The Diffiety Institute Preprint Series

Preprint DIPS–3/2006 September 11, 2006

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

The Diffiety Institute

Iterated Differential Forms II: Riemannian Geometry Revisited

by

A. M. Vinogradov and L. Vitagliano

DIPS-3/2006

                                                 arXiv:math.DG/0609287

Iterated Differential Forms II: Riemannian Geometry Revisited

A. M. VINOGRADOV AND L. VITAGLIANO

Abstract. A natural extension of Riemannian geometry to a much wider context is presented on the basis of the iterated form formalism developed in [1] and an application to general relativity is given.

In this note the formalism of iterated differential forms developed in the preceding note [1] (see also [2]) is used to decipher the conceptual meaning of Riemannian geometry in the context of differential calculus over (graded) commutative algebras. This is indispensable in order to extend Riemannian geometry to algebraic geometry and Secondary Calculus and to clarify some aspects in general relativity and, more generally, in field theory. In particular, we show that a Levi-Civita-like connection is naturally associated with a generic covariant 2-tensor whose symmetric part is nondegenerate and on this basis present new Einstein-like equations in general relativity. The super-case is briefly sketched. In the sequel the notation and definitions of [1] (see also [3, 4, 5, 6]) are used.

1. Differential form calculus and connections

Let G = (G, μ) be a grading group, 𝕜 be a field of zero–characteristic. In what follows a graded 𝕜–algebra will be always assumed associative, unitary and graded commutative (see [1]). Accordingly, a graded morphism of algebras will be always assumed unitary. The necessary facts concerning differential forms over G–graded 𝕜–algebras we shall need are summarized below along the lines of [7]. Let B be a G–graded 𝕜–algebra, Λ(B) = ⨁_{k} Λ^{k}(B) be the G ⊕ ℤ–graded 𝕜–algebra of differential forms over B and d_{B} : Λ(B) ⟶ Λ(B) the exterior differential. If P is a graded B–module, denote by D(B, P) the totality of all P –valued graded derivations of B. In a natural way D(B, P) is a graded B–module. Let φ : B ⟶ C be a morphism of G–graded 𝕜–algebras. It allows to supply any C–module Q with a structure of graded B–module which will be denoted by Q^{φ}. φ induces a morphism of G ⊕ ℤ–graded 𝕜–algebras Λ(φ) : Λ(B) ⟶ Λ(C). Denote by Λ^{1}(φ) : Λ^{1}(B) ⟶ Λ^{1}(C)^{φ} its first homogeneous component (with respect to the ℤ– grading). Λ^{1}(φ) is a morphism of B–modules. Let X ∈ D(B, Λ^{l}(C)^{φ}), l ≥ 0. The insertion of X into a differential form σ ∈ Λ(B), denoted by i_{X}σ ∈ Λ(C), is defined inductively on the degree of σ by the formula

2000 Mathematics Subject Classification. 53B20, 58A10, 58A50.

Key words and phrases. Differential Calculus, Graded Algebras, Differential Forms, Riemannian Geometry.

i_{X}(da ∧ σ) = X(a) ∧ (Λ(φ)(σ)) − (−1)^{X·a+l}((d_{C} ∘ φ)(a)) ∧ i_{X}σ, a ∈ B.

The insertion operator i_{X} : Λ(B) ⟶ Λ(C)^{φ} is a derivation of the algebra Λ(B) of bi–degree (|X|, l −1). The Lie derivative along X is defined as L_{X} =^{def} [i_{X}, d] = i_{X} ∘d_{B} + (−1)^{l}d_{C} ∘ i_{X} : Λ(B) ⟶ Λ(C)^{φ}. L_{X} is a derivation of the algebra Λ(B) of bi–degree (|X|, l). Let Λ^{1}(B) be a projective and finitely generated B–module. Then, for any X, Y ∈ D(B, Λ(B)) there exists a unique Z ∈ D(B, Λ(B)) such that [L_{X}, L_{Y}] = L_{Z}. Put Z =^{def} [X, Y]^{fn}. The bracket [·, ·]^{fn} : D(B, Λ(B)) × D(B, Λ(B)) ∋ (X, Y) ⟼ [X, Y]^{fn} ∈ D(B, Λ(B)) is called Frölicher–Nijenhuis (F–N) bracket and supplies D(B, Λ(B)) with a structure of G ⊕ ℤ–graded 𝕜–Lie–algebra. Now let φ : B_{1} ⟶ B_{2} be a morphism of G–graded 𝕜–algebras. Consider the functor

D(B_{1}, •^{φ}) : P ⟼ D(B_{1}, P^{φ}),

P being a G–graded B_{2}–module. It is represented by the graded B_{2}–module Λ^{1}(B_{1})⊗_{B_{1}} B_{2}^{φ}, i.e., for any graded B_{2}–module P and X ∈ D(B_{1}, P^{φ}) there exists a unique graded B_{2}–module homomorphism ψ_{X} : Λ^{1}(B_{1}) ⊗_{B_{1}} B_{2}^{φ} ⟶ P such that X = ψ_{X} ∘ d_{φ}, where d_{φ} : B_{1} ⟶ Λ^{1}(B_{1}) ⊗_{B_{1}} B_{2}^{φ} is a natural derivation defined by

d_{φ}b =^{def} d_{B_{1}}b ⊗ 1 ∈ Λ^{1}(B_{1}) ⊗_{B_{1}} B_{2}^{φ}, b ∈ B_{1}.

Note that Λ^{1}(φ) ∘ d_{B_{1}} : B_{1} ⟶ Λ^{1}(B_{2}) is a graded derivation, i.e., Λ^{1}(φ) ∘ d_{B_{1}} ∈ D(B_{1}, Λ^{1}(B_{2})^{φ}). Denote ψ = ψ_{Λ^{1}(φ)∘d_{B_{1}}} : Λ^{1}(B_{1}) ⊗_{B_{1}} B_{2}^{φ} ⟶ Λ^{1}(B_{2}) and put Λ_{φ}^{1}(B_{1}, B_{2}) = im ψ ⊂ Λ^{1}(B_{2}) (or, simply, Λ^{1}(B_{1}, B_{2}) if the context does not allow a confusion).

Definition 1. A connection in φ is a graded derivation ∇ : B_{2} ⟶ Λ(B_{1}) ⊗_{B_{1}} B_{2}^{φ} such that, ∇(B_{2}) ⊂ Λ^{1}(B_{1}) ⊗_{B_{1}} B_{2}^{φ} and ∇ ∘ φ = d_{φ}.

Let ∇ be a connection in φ and i_{∇} : Λ^{1}(B_{2}) ⟶ Λ^{1}(B_{1}) ⊗_{B_{1}} B_{2}^{φ} the unique graded B_{2}–module homomorphism such that ∇ = i_{∇} ∘ d_{B_{2}}. By definition i_{∇} is a left inverse of ψ. This proves that ψ is injective and Λ^{1}(B_{1}) ⊗_{B_{1}} B_{2}^{φ} ≃ Λ_{φ}^{1}(B_{1}, B_{2}). A connection ∇ in φ defines a natural transformation of functors ∇^{•} : D(B_{1}, •^{φ}) ⟶ D(B_{2}, •) (see [7]). Namely,

∇^{P} : D(B_{1}, P^{φ}) ∋ X ⟼ ∇_{X}^{P} =^{def} ψ_{X} ∘ ∇ ∈ D(B_{2}, P),

P being a G–graded B_{2}–module. By definition ∇_{X}^{P} ∘ φ = X. There are many situations when a new connection can be constructed on the basis of other ones. Some of them we need are described below.

Let ∇ : B_{2} ⟶ Λ(B_{1}) ⊗_{B_{1}} B_{2}^{φ} be a connection in φ : B_{1} ⟶ B_{2} and κ : B_{2} ⟶ B_{2} an automorphism of the algebra B_{2}. Obviously, κ : B_{2}^{φ} ⟶ B_{2}^{κ∘φ} is B_{1}–linear and, so, the map

id_{Λ(B_{1})} ⊗ κ : Λ(B_{1}) ⊗_{B_{1}} B_{2}^{φ} ⟶ Λ(B_{1}) ⊗_{B_{1}} B_{2}^{κ∘φ}.  (1)

is well-defined. The composition

∇^{κ} =^{def} (id_{Λ(B_{1})} ⊗ κ) ∘ ∇ ∘ κ^{−1} : B_{2} ⟶ Λ(B_{1}) ⊗_{B_{1}}^{κ∘φ} B_{2}

is a connection in κ ∘ φ : B_{1} ⟶ B_{2}. Let φ and ∇ be as above. ∇ extends naturally to a connection in Λ(φ) : Λ(B_{1}) ⟶ Λ(B_{2}) as follows. Consider the bi–differential algebra (Λ_{2}(B_{1}), d_{1}, d_{2}) of doubly iterated differential forms over B_{1} (see [1]). Note that the map

ν : B_{2} ∋ b ⟼ 1 ⊗ b ∈ Λ(B_{1}) ⊗_{B_{1}} B_{2}^{φ}

is a morphism of G–graded algebras and ∇ ∈ D(B_{2}, (Λ(B_{1}) ⊗_{B_{1}} B_{2}^{φ})^{ν}). According to the above construction ∇ extends to a derivation

L_{∇} : Λ(B_{2}) ⟶ Λ(Λ(B_{1}) ⊗_{B_{1}} B_{2}^{φ}).

There exists a canonical isomorphism of graded algebras,

ϕ : Λ(Λ(B_{1}) ⊗_{B_{1}} B_{2}^{φ}) ˜⟶ Λ_{2}(B_{1}) ⊗_{Λ(B_{1})} Λ(B_{2})^{Λ(φ)},

given by the formula

(ϕ ∘ d_{Λ(B_{1})⊗B_{2}})(ω ⊗ b) = d_{2}ω ⊗ b + ω ⊗ d_{B_{2}}b, ω ∈ Λ(B_{1}), b ∈ B_{2}.

Put

Λ∇ =^{def} ϕ ∘ L_{∇} : Λ(B_{2}) ⟶ Λ_{2}(B_{1}) ⊗_{Λ(B_{1})} Λ(B_{2})^{Λ(φ)}.

Then Λ∇ is a connection in Λ(φ) : Λ(B_{1}) ⟶ Λ(B_{2}). Finally, let B_{1}, B_{2}, B_{3} be G–graded algebras, φ : B_{2} ⟶ B_{3} and φ′ : B_{1} ⟶ B_{2} be homomorphisms of graded algebras and ∇ : B_{3} ⟶ Λ(B_{2}) ⊗_{B_{2}} B_{3}^{φ}, ∇′ : B_{2} ⟶ Λ(B_{1})⊗_{B_{1}} B_{2}^{φ′} be connections in φ, φ′, respectively. Similarly as above (see (1)) consider the homomorphism

id_{Λ(B_{1})} ⊗ φ : Λ(B_{1}) ⊗_{B_{1}} B_{2}^{φ′} ⟶ Λ(B_{1}) ⊗_{B_{1}} B_{3}^{φ∘φ′}.

By the universal property of Λ^{1}(B_{2}) ⊗_{B_{2}} B_{3}^{φ} there exists a unique homomorphism of graded B_{3}–modules \overline{ψ} : Λ^{1}(B_{2}) ⊗_{B_{2}} B_{3}^{φ} ⟶ Λ(B_{1}) ⊗_{B_{1}} B_{3}^{φ∘φ′} such that \overline{ψ} ∘ d_{φ} = (id_{Λ(B_{1})} ⊗ φ) ∘ ∇′. Put

□ ≡ \overline{ψ} ∘ ∇ : B_{3} ⟶ Λ(B_{1}) ⊗_{B_{1}} B_{3}^{φ∘φ′}.

Then □ is a connection in φ ∘ φ′ : B_{1} ⟶ B_{3}.

\xymatrix{B_{2} \ar[r]^{∇′} \ar[d]_{d_{φ}} & Λ(B_{1}) ⊗ B_{2}^{φ′} \ar[r]^{id⊗φ} & Λ(B_{1}) ⊗ B_{3}^{φ∘φ′} \\ Λ^{1}(B_{2}) ⊗ B_{3}^{φ} \ar[urr]^{\overline{ψ}} \\ B_{3} \ar[u]^{∇} \ar[uurr]^{□}}

Definition 2. □ is called the composition of connections ∇ and ∇′.

Abusing the notation we also denote this connection by ∇ ∘ ∇′.

2. Levi-Civita-like connections over smooth algebras

To simplify the exposition and for pedagogical reasons we shall limit to the case when the basic smooth algebra A is the smooth function algebra on a smooth manifold. Consider an n–dimensional smooth manifold M, a local chart (x^{1}, …, x^{n}) on it and put A ≡ C^{∞}(M). Let Λ = Λ_{1} be the ℤ–graded algebra of (geometric) differential forms over M and Λ_{k} = Λ(Λ_{k−1}) be the ℤ^{k}–graded algebra of k–times iterated (geometric) differential forms over M (see [8], where iterated differential forms were first introduced, and [9] for an alternative approach to iterated differential forms on super–manifolds). Denote by κ : Λ_{2} ⟶ Λ_{2} the canonical involution (see [1, 2]). Recall that κ∘d_{1} ∘κ = d_{2}, κ ∘ d_{2} ∘ κ = d_{1} and d_{1} is a natural extension of the de Rham differential on M to Λ_{2}. In the following we shall denote simply by ωω′ (rather than ω ∧ ω′) the product of the iterated forms ω and ω′. Since Λ_{2}^{1} ≡ Λ^{1}(Λ) is a projective and finitely generated Λ–module the F–N bracket in D(Λ, Λ_{2}) is well-defined. The injective A–homomorphism ι_{2} : T_{2}^{0}(M) ↪ Λ_{2} (see [1]) allows one to interpret covariant 2–tensors, in particular, (pseudo-)metric tensors, as iterated forms. This way pseudo–Riemannian geometry becomes a subject of the differential calculus over the algebra of iterated forms Λ_{∞}. This interpretation allows to associate a linear connection in the tangent bundle of M with a (possibly non–symmetric) 2–tensor whose symmetric part is non–degenerate. According to the standard approach, the Levi-Civita connection associated with a pseudo-metric is defined implicitly as the unique one satisfying certain properties. On the contrary, we define it directly by applying to the considered 2– tensor a natural operator in the algebra Λ_{∞}. This is one of numerous examples showing potentialities of the calculus of iterated forms.

Proposition 3. Let g ∈ Λ_{2} be a second iterated form of bi–degree (1, 1). Then the following two assertions are equivalent:

(1) The map D(M) × D(M) ∋ (X, Y) ⟼ (i_{Y}^{(2)} ∘ i_{X}^{(1)})(g) ∈ A ⊂ Λ is A–bilinear, κ(g) = g and the A–homomorphism ⌋_{g} : D(M) ∋ X ⟼ i_{X}^{(2)}g ∈ Λ^{1} ⊂ Λ_{2} is bijective.

(2) g = ι_{2}(g′) for a Riemannian or pseudo–Riemmannian metric g′ over M.

Proposition 3 is a corollary of proposition 4 from [1] and motivates the following new point of view on what, in reality, the Riemannian geometry is.

Definition 4. A metric over a smooth algebra A, for instance, A = C^{∞}(M), is an iterated 1–form g ∈ Λ_{2} satisfying hypothesis 1 of proposition 3.

Locally a metric g ∈ Λ_{2} looks as g = g_{μν}d_{1}x^{μ}d_{2}x^{ν} with g_{μν} = g_{νμ} ∈ A, 1 ≤ μ, ν ≤ n. This way the Riemannian geometry is embedded in the context of iterated differential forms and, after that, it is generalized rather directly to more general 2–tensors. Namely, let τ ∈ Λ_{2} be an iterated form of bi–degree (1, 1) such that τ = ι_{2}(τ′) for a covariant 2–tensor τ′ over M. Locally τ looks as τ = τ_{μν}d_{1}x^{μ}d_{2}x^{ν} with τ_{μν} ∈ A, 1 ≤ μ, ν ≤ n. τ can be uniquely written as τ = g + ω where κ(g) = g and κ(ω) = −ω. Clearly, g = ι_{2}(g′) and ω = ι_{2}(ω′), where g′ and ω′ are symmetric and skew–symmetric parts of τ′, respectively. From now on, suppose that g is nondegenerate, i.e., a pseudo-metric. Consider the iterated 2–form γ = −d_{2}d_{1}τ. It is of bi–degree (2, 2) and its local expression is

γ = ∂_{ν}∂_{β}τ_{αμ}d_{1}x^{β}d_{1}x^{α}d_{2}x^{μ}d_{2}x^{ν} + γ_{β}_{αμ}d_{1}x^{α}d_{2}x^{μ}d_{2}d_{1}x^{β} + τ_{αβ}d_{2}d_{1}x^{α}d_{2}d_{1}x^{β}.

with ∂_{μ} = \frac{∂}{∂x^{μ}} and γ_{β}_{αμ} = ∂_{μ}τ_{αβ} + ∂_{α}τ_{βμ} − ∂_{β}τ_{αμ}. Note that if ω = 0, then the γ_{β}_{αμ}’s are doubled first kind Christoffel symbols of g′. By this reason γ is naturally baptized to be the (generalized) Christoffel form. Next, the homomorphism ⌉^{g} = \frac{1}{2}⌋_{g}^{−1} : Λ^{1} ⟶ D(M) extends to Λ as an A–linear, D(A, Λ)–valued derivation of Λ of degree −1 denoted by ⌉_{1}^{g} : Λ ⟶ D(A, Λ). It should be noted that D(A, Λ) is a Λ–module and ⌉_{1}^{g} ∈ D(Λ, D(A, Λ)). Recall that with any X ∈ D(A, Λ) the insertion operator i_{X} ∈ D(Λ, Λ) is associated and consider the map

Δ : Λ ∋ σ ⟼ Δ(σ) =^{def} i_{⌉_{1}^{g}(σ)} ∈ D(Λ, Λ).

Δ is an A–linear, D(Λ, Λ)–valued derivation of Λ of degree −2. Define the map

⌉_{2}^{g} : Λ_{2} ⟶ D(Λ, Λ_{2}),

by putting for any Ω ∈ Λ_{2} and ω ∈ Λ

⌉_{2}^{g}(Ω)(ω) ≡ (−1)^{Ω·ω}i_{Δ(ω)}^{(2)}Ω.

Note that D(Λ, Λ_{2}) possesses a natural Λ_{2}–module structure and ⌉_{2}^{g} is a D(Λ, Λ_{2})–valued derivation of Λ_{2} of bi–degree (−2, −1). Locally it looks as

⌉_{2}^{g}(Ω)(ω) = (−1)^{|Ω|·(0,−1)} \frac{1}{2}g^{μν}(i_{i_{∂μ}}^{(2)}Ω)(i_{∂_{ν}}ω) ∈ Λ_{2},

where ∥g^{μν}∥ = ∥g_{μν}∥^{−1}.

Definition 5. The Λ_{2}–valued derivation Γ =⌉_{2}^{g}(γ) = −⌉_{2}^{g}(d_{1}d_{2}τ) of the algebra Λ is called the Levi-Civita connection form of τ.

This terminology is motivated by the fact that the local expression of Γ is

Γ = (d_{1}d_{2}x^{α} + Γ_{μ}_{β}^{α}d_{1}x^{β}d_{2}x^{μ}) i_{∂_{α}},

with Γ_{μ}_{β}^{α} ≡ \frac{1}{2}g^{αδ}γ_{δ}_{βμ} and in the case when ω = 0 Γ_{ρ}_{γ}^{α}’s are nothing but Christoffel symbols of g′. So, with any covariant 2–tensor whose symmetric part is nondegenerate a linear connection in the tangent bundle is associated (see below). Now consider the insertion into 2-iterated forms operator i_{Γ}^{(2)} ∈ D(Λ_{2}, Λ_{2}) corresponding to the graded form–valued derivation Γ. The composition κ ∘ i_{Γ}^{(2)} ∘ κ is again a derivation of Λ_{2}. Put

κ(Γ) =^{def} [κ ∘ i_{Γ}^{(2)} ∘ κ, d_{2}]|_{Λ} ∈ D(Λ, Λ_{2}).

The graded form–valued derivation T = Γ − κ(Γ) has the following local expression

T = T_{μ}_{β}^{α}d_{1}x^{β}d_{2}x^{μ} i_{∂_{α}}, T_{μ}_{β}^{α} = Γ_{μ}_{β}^{α} − Γ_{β}_{μ}^{α} = 3g^{αδ}∂_{[μ}ω_{βδ]}d_{1}x^{β}d_{2}x^{μ} i_{∂_{α}},

where the ω_{βδ}’s are local components of ω, i.e., locally ω = ω_{βδ}d_{1}x^{β}d_{2}x^{δ}, and indexes in square bracket are skew–symmetrized.

Definition 6. The Λ_{2}–valued derivation T of the algebra Λ is called the torsion of τ.

The graded F–N square of Γ

R = [Γ, Γ]^{fn} ∈ D(Λ, Λ_{2})

is well defined and has the following local expression

R = [Γ, Γ]^{fn} = R_{σμ}_{δ}^{α}d_{1}x^{δ}d_{2}x^{σ}d_{2}x^{μ} i_{∂_{α}},

where R_{σμ}_{δ}^{α} = ∂_{σ}Γ_{μ}_{δ}^{α} − ∂_{μ}Γ_{σ}_{δ}^{α} + Γ_{σ}_{β}^{α}Γ_{μ}_{δ}^{β} − Γ_{μ}_{β}^{α}Γ_{σ}_{δ}^{β}.

Definition 7. The Λ_{2}–valued derivation R of the algebra Λ is called the Riemann tensor of τ.

The introduced above concepts of generalized Levi-Civita connection, torsion and Riemann tensor associated with the 2–tensor τ have standard counterparts. Namely, recall that τ = ι_{2}(τ′), τ′ ∈ T_{2}^{0}(M) being a “standard” covariant 2–tensor on M. The above computations show that there is a standard connection Γ′ in the tangent bundle of M associated with τ′. Indeed, the covariant derivative along X ∈ D(M) with respect to Γ′ ∇′_{X} is well defined by

(∇′_{X}Y)(f) = (i_{X}^{(1)} ∘ i_{Y}^{(2)} ∘ Γ)(df) ∈ A ⊂ Λ_{2}.

Now, denote by T′ and R′ the “standard” torsion and Riemann curvature tensor of Γ′, respectively.

Proposition 8. For any X, Y, Z ∈ D(M) and f ∈ A

T′(X, Y)(f) = (i_{X}^{(1)} ∘ i_{Y}^{(2)} ∘ T)(df) ∈ A ⊂ Λ_{2}.
R′(X, Y)(Z)(f) = (i_{Z}^{(1)} ∘ i_{X}^{(2)} ∘ i_{Y}^{(2)} ∘ R)(df) ∈ A ⊂ Λ_{2}.

Proof. Obvious from local expression. □

In new terms geodesics of Γ′ are described as follows. Let \frac{d}{dt} : C^{∞}(ℝ) ⟶ C^{∞}(ℝ) be the “time current” field. Consider a smooth curve χ : I ⟶ M on M, I ⊂ ℝ being an interval, the algebra Λ_{2}(I) of second iterated forms on I and the pull–back homomorphism χ^{∗} : Λ_{2} → Λ_{2}(I). The geodesic curvature vector field (along χ) is the composition

K_{χ} =^{def} i_{d/dt} ∘ i_{d/dt}^{(2)} ∘ χ^{∗} ∘ Γ : Λ ⟶ Λ(I).

Proposition 9. χ is an affinely parameterized geodesic of the metric g′ if and only if K_{χ} = 0.

This is obvious from the local expression

K_{χ}(dx^{μ}) = Γ_{ρ}_{α}^{μ} \frac{dχ^{ρ}}{dt} \frac{dχ^{α}}{dt} + \frac{d^{2}χ^{μ}}{dt^{2}} .

We now describe covariant derivatives of covariant tensors in new terms. First of all note that for any l < k, Λ(Λ_{l}) ⊗_{Λ_{l}} Λ_{k} is naturally isomorphic to Λ(Λ_{l}, Λ_{k}). Therefore, a connection in the inclusion Λ_{l} ⊂ Λ_{k} (in the sense of section 1) may be understood as a derivation Λ_{k} ⟶ Λ(Λ_{l}, Λ_{k}). In the following we will adopt this point of view. Define a new graded form-valued derivation ∇ = d_{2} − Γ : Λ ⟶ Λ_{2}.∇ has local expression

∇ = d_{2}x^{μ} (∂_{μ} − Γ_{μ}_{β}^{α} d_{1}x^{β} i_{∂_{α}}).

Note that im ∇ ⊂ Λ(A, Λ).

Proposition 10. ∇ : Λ ⟶ Λ(A, Λ) is a connection in the natural inclusion φ : A ↪ Λ in the sense of section 1.

As shown in section 1, in its turn ∇ induces a connection Λ∇ in Λ(φ) : Λ ↪ Λ_{2} . Note that Λ(φ) = κ|_{Λ}. Therefore,

∇_{2} ≡ Λ(κ) ∘ ∇′ ∘ κ : Λ_{2} ⟶ Λ(Λ, Λ_{2})

is a connection in the natural inclusion Λ ⊂ Λ_{2}. Iterating the procedure we get, at the k–th step, a connection ∇_{k} in the inclusion Λ_{k−1} ⊂ Λ_{k}. Note that the existence of the connections ∇_{k} enriches noteworthy the standard calculus in Riemannian geometry

(consequences of this remark will be explored elsewhere). Denote by ∇^{k} the composition

∇^{k} = ∇_{k} ∘ ∇_{k−1} ∘ ⋯ ∘ ∇_{2} ∘ ∇_{1} : Λ_{k} ⟶ Λ(A, Λ_{k}).

∇^{k} is a connection in the inclusion A ⊂ Λ_{k}. Accordingly, we can lift A–linearly a

derivation X ∈ D(A, Λ_{k}) to a derivation ∇^{k}_{X} = i_{X}^{(k+1)} ∘ ∇^{k} ∈ D(Λ_{k}, Λ_{k}).

Definition 11. ∇^{k} is called the k–th covariant derivative associated to Γ.

As an example, we report local expression of ∇^{2}:

∇^{2} = d_{3}x^{μ}(∂_{μ} − Γ_{μ}_{β}^{α} d_{1}x^{β} i_{∂_{α}} − Γ_{μ}_{β}^{α} d_{2}x^{β} i_{∂_{α}}^{(2)} − d_{1}(Γ_{μ}_{β}^{α} d_{2}x^{β}) i_{i_{∂α}}^{(2)}).

The terminology of definition 11 is motivated by the fact that the standard covariant derivatives with respect to Γ′ of a covariant k–tensor s′ ∈ T_{k}^{0}(M) are naturally recovered

from ∇^{k}. Namely, we have

Proposition 12. Let s = ι_{k}(s′) ∈ Λ_{k} and X, Y_{1}, …, Y_{k} ∈ D(M). Then

(∇′_{X} s′)(Y_{1}, …, Y_{k}) = (i_{Y_{1}}^{(1)} ∘ ⋯ ∘ i_{Y_{k}}^{(k)} ∘ i_{X}^{(k+1)} ∘ ∇^{k})(s) ∈ A ⊂ Λ_{k}.

It is worth mentioning that, on the contrary to the the classical Levi-Civita connections, i.e., associated with metrics, the covariant derivative of a 2–tensor τ with respect

to the associated connection is, generally, different from zero, i.e., ∇^{2}τ ≠ 0. Namely, let

Γ^{g} : Λ_{1} ⟶ Λ_{2} be the (generalized) Levi-Civita connection of g and ∇^{k}^{g} the associated k–th covariant derivative. Then,

∇^{2}τ = ∇^{2}ω = ∇^{2}^{g}ω + T (ω),

where T (ω) is the iterated form locally given by

T (ω) = T_{μ[ν}^{β}ω_{α]β}d_{1}x^{α}d_{2}x^{ν}d_{3}x^{μ}.

3. Natural equations in general relativity

The possibility to associate in a canonical way a Levi-Civita-like connection with a 2–tensor τ with non–degenerate symmetric part suggests an immediate application to general relativity. Indeed, it is natural to interpret the skew-symmetric part ω of τ as a matter field. Then, it is natural as well to assume that the space, i.e., the symmetric part g of τ, is shaped by the matter in such a way that the Ricci tensor Ric(τ) of the associated with τ connection vanishes. This way one gets the natural equations in general relativity. We use here the word “natural” in order to stress that in the proposed approach the credit is definitively given to the naturalness of the background mathematical language, but not to the “physical intuition” which could be easily misleading in this context. In coordinates these natural equations look as follows. Let R = R_{μσ}_{δ}^{α}dx^{δ} ⊗dx^{σ} ⊗dx^{μ}⊗ \frac{∂}{∂x^{α}} be the associated with τ standard Riemann tensor. The Ricci tensor Ric(τ) of τ is defined as the 2–tensor locally given by Ric(τ) = R_{μδ} dx^{μ} ⊗ dx^{δ}, where R_{μδ} = R_{μα}_{δ}^{α}, μ, δ = 1, …, n (the interpretation of the Ricci tensor in terms of iterated differential forms will be discussed separately). Then the natural equations for τ read Ric(τ) = 0.

Now denote by ∇^{g} and Ric(g) the covariant differential and the Ricci tensor of the

pseudo-metric g, respectively, and put Ric(g) = R^{g}_{μδ} dx^{μ} ⊗ dx^{δ}. Then in terms of g and ω the natural equations for τ read

\begin{matrix}R^{g}_{μν} + \frac{9}{16}g^{γρ}g^{δσ}∂_{[δ}ω_{μρ]}∂_{[γ}ω_{νσ]} = 0 \\ ∇^{g}^{γ}(∂_{[μ}ω_{νγ]}) = 0 \end{matrix} .  (2)

System (2) may be understood as Einstein-like equations for g in presence of a matter field (represented by ω) satisfying a suitable constitutive equation. Indeed, it can be shown that (2) are Einstein equations in presence of a perfect irrotational fluid with a definite equation of state (details will be discussed in a separate publication). It is worth stressing that the matter in the above natural equations is treated in a simplified manner. More rich and exact versions of natural equations in the frames of the developed approach will be discussed separately.

4. Levi-Civita connections over smooth super-algebras

The previous results are straightforwardly generalized to supermanifolds. In this section, we shall sketch how Riemannian supergeometry (see, for instance, [10] and references therein) can be developed in the framework of iterated differential forms along the lines of section 2. Consider a super–manifold S, a local chart (θ^{1}, …, θ^{n}) on it and put A = C^{∞}(S). A is a ℤ_{2}–graded commutative algebra. Put ᾱ = |θ^{α}| ∈ ℤ_{2}. Let Λ = Λ_{1} be the ℤ_{2}⊕ℤ–graded algebra of (geometric) differential forms over S and Λ_{2} = Λ(Λ) be the ℤ_{2}⊕ℤ^{2}–graded algebra of doubly iterated geometric differential forms over S (see also [8]). Let g be an even doubly iterated form of bi–degree (1, 1) over A such that the map D(S) × D(S) ∋ (X, Y) ⟼ (i_{Y}^{(2)} ∘ i_{X}^{(1)})(g) ∈ A ⊂ Λ is A–bilinear, κ(g) = g and the A–homomorphism ⌋_{g} : D(M) ∋ X ⟼ i_{X}^{(2)}g ∈ Λ^{1} ⊂ Λ_{2} is bijective.

Definition 13. g is called a supermetric over S.

Locally, g = g_{μν}d_{1}θ^{μ}d_{2}θ^{ν}, g_{μν} = (−1)^{μ̄ν̄}g_{νμ} ∈ A. Moreover, the matrix ∥g_{νμ}∥ is invertible in A. Given a metric g over S the derivation ⌉_{2}^{g} : Λ_{2} ⟶ D_{Λ}(Λ_{2}) is defined exactly as above. Its local expression is

⌉_{2}^{g}(Ω)(ω) = (−1)^{|Ω|·(ν̄,0,−1)} \frac{1}{2}g^{μν}(i_{i_{∂μ}}^{(2)}Ω)(i_{∂_{ν}}ω) ∈ Λ_{2},

where Ω ∈ Λ_{2}, ω ∈ Λ, (ν̄, 0, −1,) ∈ ℤ_{2}⊕ℤ^{2} is the multi–degree of i_{∂_{ν}}, ∂_{ν} = \frac{∂}{∂θ^{ν}} and g^{μν} ∈ A, (−1)^{γ̄·ᾱ}g_{να}g^{νγ} = δ_{α}^{γ}.

Definition 14. The Λ_{2}–valued derivation Γ =⌉_{2}^{g}(γ) = −⌉_{2}^{g}(d_{1}d_{2}g) of the algebra Λ is called the Levi-Civita connection form of g.

The local expression of Γ is

Γ = (d_{1}d_{2}θ^{α} + Γ_{μ}_{β}^{α}d_{1}θ^{β}d_{2}θ^{μ}) i_{∂_{α}},

with Γ_{μ}_{β}^{α} = \frac{1}{2}g^{γα}[(−1)^{γ̄·γ̄+μ̄(μ̄+γ̄)}∂_{μ}g_{βγ} + (−1)^{γ̄·γ̄+β̄(μ̄+β̄+γ̄)}∂_{β}g_{μγ} − ∂_{γ}g_{βμ}]. Similarly, the Riemann tensor and the covariant derivative are introduced. For instance,

R = [Γ, Γ]^{fn} = R_{γβ}_{α}^{ν}d_{1}θ^{α}d_{2}θ^{γ}d_{2}θ^{β}i_{∂_{ν}}

where

R_{γβ}_{α}^{ν} = (−1)^{ᾱ(β̄+γ̄)+β̄(β̄+ν̄)}∂_{β}Γ_{α}_{γ}^{ν} − (−1)^{β̄(β̄+γ̄+ν̄)}∂_{γ}Γ_{α}_{β}^{ν}
+ (−)^{μ̄β̄+ᾱγ̄}Γ_{μ}_{β}^{ν}Γ_{α}_{γ}^{μ} − (−1)^{β̄·(ᾱ+γ̄)+μ̄γ̄}Γ_{μ}_{γ}^{ν}Γ_{α}_{β}^{μ}.

References

[1] A. M. Vinogradov, L. Vitagliano, Diffiety Inst. Preprint Series, DIPS–1/06, http://diffiety.

   ac.ru/preprint/2006/01-06abs.htm , see also arXiv:math.DG/0605113 .

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

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

[7] 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/98/07_98abs.htm .

[8] D. Kochan, P. Severa, math.DG/0307303.

[9] P. Severa, arXiv:math.DG/0606645 .
[10] O. Goertsches, arXiv:math.DG/0604143 .

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