_{α_{l−1}} ∧ d_{2}C(σ_{α_{l}}) = sΩ.

Therefore, [H_{2}, d_{2}](Ω) = Ω = (id_{Λ_{2}} − ι ∘ π)(Ω). □

The following consequence is obvious.

Corollary 7. for any k H(Λ_{∞}, d_{k}) ≃ H(Λ, d).

3. Iterated differential forms over a smooth manifold

In this and the next section our considerations are restricted to the case A = C^{∞}(M), where M is an n–dimensional smooth manifold. In this situation it is convenient to pass to the category Mod_{A}^{g} of geometric A–modules (see, e.g., [8]) and to work with representing objects of functors of differential calculus only in this category. For instance, geometric differential forms over the algebra A are nothing but standard differential forms over the manifold M. Denote by Λ_{∞}(M) (or simply Λ_{∞}) iterated geometric differential forms over C^{∞}(M). Their coordinate description is as follows. Let K = {k_{1}, …, k_{r}} ⊂ ℕ be a finite subset. For f ∈ A put d_{K}f =^{def} d_{k_{1}} ⋯ d_{k_{r}}f. If x^{1}, …, x^{n} are local coordinates on M, then Λ_{∞}(M) is locally generated as an algebra by the elements d_{K}x^{μ}, μ = 1, …, n, K ⊂ ℕ. In particular, it is easy to see that locally

d_{K}f = ∑_{\{J_{1},...,J_{l}\}} \frac{∂^{l}f}{∂x^{μ_{1}} ⋯ ∂x^{μ_{l}}} d_{J_{1}}x^{μ_{1}} ∧ ⋯ ∧ d_{J_{l}}x^{μ_{l}},

where the sum runs over all repeated indexes and all partitions {J_{1}, …, J_{l}} of K into l parts, 1 ≤ l ≤ r. Let N be an m–dimensional manifold, (y^{1}, …, y^{m}) local coordinates on N and ϕ : M ⟶ N, x^{μ} ⟼ y^{α} = ϕ^{α}(x), μ = 1, …, n, α = 1, …, m, be a smooth map. Denote by Λ′_{∞} the algebra of iterated differential forms over N and consider the homomorphism Λ_{∞}(ϕ^{∗}) : Λ′_{∞} ⟶ Λ_{∞} generated by ϕ^{∗} : C^{∞}(N) ⟶ C^{∞}(M) which, for simplicity, will be also denoted by ϕ^{∗} as for ordinary forms. Then, locally

ϕ^{∗}(d_{K}y^{α}) = d_{K}(ϕ^{∗}(y^{α})) = ∑_{\{J_{1},...,J_{l}\}} \frac{∂^{l}ϕ^{α}}{∂x^{μ_{1}} ⋯ ∂x^{μ_{l}}} d_{J_{1}}x^{μ_{1}} ∧ ⋯ ∧ d_{J_{l}}x^{μ_{l}} ∈ Λ_{∞},

where the sum runs over all repeated indexes and all partitions {J_{1}, …, J_{l}} of K = {k_{1}, …, k_{r}} ⊂ ℕ into l parts, 1 ≤ l ≤ r.

4. Covariant tensors as iterated differential forms

For any p ∈ ℕ the functor D_{A}^{p} ≡ D_{A} ∘ ⋯ ∘ D_{A} : Mod_{A}^{g} ⟶ Mod_{A}^{g} is represented by the module T_{p}^{0}(M) ≡ Λ^{1}(M)^{⊗p} of covariant p–tensors on M. For any p the map

□ : A × ⋯ × A ∋ (f_{1}, …, f_{p}) ⟼ d_{1}f_{1} ∧ ⋯ ∧ d_{p}f_{p} ∈ Λ_{∞}

is a multi–derivation, i.e., □ ∈ D_{A}^{p}(Λ_{∞}). Therefore, there exists an unique A– homomorphism ι_{p} : T_{p}^{0}(M) ⟶ Λ_{∞} such that

ι_{p}(df_{1} ⊗ ⋯ ⊗ df_{p}) = □(f_{1}, …, f_{p}) = d_{1}f_{1} ∧ ⋯ ∧ d_{p}f_{p} ∈ Λ_{∞}

for any f_{1}, …, f_{p} ∈ A.

Proposition 8. ι_{p} is injective.

Proof. Obvious from local expression. □

Proposition 8 shows that the calculus of covariant tensors over M is just a part of differential calculus over the algebra Λ_{∞} and, therefore, is not conceptually closed. In that sense the proposed embedding of tensors into Λ_{∞} may be seen as a conceptual closure of tensor calculus. Standard operations with tensors, such as multiplications, “permutations of indexes”, insertions of vector fields, Lie derivatives, etc., have proper counterparts in Λ_{∞} and this looks as follows. First, all kinds of tensor multiplications are encoded in the wedge product “∧” in Λ_{∞}. For instance, exterior and symmetric products of differentials df and dg look as d_{1}f ∧ d_{2}g − d_{1}g ∧ d_{2}f and d_{1}f ∧ d_{2}g + d_{1}g ∧ d_{2}f, respectively. Second, the embedding ι_{p} is equivariant with respect to the natural action τ_{p} of the permutation group S_{p} on T_{p}^{0}(M), i.e., ι_{p} ∘ τ_{p}(σ) = κ_{σ} ∘ ι_{p}, ∀σ ∈ S_{p}. Evaluation of a tensor field T ∈ T_{p}^{0}(M) on a p–ple of vector fields X_{1}, …, X_{p} ∈ D(M) is interpreted in Λ_{∞} by means of the formula

T (X_{1}, …, X_{p}) = (i_{X_{p}}^{(p)} ∘ ⋯ ∘ i_{X_{1}}^{(1)})(ι_{p}(T)) ∈ A ⊂ Λ_{∞}.  (3)

Indeed, if T = df_{1} ⊗ ⋯ ⊗ df_{p}, f_{1}, …, f_{p} ∈ C^{∞}(M), then

(i_{X_{p}}^{(p)} ∘ ⋯ ∘ i_{X_{1}}^{(1)})(ι_{p}(T)) = (i_{X_{p}}^{(p)} ∘ ⋯ ∘ i_{X_{1}}^{(1)})(d_{1}f_{1} ∧ ⋯ ∧ d_{p}f_{p}) = X_{1}(f_{1}) · · · · · X_{p}(f_{p})
= T (X_{1}, …, X_{p}).

In particular, the insertion of a vector field X ∈ D(M) into the l–th place of T is given by

ι_{p}(T ( · , …, · , X_{l}, · , …, ·)) = (i_{X}^{(l)} ∘ ι_{p})(T).

Similarly, the Lie derivative L_{X}T of a tensor field T ∈ T_{p}^{0}(M) along a vector field X ∈ D(M) is given by

ι_{p}(L_{X}T) = (X ∘ ι_{p})(T).

Indeed, let

T = ∑ g_{α_{1}···α_{p}}df_{α_{1}} ⊗ ⋯ ⊗ df_{α_{p}},

f_{1}, …, f_{p} ∈ C^{∞}(M). Then

(X ∘ ι_{p})(T) = X(∑g_{α_{1}···α_{p}}d_{1}f_{α_{1}} ∧ ⋯ ∧ d_{p}f_{α_{p}}) = ∑X(g_{α_{1}···α_{p}})d_{1}f_{α_{1}} ∧ ⋯ ∧ d_{p}f_{α_{p}}

+ ∑g_{α_{1}···α_{p}}d_{1}X(f_{α_{1}}) ∧ ⋯ ∧ d_{p}f_{α_{p}} + ⋯ + ∑g_{α_{1}···α_{p}}d_{1}f_{α_{1}} ∧ ⋯ ∧ d_{p}X(f_{α_{p}})

= ι_{p}(∑X(g_{α_{1}···α_{p}})df_{α_{1}} ⊗ ⋯ ⊗ df_{α_{p}} + ∑g_{α_{1}···α_{p}}dX(f_{α_{1}}) ⊗ ⋯ ⊗ df_{α_{p}}
+ ⋯ + ∑g_{α_{1}···α_{p}}df_{α_{1}} ⊗ ⋯ ⊗ dX(f_{α_{p}}))
= ι_{p}(L_{X}T).

The intrinsic characterization of covariant tensors as elements of Λ_{∞} is as follows.

Proposition 9. A homogeneous element Ω ∈ Λ_{p} ⊂ Λ_{∞} of multi–degree (1, 1, …, 1) ∈ ℤ^{p} is a covariant tensor on M, i.e., Ω ∈ im ι_{p}, iff the map

Ω̃ : D(M) × ⋯ × D(M) ∋ (X_{1}, …, X_{p}) ⟼ (i_{X_{p}}^{(p)} ∘ ⋯ ∘ i_{X_{1}}^{(1)})(Ω) ∈ A ⊂ Λ_{∞}

is A–multi–linear.

Proof. Let Ω ∈ Λ_{p} be a homogeneous element of multi–degree (1, 1, …, 1). Ω can be expressed in the form

Ω = ι_{p}(T) + ∑_{{J_{1},...,J_{l}}} g_{α_{1}···α_{p}}^{J_{1}···J_{l}}d_{J_{1}}(f_{α_{1}J_{1}}) ∧ ⋯ ∧ d_{J_{l}}(f_{α_{l}J_{l}})

where T ∈ T_{p}^{0}(M), g_{α_{1}···α_{p}}^{J_{1}···J_{l}}, f_{α_{1}J_{1}}, …, f_{α_{l}J_{l}} ∈ C^{∞}(M) and the sum runs over α_{1}, …, α_{l} and all partitions {J_{1}, …, J_{l}} of {1, …, p} into l parts, 1 ≤ l < p. Let X ∈ D(M) and s ≤ p. Without loss of generality suppose that J_{1} = {i_{1}, …, i_{r}, s}. Then,

i_{X}^{(s)}Ω =ι_{p}(T ( · , …, · , X_{s}, · , …, ·))

+ ∑_{{J_{1},...,J_{l}}} g_{α_{1}···α_{p}}^{J_{1}···J_{l}}d_{{i_{1},...,i_{r}}}(X(f_{α_{1}J_{1}})) ∧ d_{J_{2}}(f_{α_{2}J_{2}}) ∧ ⋯ ∧ d_{J_{l}}(f_{α_{l}J_{l}}).

Thus, Ω̃ is multilinear iff ∑_{{J_{1},...,J_{l}}} g_{α_{1}···α_{p}}^{J_{1}···J_{l}}d_{J_{1}}(f_{α_{1}J_{1}}) ∧ ⋯ ∧ d_{J_{l}}(f_{α_{l}J_{l}}) = 0, i.e. Ω = ι_{p}(T). □

References

[1] A. M. Vinogradov, in M. Henneaux, I. Krasil’shchik, and A. Vinogradov (Eds.), Secondary Calculus and Cohomological Physics, Contemporary Mathematics 219, AMS (1998). See also The Diffiety

   Inst. Preprint Series, DIPS 5/98, http://diffiety.ac.ru/preprint/98/05_98abs.htm .

[2] A. M. Vinogradov, Cohomological Analysis of Partial Differential Equations and Secondary Calculus, AMS “Translation of Mathematical Monographs ”series 204 (2001).

[3] A. M. Vinogradov, Soviet Math. Dokl. 13 (1972) 1058.

[4] A. M. Vinogradov, J. Soviet Math. 17 (1981) 1624.

[5] M. M. Vinogradov, Russian Math. Surveys 44 (1989) no. 3, 220.

[6] A. Verbovetsky, J. Geom. Phys. 18 (1996) 195.

[7] A. M. Vinogradov and L. Vitagliano, Dokl. Math. 73, n^{∘} 2 (2006) 169.

[8] J. Nestruev, Smooth Manifolds and Observables, Springer–Verlag (New York) 2003.

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